CHAPTER 8

# THE S-WEAK ORDER AND S-PERMUTAHEDRA

> I just knew there were stories I wanted to tell.

> – Octavia E. Butler

As I was working on the *m*-Tamari lattices by the end of my thesis, I wondered what was the equivalent of the weak order in the *m*-Tamari case. This is indeed a very natural question considering the many links between the weak order and the Tamari lattice in combinatorics, geometry and algebra. I studied the *m*-permutations, i.e., the permutations with *m* copies of each letter. I was interested in the lattice properties and at the same time, Novelli and Thibon were working on *m* versions of the Hopf algebraic structures. In this context, the notion of *metasylvester congruence* was introduced in [NT20]. This lead me to define the *metasylvester lattice* in [Pon15].

Even though I am very interested in all the geometric aspects related to my work, my own background and approach is very much on the combinatorial side. In 2014, I could “see” that this new lattice of mine was “nice” and looked like a permutahedron that had been cut into smaller pieces, but I had no idea how to formalize that property. Around this time, I attended a talk by Cesar Ceballos at Séminaire Lotharingien de Combinatoire where he presented the results he obtained with Padrol and Sarmiento on the realization of the *ν*-Tamari lattices [CPS16]. I went to talk to him right after to show him my own work and see if we could work on a geometric realization together. This resulted in a fruitful collaboration which is still going on with many fascinating results and open questions. We presented a first version of our work at FPSAC 2019 [CP19]. In 2022, we released the first paper [CP22] of a series of at least two. It covers the combinatorial aspects of [CP19] and is a generalization of [Pon15]. We are working at the moment on the second paper [CP23] where we cover the combinatorial definition of the *s*-permutahedra complex. In this Chapter, I present an overview of all these results as well as the many different leads for future work. Computations and examples are available here [PonSage22b].
## 8.1 The metasylvester lattice

As the *m*-Tamari lattice is an upper ideal of the Tamari lattice of size $n \times m$, it is a sublattice and a quotient lattice of a certain upper ideal in the weak order. It is actually more natural to look at the reversed *m*-Tamari which is a sublattice of the weak order on permutations where every letter is repeated *m* times as in Figure 8.1.

<figure><img src="image_72.png" /><figcaption>Figure 8.1: The lattice of 2-permutations (left) with metasylvester classes and the corresponding *s*-weak order.</figcaption></figure>

The *metasylvester congruence* [NT20] is a monoid congruence defined as the transitive closure of the following rules

$$ \dots ac \dots a \equiv \dots ca \dots a \qquad (8.1) $$

$$ \dots b \dots ac \dots b \dots \equiv \dots b \dots ca \dots b \dots \qquad (8.2) $$

where $a < b < c$. As an example, the classes on Figure 8.1 are the red shapes. As both local moves require to have a letter repeated, when $m = 1$, on the classical weak order, each permutation is its own class. Besides, a *sylvester* class as defined by (2.2) in Section 2.2 is a union of metasylvester classes.

The metasylvester congruence is not a lattice congruence. Indeed, it is not compatible with the meet operation. For example, take $\sigma = 12132$ and $\mu = 131223$. We have that $\sigma \equiv 332112$ and $\mu \equiv 311223$. But $\sigma \wedge \mu = 112323$ is not equivalent to $332112 \wedge 311223 = 311223$. Nevertheless, the join operation is compatible and actually the maximal elements of the class have a nice characterization. They are *Stirling permutations*, i.e., permutations avoiding the pattern 121. They form a sublattice of the weak order and are in bijections with *m*-decreasing trees. In [Pon15] I call this lattice the metasylvester lattice. This is actually a special case of the *s*-weak order which I defined with Ceballos in [CP19] and [CP22].
## 8.2 The s-weak order

Let $s$ be a sequence of $s$ non negative integers. We define the *s-decreasing trees*. They are ordered rooted trees on $n$ nodes labeled 1 to $n$ such that the node $i$ has $s(i)+1$ children and so that the labels are decreasing from the root to the leaves. In particular, there are $(1+s(n))(1+s(n)+s(n-1))\cdots(1+s(n)+\cdots+s(2))$ s-decreasing trees for a given sequence $s$. When $s$ does not contain any zeros, then s-decreasing trees are in bijection with s-permutations: permutations where each letter $i$ appears $s(i)$ times and avoiding the pattern 121. The bijection is illustrated on Figure 8.2.

<figure><img src="image_73.png" /><figcaption>Figure 8.2: An s-decreasing tree and the corresponding s-permutation.</figcaption></figure>

When $s = (m, m..., m)$, the s-permutations are the maximal elements of the metasylvester classes and so we have a lattice structure. This generalizes easily for all $s$ without zeros: the s-permutations form a sublattice of the weak order. This is not the case when $s$ contains zero but we can still extend the definition. We generalize the notion of *tree inversion* which was already introduced in [Pon15]. Basically, the tree-inversions are a *multiset* where each inversion $(b, a)$ appears with a certain *cardinality* $\#(b, a)$. The cardinality of $(b, a)$ expresses where is $a$ regarding $b$ in the tree. If it is to the left or in the first subtree of $b$, then $\#(b, a) = 0$. If it is to the right of $b$ or in the last subtree, we have $\#(b, a) = s(b)$. The intermediate values correspond to the case where $a$ belongs to a *middle child* of $b$. We show an example on Figure 8.3.

In particular, when $s = (1, 1, ..., 1)$, then s-decreasing trees are decreasing binary trees, in bijection with permutations. In this case, the tree-inversions are the classical value inversions of permutations: each cardinality $\#(b, a)$ can either be 0 (no inversion, $a$ is before $b$) or 1 ($b$ is before $a$). Another interesting case is when $s$ contains some zeros: if $s(b) = 0$, then the node $b$ has one child and $\#(b, a) = 0$ for all $a < b$, independently of the position of $a$.

The tree-inversions form a multiset and we can generalize the notion of inclusion of classical sets. For two multisets $T$ and $S$, we have $T \subseteq S$ if, for all $a < b$, $\#_T(b, a) \le \#_S(b, a)$. This gives a partial order on s-decreasing trees which we call
<figure><img src="image_74.png" /><figcaption>Figure 8.3: An *-decreasing tree and its tree-inversions.</figcaption></figure>

the *-weak order as it is a direct generalization of the classical weak order. See some examples on Figure 8.4.

**Theorem 14** (Theorem 1.21 of [CP22]). *The *-weak order is a lattice.*

Our proof is based on a generalization of the notion of a *transitive relation* to a multiset of inversions: it is transitive if for all $c > b > a$, $\#(b, a) > 0$ implies that $\#(c, a) \ge \#(c, b)$. In the classical case where cardinalities are either 0 or 1, this indeed says that if $(b, a)$ and $(c, b)$ are inversions, we have $\#(c, a) = 1$: $(c, a)$ is also an inversion. A multiset of inversions is an *s-tree inversion set* if it is *transitive* and *co-transitive* (the non-inversions are also transitive). We actually call this second property the *planarity* of the set, it translates to: for all $c > b > a$ such that $\#(c, a) = i$, either $\#(b, a) = s(b)$ or $\#(c, b) \ge i$. It is easy to see that the tree-inversions of an *-decreasing tree* satisfy these properties as illustrated on Figure 8.5. We prove that they are sufficient properties and give an explicit construction to recover the tree from the tree-inversions.

To prove that the *-weak order is a lattice*, we then generalize the notions of *union* and *transitive closure* to multisets of inversions. The join of two *-decreasing trees* is obtained by taking the transitive closure of the union of their tree-inversions. We also characterize the *tree-ascents*, i.e., the cover relations of the poset. Basically, a cover relation corresponds to moving a node *i* to the right. There are some technicalities in *when* and *how* we can do this move: nodes move by taking along their *middle children* but leave behind their *left child*. The nodes are not allowed to move if they have a *strict right child*. For example, on the tree of Figure 8.3, 1 and 2 are allowed to move to the right of 5 but not 3 because 2 is a strict right child of 3 (while 1 is not a strict right child of 2 because $s(2) = 0$).

Furthermore, we prove that the *-weak order is polygonal, semidistributive and*
<figure><img src="image_75.png" /><figcaption>Figure 8.4: Examples of s-weak lattices.</figcaption></figure>
<figure><img src="image_76.png" /><figcaption>Figure 8.5: Illustration of the transitivity and planarity conditions on s-tree inversion sets.</figcaption></figure>

congruence uniform. These are all interesting lattice properties that are defined in particular in [Rea16]. It implies some properties on the lattice congruences of the s-weak order. Our proof is based on the notion of $\mathcal{HH}$ lattices found in [CLM04]. It also uses some useful properties of the intervals of the s-weak order found in [Lac22] who studies some topological properties of the lattice based on our initial extended abstract [CP19].

## 8.3 Link with the $\nu$-Tamari lattice

In [Pon15], I claimed that the $m$-Tamari lattice was both a sublattice and a quotient lattice of the metasylvester lattice. In [CP22], we generalize and prove this result on the $s$-weak order. We define the *s-Tamari trees*: there are the trees such that $\#(c, a) \le \#(c, b)$ for all $a < b < c$. This means that a node $a$ cannot move faster to the right than a node $b$ of greater value.

**Theorem 15** (Theorem 2.2 of [CP22]). *The set of s-Tamari trees form a sublattice of the s-weak order.*

This is illustrated on Figure 8.6. We give explicit descriptions of the cover relations and prove that this is isomorphic to a certain $\nu$-Tamari lattice as defined in Section 2.9 where $\nu$ is simply the reversed sequence $s$.

Besides, when $s$ does not contain any zeros, then the $s$-Tamari lattice is a quotient lattice of the $s$-weak order and we give direct characterization of the congruence classes. In particular, the minimal elements of the classes are the $s$-decreasing trees.
<figure><img src="image_77.png" /><figcaption>Figure 8.6: *s*-decreasing trees and *s*-Tamari trees for *s* = (0, 2, 3). On the left picture, the non *s*-Tamari trees are in black.</figcaption></figure>

## 8.4 Pure intervals and *s*-permutahedra

Looking at the examples of Figure 8.4, a geometrical structure seems to appear. Indeed, the lattices can be drawn in a way that “cuts” the plane into smaller polygonal areas. These polygons are glued together in a “nice” way and form a bigger polygon. All examples of Figure 8.4 are for *n* = 3. For bigger *n*, we notice the same phenomenons of cutting a polytope of dimension *n* − 1 into smaller polytopal cells. This is what we call a *polytopal complex*. The road from *seeing* to *proving* turned out to be a much longer path than we initially thought and at this date, we still have no final proof that the *s*-weak order defines a polytopal complex. Nevertheless, in [CP19] and [CP23] we define what we call a *combinatorial complex: the s-permutahedron* as shown on Figure 8.7.

In [CP22], we proved that the *s*-weak order is a *polygonal lattice*. This means that if a tree *T* is covered by two trees *S* and *S*', then the interval between *T* and *S* ∨ *S*' is a *polygon*: it is formed by two chains that meet only in their extremities. The polygons that appear have been characterized by Lacina [Lac22]: there are either squares, hexagons or pentagons. We believe that this generalizes to other dimensions: if a tree *T* is covered by *k* trees *S*₁, ..., *S*ₖ, then the interval between *T* and ∨<sub>i=1</sub><sup>*k*</sup> *S*ᵢ is the skeleton of a *k*-dimensional polytope which is a
<figure><img src="image_78.png" /><figcaption>Figure 8.7: The $s$-permutahedron complex for $s = (0, 2, 2)$.</figcaption></figure>

<figure><img src="image_79.png" /><figcaption>Figure 8.8: Example of a pure interval. On the left, the minimal tree with colored tree-ascents and on the right the corresponding interval.</figcaption></figure>

generalized permutahedra.

This is what we call a *pure interval* of the *s*-weak order. More precisely, a pure interval is defined by an *s*-decreasing tree $T$ with a set of selected tree-ascents. Each tree-ascent gives a covering element $S$ and the pure interval is the interval between $T$ and the join of the covering elements $S$. We show an example on Figure 8.8. Our paper [CP23] studies the combinatorial properties of the pure
intervals. The *s*-permutahedra is the set of pure intervals on which we can define notions of *containment*, *intersection* and *dimension*.

The “dimension” of a pure interval is the number of selected tree-ascents. At this stage, this is a formal notion which should later correspond to the actual dimension of the polytope.

Not all intervals are pure intervals (just like not all intervals of the weak order are faces of the permutahedron). The first question we ask is then: if $T \prec_S S$, how can we tell if this corresponds to a pure interval? We prove that there is a simple characterization using the *variations* of the interval, i.e., the inversions $(b, a)$ such that $\#_T(b, a) < \#_S(b, a)$. We define a notion of *essential variation* and show in particular that the selected tree-ascents of the pure interval are the *minimal essential variations* of the interval. This characterization leads to our main result in this paper.

**Theorem 16** (from [CP23]). *The intersection of two pure intervals is a pure interval.*

Besides, the dimension of the intersection is always smaller than or equal to the dimensions of the initial pure intervals. We reach equality only if one pure interval is already included in the other. You can check this property on Figure 8.7: the 2-dimensional faces can intersect only in an edge (pure interval of dimension 1) or in a tree (pure interval of dimension 0). To prove that the *s*-permutahedra is indeed a polytopal complex, we need to prove further that each pure interval can be realized as a polytope. We have serious leads for this result which I explain in Section 8.5. We hope to develop them in a sequel paper.

We can extend the definitions of pure intervals to the *s*-Tamari lattice and thus define the *s*-associahedron (see Figure 8.9). In this case, we can actually prove that it is a polytopal complex through a direct bijection with the *v*-associahedron of Ceballos, Padrol and Sarmiento [CPS16].

<figure><img src="image_80.png" /><figcaption>Figure 8.9: The *s*-associahedron complex for $s = (0, 2, 2)$.</figcaption></figure>
<figure><img src="image_81.png" /><figcaption>Figure 8.10: Some realizations for *s*-permutahedra and *s*-associahedra.</figcaption></figure>

We express a conjecture at the end of [CP19] which should also appear in [CP23].

**Conjecture 17** (from [CP19] and [CP23]). *A geometric realization of the *s*-permutahedron as a polytopal subdivision of permutahedron exists in all dimensions.*

By removing some facets, one should obtain a realization of the *s*-associahedron. We have such a realization for dimensions 2 and 3 as we show on Figure 8.10. More can be found on this webpage<fnref n="1" />.

## 8.5 Perspectives

Our work on the *s*-weak order and *s*-permutahedra is still in progress. We are finishing our second paper at the moment and hope to have at least 2 more. Besides, our initial extended abstract [CP19] has raised the interest of the scientific community. For example, Lacina [Lac22] explored some topological properties of the lattice. At the moment, the group of [GDMP$^+$23] is working on a positive

<footnote n="1">¹https://www.lri.fr/~pons/static/spermutahedron/</footnote>
solution of our conjecture. I present here the current state of our (and other's)
research and the perspectives we would like to explore.

**8.5.1 Ascentopes**

We do not prove yet in [CP23] that the *s*-permutahedra is a polytopal complex. Nevertheless, we actually have a polytopal construction for each pure interval which we call the *ascentope*. An ascentope is a certain removahedron defined using an *s*-decreasing tree *T* and a set of selected tree-ascents. Experimentally, our construction gives a geometrical realization of the pure intervals. See Figure 8.11 for an example in 3 dimensions. We can compute the skeleton of the polytope and check that it corresponds to the Hasse diagram of the interval. We also computed *f*-vectors for big examples such as the one of Figure 8.12. The *f* vector gives the number of faces in each dimension. As we understand the faces combinatorially, we can check that the geometric results are consistent with our combinatorial construction: it is the case on all the examples we have computed.

<figure><img src="image_82.png" /><figcaption>Figure 8.11: A pure interval and its corresponding ascentope</figcaption></figure>

The next step is obviously to prove that this construction is indeed a realization
of the pure intervals and to study the properties of this family of polytopes. Note
that the ascentope realization does not solve the more general conjecture which
requires to construct the polytopal complex as a polytopal subdivision. Indeed,
there is no indication that the ascentopes can be “glued” together. Nevertheless,
it would prove that the *s*-permutahedra is indeed a polytopal complex for all *s*.

**8.5.2 Solving the conjecture**

Since 2019 and [CP19], I have been mentioning the conjecture of the polytopal
subdivision realization a few times. In particular, I had started discussing this
<figure><img src="image_83.png" /></figure>

$$f \text{ vector} = (1, 2178, 9801, 19008, 20790, 14082, 6099, 1680, 282, 26, 1)$$

<caption>Figure 8.12: A pure interval of dimension 9 with the *f* vector of its ascentope.</caption>

with Éva Philippe who is a PhD student in Paris (under the supervision of Arnau Padrol). Around the same time, Rafael González D'León launched a research project to strengthen the links between France and South American research in combinatorics. As I was traveling to Colombia for the ECCO conference in 2022, I planed a few extra days of research over there. I also presented the conjecture during an “open problem” session at the conference.

Following this, Rafael González suggested that we look at the problem through the lens of *flow polytopes*. These are certain polytopes defined by *flows* on a graph which can be used to obtain *triangulations* and *polytopal subdivisions* [MM19]. This looked very promising and I invited Éva Philippe and my PhD student Daniel Tamayo to join the research discussions. Together with an extended group of researchers, they came up with a certain flow polytope which gave a dual realization of the *s*-permutahedra. Even though the proof is not finalized yet, they were able to show me explicit bijections and coordinates to construct the polytopal subdivision. Their paper is in writing [GDMP+23] and would partially solve the conjecture.

The construction of [GDMP+23] only works for an *s* which do not contain zeros. The next natural step would then be to see if the result can be extended to the zero case. Besides, we also conjectured that the realization should give a realization of the *s*-associahedron. Is it the case? Can it be proved using *flows*? What is the link with the current realization of Ceballos, Padrol and Sarmiento [CPS16]? In both cases, Tropical geometry seem to be necessary to compute the vertices coordinates. Is there a realization which does not rely on this?

Besides, what are the polytopes we obtain for each pure interval? Are they removahedron or even generalized permutahedron? Are they related to the ascentopes that we have been defining with Cesar Ceballos?

Finally, the dual of the *s*-permutahedron seems to have interesting properties. It looks like a certain *Shi-arrangement* where some walls have been removed. We would like to study this construction to better understand its properties.
8.5.3 Lattice quotients and geometric realizations

We have proved that the *s*-Tamari lattice is a lattice quotient of the *s*-weak order when *s* does not contain any zeros. What are the other lattice quotients? In particular, Tamayo, Philippe and Pilaud have been working on some *s*-Permutrees with interesting combinatorial and geometrical properties. When *s* contains some zeros, the Tamari case defines another lattice which is not isomorphic to *ν*-Tamari: what is it? What are the elements counted by? Does it have an interpretation in terms of paths?

8.5.4 Other groups and other lattices

An natural question is also to look at other Coxeter groups. For this, the per-
spective of the generalized Shi-arrangement would be a good starting point. Do
we also find polytopal subdivision? This is the case in type *B* for the Ceballos,
Padrol and Sarmiento realization of the *ν*-Tamari lattices [CPS16]. More gen-
erally, our approach through pure intervals works in many other combinatorial
lattices with nice geometric realization. Is there a way we can extend our proof
and characterizations to these other cases?
# Part V

## Experimental Approach
<figure><img src="image_84.png" /></figure>
# CHAPTER 9

## EPISTEMOLOGY OF THE EXPERIMENTAL APPROACH

*To me programming is more than an important practical art. It is also a gigantic undertaking in the foundations of knowledge.*

– Grace Hopper

In 2010, I started a research internship at the end of my master degree that would eventually lead to my PhD. My first assignment was to attend a workshop on the open source mathematical software **SageMath**. At the time, I had no experience with **SageMath** nor even with the python language (on which **SageMath** is based). Nevertheless, with a strong background in computer science, I had no difficulties to learn the system. By the end of my internship, I had actually developed the first draft of what is now an external package on bases for multivariate polynomials [PonSage16].

Already, at this early stage of my research career, it felt natural to me to approach mathematical knowledge and research through experimentation and code. Indeed, it has since become an essential part of my research methodology. In other words: I have written some code related to all my research and all my papers. Computer exploration and experimentation is as essential to me as having a pen and paper. Moreover, I believe that, to a certain degree, experimentation is part of any mathematical research, whether a computer is used or not. On the other hand, as it is used on early stages of the research, it can often be absent of the final result. This creates a difficult problem for historians and philosophers who try to understand *how* mathematical knowledge is created. For us mathematicians, and especially mathematicians who use computers, this creates another challenge: how do we value this work? In particular, how do we value the necessary technical work to create the tools that are needed?

A first step, in my opinion, is to make this work visible and to explain the experimental process in mathematics. Indeed, it has appeared to me that many people do not associate experimentation with mathematics and do not really understand what we mean by that. And probably, different mathematicians have different practices and mean different things. This has led me to have an introspective
approach of my own methods and to discuss it with historians and philosophers. In this context I was invited by Emmylou Haffner to give a talk at a seminar of mathematical history and philosophy in SPHERE [PonMisc22]. Emmylou Haffner studies ancient mathematician's drafts to understand the creative process [Haf22] like in this post where she looks at Elie Cartan's drafts [Haf17]. The seminar was coordinated by Arilès Remaki whose thesis focuses on Leibnitz [Rem21] and especially on the combinatorial explorations that led to his work. Looking at Leibnitz's drafts with combinatorial pictures and computational tables makes an interesting connection with our own exploratory methods now enhanced by computer.

In this chapter, I use the research example that I presented in [PonMisc22] to explain my research approach and unfold the early stages of research creation through experimentation using both the computer and “pen and paper”. Then I give an overview of the research code that I have written for the past ten years and of the good practices that I have put in place to share, reuse and valorize this work.

## 9.1 The experimental process

When I gave this talk at the SPHERE seminar, I was in the middle of a research project that led to what I presented in Chapter 5. In particular, I was looking at a certain lattice structure on triangular Dyck paths which would give a combinatorial interpretation to some Schur expansions computed by François Bergeron. I decided to present this problem as a typical example of research investigation.

### 9.1.1 To understand objects

Looking at my drafts, it became apparent that my first task had been to “understand” the paper [BM22] that Bergeron presented to me when I arrived in Canada. On Figure 9.1, I show some initial drafts kept in my written notes. We see that I am recomputing some of the examples of [BM22] to familiarize myself with the material. I also started to code at this stage. I implemented several classes to represent triangular partitions and triangular Dyck paths along with methods to compute the properties described in the paper. For example, below is a piece of code which is part of the triangular partition class to test if another partition is *similar* in the sense of [BM22]

```python
def is_sim_cell(self, i, j, tau):
    return self.cell_min_slope(i, j) < tau.mean_slope() \  
    and tau.mean_slope() <= self.cell_max_slope(i, j)
```

```python
def simcells(self, tau):
    for i, j in selfcells():
        if self.is_sim_cell(i, j, tau):
            yield (i, j)
```
<figure><img src="image_85.png" /><figcaption>Figure 9.1: Pictures of early drafts of triangular partitions and Dyck paths</figcaption></figure>
```python
def is_similar(self, tau):
    return all(self.is_sim_cell(i, j, tau) \ 
               for i, j in self cells))
```

The constant back and forth between my manual computations and examples and the automated ones is what ensures me that I am progressing on solid grounds. The manual computation and often manual enumeration of objects gives me the time to understand the processes to the point where I can code and automate the task. Doing so assures me that I have not overlooked a crucial detail or misunderstood a definition. Besides, sometimes the error is not mine. In the early paper draft printed to me by Bergeron, a typo had replaced a “≤” sign by a “<” sign: I was able to spot it when I tested the definition.

### 9.1.2 To eliminate false statements and generate counter example

Once I had read the paper, I wanted to study lattice structures on triangular Dyck paths to see if I could find a *q*,*t* enumeration of intervals that would be symmetric and Schur positive (as in the classical and *m*-Tamari cases). As there was a general definition of Tamari in [PV17], my first attempt was to look at it and see if it “worked”. I manually computed the first non-trivial example (see Figure 9.2) and noticed that, in this case, I had a *q*,*t* symmetry.

Here, the manual computation helps me understand how the *v*-Tamari lattice translates into triangular partitions and gives me some initial insights on how the *q*,*t* symmetry could work. Of course, the immediate next step is to implement this. Below are the methods I used to generate the *v*-Tamari lattice of triangular Dyck paths.

```python
def path_tamari_rotate(self, line):
    L = self.skew_partition().row_length}s
    p = list(self.path())
    v = L(line]
    p(line]-=1
    i = line - 1
    while i >= 0 and L[i] > v:
        p[i] -= 1
        i -= 1
    return TriangularDyckPath(self.path(), p)
```

```python
def path_tamari_up(self):
    for l, c in self.path().corners():
        yield self.path_tamari_rotate(l)
```
<figure><img src="image_86.png" /><figcaption>Figure 9.2: A manual computation of $q, t$ symmetry on intervals of $\nu$-Tamari</figcaption></figure>

Once I had this, it quickly appeared that the $\nu$-Tamari lattice did not give the desired $q, t$ interval enumeration. It is actually very common that the computer exploration is here to *save us time* by rejecting false hypothesis early on. Not only does it tell us that some ideas are just wrong but it also exhibits interesting counter examples. In this case, the partition (3, 2, 2, 1) was the first truly interesting example where $\nu$-Tamari would fail to provide a symmetric $q, t$ enumeration of intervals. I could then concentrate my efforts on this particular partition.

The partition (3, 2, 2, 1) has 23 subpartitions and we expect to find 161 intervals in the “good” lattice: 161 is the result of a certain expansion of a Schur functions in 3 variables $q, t, r$ and computing the polynomial for $q = t = r = 1$. This is exactly a case which is “in between” manual and automatic exploration. Indeed, 23 elements is rather small and manageable by hand while 161 starts to be a bit too big. My research at this point illustrates that. I was literally looking for the right lattice by manually guessing what the correct cover relations might be using my intuition and some indication from the Schur computation (see Figure 9.3). I would then hard code the lattice inside my program and compute the $q, t$ enumeration of intervals.

Doing so, I was actually able to find a working structure for (3, 2, 2, 1) but could not deduce a general rule. But the multiple attempts has left me with a better knowledge of what works and what does not.
<figure><img src="image_87.png" /><figcaption>Figure 9.3: Manual attempt to obtain the desired lattice structure on partition (3, 2, 2, 1).</figcaption></figure>

### 9.1.3 To find ideas and emit conjectures

Even though we are often wrong, we are thankfully sometimes right. There again, conjectures arise from a combination of manual and computer testing. By manually enumerating triangular Dyck paths along with their similar cells as in Figure 9.4, I came up with the idea of the *deficit* that I explain in Section 5.2.2. Only by computer exploration was I able to “check” that it was true with the following code.

```python
def test_deficitells (tdp):
    return all(tdp.is_sim_cell(i, j) != tdp.is_deficit_cell(i, j) \
               for i, j in tdp.path().cells())
```

```python
def test_deficitells_partition (p):
    for tdp in p.triangular_dyck_paths():
        if not test_deficitells (tdp):
            return tdp
    return True
```

# tested 3..12
def test_all_partitions_deficit:
    for p in TriangularPartitions(n):
        print(p)
        r = test_deficit:
        if r != True:
            return r
    return True

<figure><img src="image_88.png" /><figcaption>Figure 9.4: Manual enumeration of triangular Dyck paths to define the tableau
deficit.</figcaption></figure>

This was proved to be true by Loïc Le Mogne in our paper [LMP23] (this is an
extended abstract, the complete proof will actually appear in our future complete
paper).

Similarly, the final conjecture of [LMP23] which I explain in Section 5.3 has
also been made possible by computer exploration. For example, the following
function tests that the *v*-Tamari enumeration of intervals is Schur positive for all
triangular partitions whose top-down tableau is *sim-sym*.

# tested 3..21
def test_schur_pos PRVsimsym(n, q, t):
    for tr in TriangularPartitions(n):
        d = PRV_q_t_distribution(tr)
```python
if test_symmetry_distribution(d):
    print(tr)
    P = tr.path_tamari_lattice()
    d2 = PRV_poset_distribution(tr, P)
    pol = sum(d2[k]*q**k[0]*t**k[1] for k in d2)
    S = schur.from_polynomial pol)
    if not all(c>0 for p, c in S if len(P) <= 2):
        return tr

return True
```

The upper line is a comment which indicates which sizes have been tested. My research code is often full of testing functions with comments to keep track of positive / negative results. I also wrote the following function which tested that the Schur expansion I had corresponded to the actual Schur functions that had been sent to me by Bergeron.

```python
def test_PRV_polynomials(pols, q, t):
    wrong = []
    for key in pols:
        tr = TriangularPartition(key)
        d = PRV_q_t_distribution(tr)
        if test_symmetry_distribution(d):
            print(tr)
            P = tr.path_tamari_lattice()
            d2 = PRV_poset_distribution(tr, P)
            pol1 = sum(d2[k]*q**k[0]*t**k[1] for k in d2)
            pol2 = pols[key].expand(3)(q, t, 1)
            if not pol1 == pol2:
                wrong.append(tr)
    return wrong
```

The interesting anecdote is that this function did not give a positive answer. Indeed, two of the Schur functions did not correspond. But it actually appeared that those Schur functions are very difficult to compute on the algebraic side. The functions I had received from Bergeron were actually missing some terms that he could not get at the time using his computational methods. I was able to guess the missing terms using my conjecture. This not only strengthened my belief that the conjecture was true but also gave indications that the combinatorial computation (as inefficient as it is) would actually turn out to be more efficient than the algebraic computation which requires extensive computational power to reduce very big matrices.

### 9.1.4 To solve and to prove

As we saw, computer exploration helps us sort out positive and negative results before we even try to prove anything. This is of course useful at the investigation
stage but not only. Indeed, once we have a result that *seems* to be right and we
are trying to prove it, we need to polish our statement as much as possible. Can
we reduce the hypothesis? This other statement seems to *just* be a rephrasing,
is this the case? I believe my statement *A* implies *B*, does the computer confirm
that *B* is true? Finding the route to the final proof, we might extract lemmas or
other statements and the computer tests will be of great assistance. I consider
that I have a certain pool of objects on which I can experiment to observe the
results and guide me through my understanding<fnref n="1" />.

## 9.2 What to do with the code?

My current file for research code on triangular partitions has more than 2500
lines, not including the code written by my student Loïc Le Mogne. The code
related to [CP22] and [CP23] is about 4500 lines. This contains implementations
of the objects at stake and many test functions, failed attempts, leads, and so on.
Very much like a draft paper, it is often badly written, with few comments and
explanations. What is the value of this code? Does it have any and how can we
use it?

### 9.2.1 Contributions to open-source

Computer exploration and experimental mathematics has developed with the gen-
eral access to computer systems. As many mathematicians started to code, at-
tempts to *share* the produced code has arisen. I have been using the **SageMath**
computer system since the beginning of my thesis. I believe in the open-source phi-
losophy that the code we produce should be made accessible to all. Some objects
are wildly used in combinatorics: permutations, Dyck paths, binary trees, and so
on. They can be used in **SageMath** as the result of previous developments by other
researchers. **SageMath** actually contains the basic grounds for most mathematical
topics as can be read in [ZFM+13].

I have myself contributed to **SageMath** whenever I felt that the code I had de-
veloped could be useful to others. My larger contribution is the implementation of
Tamari interval-posets [PonSage14] following my paper [CP15]. Providing a direct
implementation into **SageMath** was very useful in a research perspective: it helped
Tamari interval-posets to become a major objects in the study of Tamari intervals
as it could be used directly for computations. The initial class has been enriched
by others, especially Frédéric Chapoton who works on this subject. Following
my second paper [Pon19], we added together the *rise-contact* involution which
I had implemented during my investigations [CPSage18]. Previously, I had also
contributed to several methods on binary trees [PonSage13]. These are just some
examples, I have authored about 15 contributions and participated in about 50.

<footnote n="1">¹Note that I am not talking about formal proof assistants. This is another area altogether which I have not had time to explore.</footnote>
9.2.2 Other good practices

The road from writing research code for oneself and contributing to an open-
source software (or develop a full independent software) is not easy. For one
thing, much of the code we write for research does not belong to a general soft-
ware like **SageMath**. A big portion of the code corresponds to failed attempts, or
constructions that would eventually emerge in a different form in the final prod-
uct. At this stage, it is not always easy to identify *what* part of our code should
belong the common shared knowledge of a software like **SageMath**. Besides, the
code we write for research almost never follows the coding standard of a final
**SageMath** contribution or of a software that could be distributed. There is often
a long road of cleaning and documenting and testing. This takes a very long time
that we, realistically, might not have. The result is that some useful, imperfect,
code might never get shared.

Even if the quality is low, I believe that research code is worth sharing, espe-
cially when it leads to a published paper. Indeed, if I used this code for my own
investigation, it could be useful for others. The objects used in the paper might
turn out to be reused in another paper. Somebody else might decide to turn them
into a proper software contribution. Or maybe, another researcher just needs to
compute extra examples for their own purpose. This is why I have taken the habit
to publish the research code along with the paper even if it is not a proper contri-
bution to an existing software or a finished product. I first started a single GitHub
repo where I would just post my research scripts organized by thematics along with
some examples and presentations written with Jupyter notebooks [PonSage18].

I provided an open license directly on the GitHub repo making it explicit that the code can be reused and, for example, integrated into another open-source software. To be fully efficient, one also needs to provide some *environment* information about *how* to run the code. I thus chose to include a *Docker* file which contains for example what **SageMath** version has been used to write the code.

It became apparent that having only one repo for everything was not practi-
cal: I would often upgrade **SageMath** but not fix all my previous codes and the
environment information would then be out of date. For my latest papers, I chose
to create one repo per paper: this is what I did for [PonSage22] as an annex
to [Pon22]. I also have a specific code repo for this document [PonSage23]. Fi-
nally, when I published the code related to [CP22], I decided to create a specific
archive of the code at the time of publication [PonSage22b] using **Zenodo**. Indeed,
as this code is still prone to be changed, it provides a link to a specific state of
the software.

As a summary, here is a list of codes related to my different papers:

* Code related to Tamari interval-posets ([CP15] and [Pon19]) is available on my general repo [PonSage18] with multiple demo notebooks. Some of it has been turned into **SageMath** contributions especially in [PonSage14] and [CPSage18]).
* Code related to [PP18, CPP19, PP20] with a demo notebook is available on my general repo [CPSage18].

* Code related to [Pon22] with a demo notebook is available on a specific repo [PonSage22].

* Code related to [CP22, CP23] with multiple demos is available on a specific repo and as a Zenodo archive [PonSage22b].

Note that the code related to [LMP23] that I mention in Section 9.1 is not yet published. Indeed, only an extended abstract has been published and the final paper is still in the making. Besides, Loïc Le Mogne also wrote some code and I would like to merge our contributions.

One reason for sharing this code is *re-usability* which is why I make the effort of providing a running environment and often some demo notebooks that I edit as I write the corresponding papers. Speaking with historians and philosophers of mathematics, I also see another purpose. This “unclean” code is an equivalent of an early draft. It contains mistakes, failed attempts, ideas that sometimes were never fully developed. In other words, it contains some information about the process itself. Whether we share those codes or not, they hold an epistemic value and should at least be kept.
> Se vouloir libre, c'est aussi vouloir les autres libres.

> – Simone de Beauvoir, *Pour une morale de l'ambiguïté*.

In the past 10 years, I have been a promoter of the software **SageMath** and more generally of open-source development and free software. This has also led me to think more of my role as a scientist and the science community we build. In 2018, I gave the opening keynote at the programming event **PyCon** Fr. I developed the idea that scientists and open source developers share a common ethic and motivation: we want to create knowledge and share it, we want contribute and to collaborate, we want our work to last to be re-used and improved.

Following this idea, it is important for me to take an active part in science communication and outreach at different levels. Within the math community, I work at promoting open-source software and programming knowledge in general. Within the scientific community at large, I have made presentations to explain the process of experimental mathematics. I also talk to general audiences and especially to middle and high schools to explain and share what research in mathematics and computer science mean. In this chapter, I give a general overview of this work.

### 10.1 Advocate for open science

Open science is the general idea that science productions should be made accessible to all: papers, data, software, etc. In mathematics and computer science, it concerns especially open-source software. Between 2015 and 2019, I have been part of the **OpenDreamKit** project which was coordinated in my university Paris-Saclay and included 18 institutions and about 50 participants. We received significant funding that was used mostly for two things: hiring full paid developers to work on open-source software and organizing events within the different communities to foster collaboration and development.

I was the local coordinator for Paris-Saclay as well as the work package leader for “Community Building, Training, Dissemination, Exploitation, and Outreach”.
In the 4 years of the project, we were part of 110 events: either organized by members of the project or where they participated in a significant way. I was myself part of 23 events. For example, I co-organized the event "Free Computational Mathematics at CIRM" in 2019. The project also included 3 reviews in front of a European commission in which I took an active part presenting the activities of the project.

My implication as a promoter of open science goes beyond my role in the project. I have organized several **Sage Days** workshop, even before the beginning of **OpenDreamKit**. In 2015, I organized a joint event at **PyCon US** to help build relations between the **SageMath** community and the open-source community at large. At this occasion, I gave a presentation at the main conference to explain the concept of experimental mathematics [PonMisc15] to many non-mathematicians.

I also participate in the effort to build a better scientific ecosystem, especially concerning scientific publications. I was an editor for the *Journal of Open Source Software* between 2019 and 2022 to help better recognize the role of open-source development in scientific progress. I have also accepted a role as *backend editor* for the new journal *Combinatorial Theory* which is owned directly by its editorial board and charge no fees neither for readers nor writers.

## 10.2 Talking about research to non researchers

I also invest myself in promoting and explaining research to non researchers, especially in schools. I took part several times to the MATH.en.JEANS project either by giving research talks to the schools, or by proposing research projects to pupils and meeting them throughout the year. Besides, I gave several general audience presentations and was invited to create an educational video with a professional team for the YouTube channel *Le Myriogon* [Myr21]. I presented a simple bijection on permutations that I explained using playing cards as you can see on Figure 10.1.

<figure><img src="image_89.png" /><figcaption>Figure 10.1: *Des Cartes bien à leur place* for *Le Myriogon* youtube channel</figcaption></figure>
## 10.3 Building a more inclusive community

An open-source software is often free in the sense of “free of charge” (French word: “gratuit”) and always free in the sense “free to be reused and shared” (French word: “libre”). But this is not enough to make it equally accessible to all. This is also true for mathematics and computer science in general. Technical and societal barriers exist which affect certain demographics more than others.

Women especially suffer from negative stereotypes early on in relations to science and mathematics. This affects their own confidence and ability to solve problems [DCL02], the way they are perceived by others [MDB+12], and their career choices [NAV21]. Danica Savonick and Cathy N. Davidson have collected several studies on gender bias in academia and its combinations with other bias such as racism [SD16]. As a result, women are largely underrepresented in mathematics and computer science [BW18]. I have myself expressed the burden of *being the only woman* in a blog post [PonMisc17b]. This has led me to take different actions. I have participated in many events such as the exhibition organized by *Femmes et mathématiques* [FetM22] to help make women in science more visible, especially to young girls. But I am also interested in building a better environment for current women scientists. For example, I run the *PyLadies* parisian chapter between 2016 and 2020 where I organized weekly meetup with many women developers. I noticed that even at a very high level, women lack confidence in their ability to code and would be reluctant to join coding events such as *Sage Days* which often show a worst gender disparity than classical mathematics events. As part of the *OpenDreamKit* project, I organized in 2017 and 2019 *Women in Sage* events which were *Sage Days* workshops targeted at women. They were a great success. You can see some pictures on Figure 10.2 and read the reports I published online [PonMisc17c, PonMisc19].

Actual borders are physical barriers that stop people from doing research. Indeed, most resources are concentrated in “western” countries. In many parts of the world, students and researchers lack funding, basic infrastructure or even access to advanced education. This is visible in open-source software: see the big African gap on the **SageMath** developer map on Figure 10.3. In 2019, two Nigerian women were supposed to join us at our *Women in Sage* event organized in Archanes, Greece. Their trip was fully paid by the **OpenDreamKit** project but they could not get their Visa in time. Nevertheless, the same year we had 3 Nigerian researchers coming to our other event “Free Computational Mathematics at CIRM”. They enjoyed the event very much and we decided to co-organize **Sage Days** in Nigeria the following summer. I did not attend myself as I was literally having a baby the same week. The project funded the trip of some **SageMath** instructors and relied on Ibadan university for local organization. It was a great success (see [Bra19]) and also an indicator for the need of such events in West Africa as well as a lesson for us **SageMath** developers on how we can improve our software to allow an easier access.
<figure><img src="image_90.png" /><figcaption>Figure 10.2: Pictures taken at Women in Sage in 2017 and 2019</figcaption></figure>

<figure><img src="image_91.png" /><figcaption>Figure 10.3: The SageMath developer map taken from https://www.sagemath.org/ in June 2023</figcaption></figure>

In 2021, I co-organized and participated remotely to a *Women in Sage* event in Senegal organized by Eliza Lorenzo. She was a participant to the *Women in Sage* event I held in Paris in 2017. I had advertised the event to my contacts in Nigeria and some researchers from Ibadan were able to attend. This shows that these events create a good dynamic that can help West African researchers to reinforce their scientific network. Thanks to these events, I formed connections with two researchers: Ini Adinya from Nigeria and Olivia Nabawanda from Uganda with whom I try to build a scientific collaboration despite the systemic difficulties.

Thanks to **OpenDreamKit**, I was also able to attend the event **ECCO** (Colombia) in 2016 and 2018 as a **SageMath** instructor. These were extremely reward-
ing experiences (see [PonMisc16, PonMisc18]). The Colombian community in combinatorics has made great work in building an inclusive space and bringing Colombian and international students together. Following these events, I became a co-organiser for ECCO 2022, especially in charge of the CIMPA application and funding. I am now part of the *ECCO Steering committee* whose role is to organize future events. Besides, ECCO 2018 is where I met Daniel Tamayo Jiménez who is now finishing his PhD under my supervision and joined supervision of Vincent Pilaud.

## 10.4 Political mathematics

There is a general belief that mathematics is *politically neutral*. Indeed, a theorem expresses a certain truth under some hypothesis, can it be racist or sexist? I do not believe this is the right question. Theorems are written by people in a certain context. Us, mathematicians, are part of a society which is, by nature, political. In [God01, God03], Roger Godement writes extensive preambles and notes to its books on Mathematical analysis. He questions the scientific responsibility of scientists in the creation of deadly weapons such as nuclear bombs. He also shows how political contexts and decisions impact what kind of science is considered *good* or *interesting* and thus developed.

In algebraic combinatorics, the long term applications of what we do, if they exist, may seem too far in the future to raise our concern. But this should not blind us in the illusion that what we do has no political implication. Defending fundamental science and the pursuit of knowledge for its own sake is already a political statement. Defending open-source software and open science is a political statement. Besides, there is the question of *how* we do mathematics, *who* do we teach it to, *how* do we teach it, etc. Federico Ardila develops this idea in [Ard20] and asks important questions about power.

Without any advanced statistical examination, I can say that I teach computer science mostly to young men and thus I am participating in a system which deprives young women of a useful knowledge in our society (I have chosen the gender disparity but, of course, many other inequalities are just as problematic). The academic system we are part of has some positive values and qualities and also some important setbacks. Of course, we cannot solve it all by ourselves. But we can choose to place our scientific activity in the general context of society and not see it as neutral. We can think about what we do and improve when we can. This is the choice I am always trying to make. This is also why I chose to include this last chapter. Not only does it relay some important activities I have run for the last 10 years but I consider these activities to be part of my scientific commitment just as much as my mathematical results.
*Truth is a matter of the imagination.*

– Ursula K. Le Guin, *The Left Hand of Darkness*.

I consider algebraic combinatorics to lie at the intersection of algorithmics and mathematics. My goal is to understand the inner structure of objects to exhibit the connection between topics that seem distant in nature. This is the case with the Tamari lattice and the Tamari intervals especially which I present in Chapters 3, 4, and 5. They appear in the context of planar maps as well as in representation theory. If I sometimes answer questions as in [Pon19], most of my papers actually *create* new structures, like permutrees [PP18], or the *s*-weak order [CP22]. My hope is that those new objects deepen our understanding of the permutahedron and associahedron and create new points of views to explore.

My motivation comes from difficult open questions. I have mentioned the *q*,*t*-symmetry of Catalan numbers in Chapter 5. To finish this document, I present two other open problems which have caught my attention recently. The Tamari lattice and its generalization the *v*-Tamari lattice can be understood in the more general framework of *pipe dreams* and *subword complexes* [CLS14]. More precisely, pipe dreams are certain objects associated to permutations (each permutation gives a collection of pipe dreams). All pipe dreams of a given permutation are related through *flips* and we can define a poset of increasing flips. For specific permutations, this gives the Tamari and *v*-Tamari lattices. In general, this is not a lattice but, in [Rub12], Rubey conjectures that it is indeed a lattice when flips are restricted to *chute moves*. Beside, interesting geometrical structures appear which look like polyhedral complexes based on pure intervals like in our work on *s*-permutahedra [CP23].

The second problem is related to the *flip distance* of triangulations. In [STT88], Sleator, Tarjan, and Thurston prove a fundamental result on the associahedron. The diameter, *i.e.*, the maximal rotation distance between two binary trees, is linear in the dimension *n*. The exact value was computed by Pournin in [Pou14]. But it leaves open the following: what is the complexity of computing the minimal rotation distance between two given trees? This is the same as computing the minimal flip distance between two triangulations of a convex polygon. When the polygon is not convex, it is NP-complete [AMP15] but the proof cannot be
extended to the convex case. Besides, remember that the associahedron is *in between* the permutahedron and the cube. In the permutahedron, the diameter is the maximal number of inversions, $\frac{n(n-1)}{2}$. In the cube, it is *n*. The minimal distance is also easy to compute in both those polytopes. Indeed, the distance between two permutations $\sigma$ and $\mu$ is the number of inversions of $\mu\sigma^{-1}$, and can be computed in $O(n^2)$. In the cube, the distance between two binary sequences is just the number of bits which are different. So it is computed in $O(n)$ where *n* is the size of the sequence. It is interesting to notice these questions are easy in those two polytopes but difficult for the associahedron. What about permutreehedra and other generalized permutahedra? Can we use what we understand from the relations of these polytopes with the permutahedron and the cube to study those questions?

Solving such problems is a long term goal that fuels my current research. With each attempt, come new ideas that will flourish in unexpected manners, adding to the general understanding of the underlying structures. Besides, in addition to working on these questions myself, I believe that my role is to foster the scientific environment that makes progress possible. Encourage seminars, collaborations, workshops, open discussions, and freedom of investigation. This is the role I take especially as PhD advisor. With each of my students, I have tried to find the right balance between providing problems and questions and giving them the opportunity to express their own ideas and build their own research path. When I ask an open question, I do not always expect a final answer. I expect attempts and ideas. Then I am here to guide them through it, in a collaborative manner, to see where it could lead us. Sometimes, it eventually answers the original question, sometimes it opens new routes that we had not seen beforehand.

Someone told me an anecdote that they were once asked "In your field, what do you think the next *breakthrough* is going to be?". I do not believe this is an interesting question. Breakthroughs do not happen because we chase after them (and their very nature makes them impossible to predict). They happen because we let our mind open to new ideas and keep inventing new truths.
GLOSSARY

CIMPA stands for “Centre International de Mathématiques Pures et Appliquées”. It is an originally French organization (now funded by France, Spain, Norway, and Switzerland) whose goal is to promote mathematical research in developing countries. 131

Docker is a system that provides easy to deploy virtual machines called *containers* with specific environments. A *Docker file* is a simple configuration file that describes the needed environment (software versions, installation commands, etc.). 124

ECCO stands for “Encuentro Colombiano De Combinatoria”. It is a summer school organized in Colombia every other year welcoming both Colombian and international students to follow advanced lectures in combinatorics. 130, 131

GitHub is a private online platform used to host code and text-based projects using the *git* version management system. 124

Jupyter is an open-source software used to create interactive web-based notebooks as computing environments in many languages. 124

MATH.en.JEANS is a French organization which organizes mathematical research projects in schools involving researchers and pupils. The researchers propose mathematical problems on which the pupils work throughout the year. Once a year, pupils present their results in special events. 128

OpenDreamKit stands for “Open Digital Research Environments Toolkit for the Advancement of Mathematics”. It is a H2020 European project which funded many different open-source software and associated communities in mathematics. https://opendreamkit.org/. 127–130

PyCon is the name given to general conferences about the Python programming language. The main event is the yearly event PyCon US which often welcomes a few thousand participants. PyCon Fr is also organized every year with a few hundred participants. 127, 128
**PyLadies** is an international mentorship program to support women who code in python. 129

**Sage Days** are workshop dedicated to the software **SageMath**. They often include introduction talks and also development time around specific topics.. 128, 129

**SageMath** is a free open source mathematical software created in 2005 and mostly written in python. It includes a wide range of mathematics and is built on top of many other open-source packages. Its stated mission is to “create a viable free open source alternative to Magma, Maple, Mathematica and Matlab”. It has been developed mostly by researchers, based on free contributions evaluated through code-reviews. 2, 9, 13, 26, 27, 41, 51, 115, 123, 124, 127–130

**Zenodo** is a general-purpose open repository which allows researcher to deposit research data. It provides a version number and a DOI. 124, 125
SELECTED LIST OF PUBLICATIONS

**International journals**

[PPTJ23] Vincent Pilaud, Viviane Pons, and Daniel Tamayo Jiménez. Permutree sorting. *Algebraic Combinatorics*, 6(1):53-74, 2023. doi:10.5802/alco.249.

[PP20] Vincent Pilaud and Viviane Pons. The Hopf algebra of integer binary relations. In *Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 1*, pages 299-344. European Mathematical Society Publishing House, February 2020. doi:10.4171/204-1/8.

[CPP19] Grégory Chatel, Vincent Pilaud, and Viviane Pons. The weak order on integer posets. *Algebraic Combinatorics*, 2(1):1-48, 2019. doi:10.5802/alco.36.

[Pon19] V. Pons. The Rise-Contact involution on Tamari intervals. *The Electronic Journal of Combinatorics*, 26(2):P2.32, 2019. doi:10.37236/7698.

[PP18] V. Pilaud and V. Pons. Permutrees. *Algebraic Combinatorics*, 1(2):173-224, 2018. doi:10.5802/alco.1.

[CP15] Grégory Châtel and Viviane Pons. Counting smaller elements in the Tamari and m-Tamari lattices. *Journal of Combinatorial Theory, Series A*, 134:58-97, 2015. doi:10.1016/j.jcta.2015.03.004.

**International conference proceedings**

[LMP23] Loïc Le Mogne and Viviane Pons. Deficit and (q,t)-symmetry in triangular Dyck paths. In *35th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2023)*, volume 89B, page Art. #21, Davis, U.S., 2023. Séminaire Lotharingien de Combinatoire.

[PPTJ21] V. Pilaud, V. Pons, and D. Tamayo Jiménez. Permutree sorting. In *33rd International Conference on Formal Power Series and Algebraic*
*Combinatorics* (FPSAC 2021), volume 85B, page Art. 31. Séminaire Lotharingien de Combinatoire, 2021.

[CP19] C. Ceballos and V. Pons. The s-weak order and s-permutahedra. In *31st International Conference on "Formal Power Series and Algebraic Combinatorics"* (FPSAC 2019), volume 82B, page Art. 76, Hanover, United States, 2019. Séminaire Lotharingien de Combinatoire.

[PP18b] Vincent Pilaud and Viviane Pons. Algebraic structures on integer posets. In *30th International Conference on Formal Power Series and Algebraic Combinatorics* (FPSAC 2018), volume 80B, page Art. 61, Hanover, United States, 2018. Séminaire Lotharingien de Combinatoire.

[PP17] V. Pilaud and V. Pons. Permutrees. In *The European Conference on Combinatorics, Graph Theory and Applications (EUROCOM'17)*, volume 61, pages 987-993. Electronic Notes in Discrete Mathematics, 2017. doi:10.1016/j.endm.2017.07.063.

[Pon15] Viviane Pons. A lattice on decreasing trees : The metasylvester lattice. In *27th International Conference on Formal Power Series and Algebraic Combinatorics* (FPSAC 2015), DMTCS Proceedings, pages 381-392. Discrete Mathematics and Theoretical Computer Science, 2015.

[CCP14] F. Chapoton, G. Chatel, and V. Pons. Two bijections on tamari intervals. In *26th International Conference on Formal Power Series and Algebraic Combinatorics* (FPSAC 2014), DMTCS Proceedings. Discrete Mathematics and Theoretical Computer Science, 2014.

[CP13] G. Chatel and V. Pons. Counting smaller trees in the Tamari order. In *25th International Conference on Formal Power Series and Algebraic Combinatorics* (FPSAC 2013), DMTCS Proceedings, pages 433-444. Discrete Mathematics and Theoretical Computer Science, 2013.

## Preprints

[CP23] Cesar Ceballos and Viviane Pons. The s-weak order and s-permutahedra II: The combinatorial complex of pure intervals. 2023. arXiv:2309.14261.

[CP22] Cesar Ceballos and Viviane Pons. The s-weak order and s-permutahedra I: combinatorics and lattice structure. 2022. arXiv:2212.11556.

[Pon22] Viviane Pons. A description of the Zeta map on Dyck paths area sequences. 2022. arXiv:2205.06375.
## Software development

[PonSage23] Viviane Pons. HDR-notebooks, March 2023. https://github.com/VivianePons/HDR-notebooks.

[PonSage22] Viviane Pons. An implementation of the Zeta map on Dyck paths area sequences. https://github.com/VivianePons/zetaAreaseq.

[PonSage22b] Viviane Pons. SageMath code and demo for s-weak order and s-permutahedra. Zenodo, December 2022. doi:10.5281/zenodo.7459784.

[PonSage18] V Pons. Public open-source SageMath research code and notebooks, 2018. https://github.com/VivianePons/public-notebooks.

[CPSage18] F. Chapoton and V. Pons. SageMath ticket: Some new methods for Tamari interval-posets, 2018. http://trac.sagemath.org/ticket/24873.

[PonSage16] V. Pons. SageMath external package implementing the abstract ring of multivariate polynomials with several bases, 2016. https://github.com/VivianePons/multipolynomial-bases.

[PonSage14] V. Pons. SageMath ticket: Interval-posets of tamari, 2014. http://trac.sagemath.org/ticket/15683.

[PonSage13] V. Pons. SageMath ticket: Adding new combinatorial maps to binary trees, 2013. http://trac.sagemath.org/sage_trac/ticket/14123.

## Other

[PonMisc22] Viviane Pons. Expérimentation manuelle et par ordinateur dans la recherche en combinatoire, 2022. Exposé au séminaire d'histoire et philosophie des mathématiques du laboratoire SPHERE. http://www.sphere.univ-paris-diderot.fr/spip.php?article2708.

[Myr21] Barcelo, Stephen, Chloé Bouchaour, Robin Jamet, Mickaël Launay, Roger Mansuy, and Viviane Pons. Des cartes bien à leur place - Viviane Pons - Le Myriogon, November 2021. Vidéo pour Le Myrogon. https://www.youtube.com/watch?v=RcXmhKF9ewo.

[PonMisc19] Viviane Pons. Report on second Women In Sage · OpenDreamKit. https://opendreamkit.org/2019/06/28/WomenInSage/.

[PonMisc18] Viviane Pons. I want to do science the Colombian way, 2018. http://openpyviv.com/2018/06/30/colombian-way/.
[PonMisc17] Viviane Pons. Ordonner les ordres : un treillis sur les ordres partiels. *Interstices*, July 2017. https://interstices.info/ordonner-les-ordres-un-treillis-sur-les-ordres-partiels/.

[PonMisc17b] Viviane Pons. Being the only woman - OpenPyViv, 2017. http://openpyviv.com/2017/04/10/OnlyWoman/.

[PonMisc17c] Viviane Pons. Report on WomenInSage · OpenDreamKit, 2017. https://opendreamkit.org/2017/04/06/WomenInSage/.

[PonMisc16] Viviane Pons. An Inclusive Maths Conference: ECCO 2016, 2016. Blog post published on AMS Blogs. https://blogs.ams.org/matheducation/2016/08/22/an-inclusive-maths-conference-ecco-2016/.

[PonMisc15] Pons, Viviane. Experimental pure mathematics using Sage, April 2015. PyCon talk. https://www.youtube.com/watch?v=3LZiZKgVjaU.
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