# Polychrony as Chinampas
## 6 Combinatorial Description of Chinampas
### Algorithms for Activated Vertices
Result: Boolean explaining if the vertex \(n\) will be activated at time \(t_{0}\); if\((n,t_{0})\) in \(PV\)then return True ;  end iforderedListOfPyramids = \(buildPymarids(PV)\); forpyramid in orderedListOfPyramidsdo  deltaT=(\(t_{0}-pyramid.t\)); if\(pyramid.1P+deltaT\leqslant n\leqslant pyramid.rP\) and \(pyramid.t\leqslant t_{0}\leqslant n-pyramid.1P+pyramid.t\)then /* (\(n,t_{0}\)) in a pyramid */  return True; else if\(pyramid.t>t_{0}\)then  break;  end if end return False;
```

**Algorithm 4**Will_vertex_be_activated

## 7 Triangular Sequences

We study the chinampas obtained by stacking several \(pyramid(2)\) instances below \(pyramid(n)\) for \(n\geqslant 4\). For ease of exposition, we define _roots_ as the sequences of \(pyramid(2)\) stacked on top of each other. Note that \(pyramid(n)\) with \(n\geqslant 4\) can have multiple roots. For simplicity, we let \(pyramid(n)\) have \(n=3K\) and \(K\in\mathbb{N}\), and all roots had the same number \(R\) of \(pyramid(2)\). **Example 3**. : _Consider the two extreme cases for \(pyramid(6)\), whose roots are depicted in Figure 17. In Figure 17a, the roots are formed by stacking \(pyramid(2)\) vertically, while in Figure 17b, the \(pyramid(2)\) instances are stacked along diagonals._

Figure 17: Roots in \(pyramid(6)\). We show all activated vertices except those above the primary vertices of \(pyramid(6)\). #### Ehrhart Series and Order Polynomials

The order polynomial \(\Omega(P,x)\) of a partially ordered set (poset) \(P\) was introduced by [PERSON] [22]. The polynomial evaluated on \(n\) returns the number of labels \(\Omega(P,n)\) on the poset \(P\), which preserves the order. Similar to the construction of \(T_{R,n}\), we can construct a poset \(P_{R}\) containing only the symbol \(\geq\) by subtracting \(i-1\) units from the column \(i\) from right to left, where \(i\in[1,\cdots,R]\). The poset only depends on the variable \(R\) and not on the variables \(k\) and \(n\):

**Lemma 3**. : _Consider chinamps with roots of multiplicity \(n=3K\), where each root with \(R\) is a copy of pyramid\((2)\). Then, the number of such chinamps is \(\Omega(P_{R},k+1)\)._

Proof. : The number of triangular sequences \(T_{R,n}\) is \(\Omega(P_{R},k+1)\). This connection with combinatorics allowed us to determine the properties of the generating functions. For a poset \(P\), one can associate the order polytope [23; 24]\(Poly(P)\). Then, the generating function of the order polynomial is the variable \(x\) times the Ehrhart series. For example, when \(R=3\), the triangular sequence corresponds to the poset \(\{a<b<c<d<e,b<f<d\}\), and we obtain the generating function \(\frac{-x}{(1-x)^{b}}+2\frac{x}{(1-x)^{y}}\). Ehrhart series of order polytopes are known to be of the form \(\frac{h^{\#}(x)}{(1-x)^{d+1}}\), or in our case \(d=\frac{R(R+1)}{2}\). The term \(h^{*}(x)\) is a polynomial of a degree of at most \(d\), where its coefficients satisfy the Dehn-Sommerville equations and are unimodal. The previous result relates the enumeration of chinamps with polytopes of the form \(Poly(P_{R})\). We counted chinamps with the help of Mathematica [25] and a topological version of the calculus of species [26]. These calculations gave us evidence of the validity of Lemma 3 and led to the concept of triangular sequences. For details on the use of Mathematica for counting order polytopes, see [27]. ## 8 Conclusions

In this paper, we introduced activation diagrams, chinamps, and pyramids to study the effect of signals on the vertices of a nonlinear signal flow path. Furthermore, we demonstrated that pyramids are the simplest possible activation diagrams. Finally, we presented a deterministic algorithm to construct chinamps out of pyramids (see Remark 5). Chinamps were conceived to serve as an idealized model for cascades, with sequences of neural spikes forming a polychrony group. To support our conclusions, we developed an optimal code to answer the following introductory questions: \"Will a fixed vertex be activated at a particular time?