algebraic and combinatorial structures on pairs of twin binary trees abstract we give a new construction of a hopf algebra defined first by reading 2005 rea05 whose bases are indexed by objects belonging to the baxter combinatorial family i e baxter permutations pairs of twin binary trees etc our construction relies on the definition of the baxter monoid analog of the plactic monoid and the sylvester monoid and on a robinson schensted like correspondence and insertion algorithm indeed the baxter monoid leads to the definition of a lattice structure over pairs of twin binary trees and the definition of a hopf algebra the algebraic properties of this hopf algebra are studied and among other multiplicative bases are provided and freeness and self duality proved

hopf algebra of the planar binary trees is a sub hopf algebra there is a basis qn of soln such that the composite map k qn soln k sn has the following property its linear dual k sn k qn is induced by a set theoretic map sn qn in between sn and qn there is an intermediate set yn which is made of planar binary trees with n vertices for instance for n 3 the number of elements in s3 y3 and q3 is 6 5 and 4 respectively the projection from sn to qn is the composite of two maps sn w yn w qn article no ai981759

the ladder problem painleve integrability and explicit solution we consider the n dimensional ladder system that is the homogeneous quadratic system of first order differential equations of the form where aij i 1 j i j 1 n introduced by imai and hirata 2002 preprint nlin si 0212007 v1 3 the ladder system is found to be integrable for all n in terms of the painleve analysis and its solution is explicitly given

a categorification of biclosed sets of strings abstract we consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type palu pilaud and plamondon proved that the collection of all biclosed sets of strings forms a lattice and moreover that this lattice is congruence uniform many interesting examples of finite congruence uniform lattices may be represented as the lattice of torsion classes of an associative algebra we introduce a generalization the lattice of torsion shadows and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick finite congruence uniform lattices admit an alternate partial order known as the core label order in many cases the core label order of a congruence uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra analogous to torsion shadows we introduce wide shadows and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows

intervals of balanced binary trees in the tamari lattice we show that the set of balanced binary trees is closed by interval in the tamari lattice we establish that the intervals t t where t and t are balanced binary trees are isomorphic as posets to a hypercube we introduce synchronous grammars that allow to generate tree like structures and obtain fixed point functional equations to enumerate these we also introduce imbalance tree patterns and show that they can be used to describe some sets of balanced binary trees that play a particular role in the tamari lattice finally we investigate other families of binary trees that are also closed by interval in the tamari lattice

improved mixing time bounds for the thorp shuffle and l reversal chain we prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards then we use it to obtain improved bounds for two previously studied models e thorp introduced the following card shuffling model in 1973 suppose the number of cards n is even cut the deck into two equal piles drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip then drop from the other pile continue this way until both piles are empty we obtain a mixing time bound of o log 4 n previously the best known bound was o log 29 n and previous proofs were only valid for n a power of 2 we also analyze the following model called the l reversal chain introduced by durrett there are n cards arrayed in a circle each step an interval of cards of length at most l is chosen uniformly at random and its order is reversed durrett has conjectured that the mixing time is o max n n 3 l 3 log n we obtain a bound that is within a factor o log 2 n of this the first bound within a poly log factor of the conjecture

structure of the loday ronco hopf algebra of trees abstract loday and ronco defined an interesting hopf algebra structure on the linear span of the set of planar binary trees they showed that the inclusion of the hopf algebra of non commutative symmetric functions in the malvenuto reutenauer hopf algebra of permutations factors through their hopf algebra of trees and these maps correspond to natural maps from the weak order on the symmetric group to the tamari order on planar binary trees to the boolean algebra we further study the structure of this hopf algebra of trees using a new basis for it we describe the product coproduct and antipode in terms of this basis and use these results to elucidate its hopf algebraic structure in the dual basis for the graded dual hopf algebra our formula for the coproduct gives an explicit isomorphism with a free associative algebra we also obtain a transparent proof of its isomorphism with the non commutative connes kreimer hopf algebra of foissy and show that this algebra is related to non commutative symmetric functions as the commutative connes kreimer hopf algebra is related to symmetric functions

weak bruhat order on the set of faces of the permutohedron and the associahedron abstract we define a partial order on the set of faces of the permutohedron which extends the weak bruhat order of the symmetric group the restriction of this order gives a partially ordered set structure to the set of faces of the associahedron which extends the tamari order defined on the set of planar binary rooted trees these orders are used to describe associative algebra structures on the vector spaces spanned by the set of faces of the permutohedra on one hand and on the vector space spanned by the set of planar rooted trees on the other hand our results extend the description of the dendriform algebra structure on the spaces spanned by permutations and by planar binary trees in terms of the weak bruhat order and the tamari order respectively obtained by j l loday and the second author in j l loday m ronco order structure and the algebra of permutations and of planar binary trees j algebraic combin 15 2002 253 270

the complete symmetry group of the generalised hyperladder problem abstract we further consider the n dimensional ladder system that is the homogeneous quadratic system of first order differential equations of the form x i x i j 1 n a ij x j i 1 n where aij i 1 j i j 1 n introduced by imai and hirata nlin si 0212007 we establish the most general system of first order ordinary differential equations invariant under the algebra which characterises the ladder system of imai and hirata and the algebra of minimal dimension required to specify completely this most general system we provide the complete symmetry group of the generalised hyperladder system and discuss its integrability

on the complete symmetry group of the classical kepler system a rather strong concept of symmetry is introduced in classical mechanics in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey accordingly a complete symmetry group realization in mechanics must be endowed with the following two features 1 the group acts freely and transitively on the manifold of all allowed motions of the system 2 the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group this program is applied successfully to the classical kepler problem since the complete symmetry group for this particular system is here obtained the importance of this result for the quantum kinematic theory of the kepler system is emphasized

ideals of quasi symmetric functions and super covariant polynomials for sn the aim of this work is to study the quotient ring r n of the ring q x 1 x n over the ideal j n generated by non constant homogeneous quasi symmetric functions we prove here that the dimension of r n is given by c n the n th catalan number this is also the dimension of the space sh n of super covariant polynomials that is defined as the orthogonal complement of j n with respect to a given scalar product we construct a basis for r n whose elements are naturally indexed by dyck paths this allows us to understand the hilbert series of sh n in terms of number of dyck paths with a given number of factors

crystals and trees quasi kashiwara operators monoids of binary trees and robinson schensted type correspondences abstract kashiwara s crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components the celebrated plactic monoid the monoid of young tableaux arises in this way from the crystal graph for the q analogue of the general linear lie algebra gl n and the so called kashiwara operators interact beautifully with the combinatorics of young tableaux and with the robinson schensted knuth correspondence the authors previously constructed an analogous quasi crystal structure for the related hypoplactic monoid the monoid of quasi ribbon tableaux which has similarly neat combinatorial properties this paper constructs an analogous crystal type structure for the sylvester and baxter monoids the monoids of binary search trees and pairs of twin binary search trees respectively both monoids are shown to arise from this structure just as the plactic monoid does from the usual crystal graph the interaction of the structure with the sylvester and baxter versions of the robinson schensted knuth correspondence is studied the structure is then applied to prove results on the number of factorizations of elements of these monoids and to prove that both monoids satisfy non trivial identities

mixing of the symmetric exclusion processes in terms of the corresponding single particle random walk we prove an upper bound for the e mixing time of the symmetric exclusion process on any graph g with any feasible number of particles our estimate is proportional to trw g ln v e where v is the number of vertices in g and trw g is the 1 4 mixing time of the corresponding single particle random walk this bound implies new results for symmetric exclusion on expanders percolation clusters the giant component of the erdos renyi random graph and poisson point processes in rd our technical tools include a variant of morris s chameleon process

combinatorics of cyclic shifts in plactic hypoplactic sylvester baxter and related monoids abstract the cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift this paper examines the cyclic shift graphs of plactic like monoids whose elements can be viewed as combinatorial objects of some type aside from the plactic monoid itself the monoid of young tableaux examples include the hypoplactic monoid quasi ribbon tableaux the sylvester monoid binary search trees the stalactic monoid stalactic tableaux the taiga monoid binary search trees with multiplicities and the baxter monoid pairs of twin binary search trees it was already known that for many of these monoids connected components of the cyclic shift graph consist of elements that have the same evaluation that is contain the same number of each generating symbol this paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank n case for the hypoplactic monoid this is n 1 for the sylvester and taiga monoids at least n 1 and at most n for the stalactic monoid 3 except for ranks 1 and 2 when it is respectively 0 and 1 for the plactic monoid at least n 1 and at most 2 n 3 the current state of knowledge including new and previously known results is summarized in a table

a set operad of formal fractions and dendriform like sub operads abstract we introduce an operad of formal fractions abstracted from the mould operads and containing both the dendriform and the tridendriform operads we consider the smallest set operad contained in this operad and containing four specific elements of arity two corresponding to the generators and the associative elements of the dendriform and tridendriform operads we obtain a presentation of this operad by binary generators and quadratic relations and an explicit combinatorial description using a new kind of bi coloured trees similar results are also presented for related symmetric operads

the algebra of binary search trees we introduce a monoid structure on the set of binary search trees by a process very similar to the construction of the plactic monoid the robinson schensted insertion being replaced by the binary search tree insertion this leads to a new construction of the algebra of planar binary trees of loday ronco defining it in the same way as non commutative symmetric functions and free symmetric functions we briefly explain how the main known properties of the loday ronco algebra can be described and proved with this combinatorial point of view and then discuss it from a representation theoretical point of view which in turns leads to new combinatorial properties of binary trees

duality between quasi symmetrical functions and the solomon descent algebra abstract the ring qsym of quasi symmetric functions is naturally the dual of the solomon descent algebra the product and the two coproducts of the first extending those of the symmetric functions correspond to a coproduct and two products of the second which are defined by restriction from the symmetric group algebra a consequence is that qsym is a free commutative algebra

symmetries of first integrals and their associated differential equations the relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point contact for third order symmetries to and from nonlocal symmetries it is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries it is evident that nonlocal symmetries should be given more attention in applications

characterisation of the algebraic properties of first integrals of scalar ordinary differential equations of maximal symmetry abstract we undertake a study of the first integrals of linear n th order scalar ordinary differential equations with maximal symmetry we establish patterns for the first integrals associated with these equations it is shown that second and third order equations are the pathological cases in the study of higher order differential equations the equivalence of contact symmetries for third order equations to non cartan symmetries of second order equations is highlighted

tree expansion in time dependent perturbation theory the computational complexity of time dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters combinatorial sequences goldstone and other feynman type diagrams etc we show that a very efficient perturbative expansion both for theoretical and numerical purposes can be obtained through an original parametrization by trees and generalized iterated integrals we emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics these tools are applied to the adiabatic approximation and the effective hamiltonian we prove perturbatively and nonperturbatively the convergence of morita s generalization of the gell mann and low wave function we show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms finally we recover the rayleigh schrodinger time independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree

on the systematic approach to the classification of differential equations by group theoretical methods complete symmetry groups enable one to characterise fully a given differential equation by considering the reversal of an approach based upon complete symmetry groups we construct new classes of differential equations which have the equations of bateman monge ampere and born infeld as special cases we develop a symbolic algorithm to decrease the complexity of the calculations involved

cambrian hopf algebras abstract cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the classical conditions for binary search trees similar to binary trees for the tamari lattice cambrian trees provide convenient combinatorial models for n reading s cambrian lattices of type a based on a natural surjection from signed permutations to cambrian trees we define the cambrian hopf algebra extending j l loday and m ronco s algebra on binary trees we describe combinatorially the products and coproducts of both the cambrian algebra and its dual in terms of operations on cambrian trees we then construct the baxter cambrian algebra which extends s law and n reading s baxter hopf algebra on rectangulations and s giraudo s equivalent hopf algebra on twin binary trees