Kazhdan–Lusztig polynomials of boolean elements

We give closed combinatorial product formulas for Kazhdan–Lusztig polynomials and their parabolic analogue of type q in the case of boolean elements, introduced in (Marietti in J. Algebra 295:1–26, 2006), in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements.     

Interprétation combinatoire des moments négatifs des valeurs de fonctions L au bord de la bande critique

We give a combinatorial interpretation of the negative moments of the values at the edge of the critical strip of the L functions of modular forms of GL(2) and GL(3). We deduce some results about the size of these numbers.     

A compehensive study of $${\vavec{}}$$-Dowling polynomials

After his extensive study of Whitney numbers, Benoumhani introduced Dowling numbers and polynomials as generalizations of the well-known Bell numbers and polynomials. Later, Cheon and Jung gave the r-generalization of these notions. Based on our recent combinatorial interpretation of r-Whitney numbers, in this paper we derive several new properties of r-Dowling polynomials and we present alternative proofs of some previously known ones.     

Reversion of power series and the extended Raney coefficients

In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.

     

An integrable semi-discrete equation and combinatorial numbers with their combinatorial interpretations

A Hankel type determinant solution for an integrable semi-discrete equation is presented. As an application, the relations between the solution and combinatorial numbers are discussed, which lead to new combinatorial numbers. The so-called Motzkin-like numbers are obtained, and the corresponding combinatorial interpretation is given. Additionally, it is also shown that some lattice paths have connections with the special solution.     

Stirling polynomials

An investigation is made of the polynomials f_k(n) = S(n + k, n) and g_k(n) = (?1)^ks(n, n ? k), where S and s denote the Stirling numbers of the second and first kind, respectively. The main result gives a combinatorial interpretation of the coefficients of the polynomial (1 ? x)^2k+1Σ_n=0^∞f_k(n)xⁿ analogous to the well-known combinatorial interpretation of the Eulerian numbers in terms of descents of permutations.     

Un -tableau d’Euler

In [R. Clarke, G.N. Han, J. Zeng, A combinatorial interpretation of the Seidel generation of q-derangement numbers, Ann. Comb. 1 (1997) 313–327] Clarke, Han and Zeng introduced a generalized Euler’s difference table. In this paper, we add a third variable and give a combinatorial interpretation of this generalization.     

From Bruhat intervals to intersection lattices and a conjecture of Postnikov

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w∈Sn is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups.A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.     

Identities in combinatorics III: Further aspects of ordered set sorting

Given two multi-sets of non-negative integers, we define a measure of their common values called the crossing number and then use this concept to provide a combinatorial interpretation of the q-Hahn polynomials and combinatorial proofs of the q-analogs of the Pfaff-Saalschutz summation and the Sheppard transformation.     

Parabolic Kazhdan–Lusztig polynomials for quasi-minuscule quotients

We study the parabolic Kazhdan–Lusztig polynomials for the quasi-minuscule quotients of Weyl groups. We give explicit closed combinatorial formulas for the parabolic Kazhdan–Lusztig polynomials of type q. Our study implies that these are always either zero or a monic power of q, and that they are not combinatorial invariants. We conjecture a combinatorial interpretation for the parabolic Kazhdan–Lusztig polynomials of type −1.     

On cographic regular matroids

The presentation of alternating permutatioas via labelled binary trees is used to define polynomials H_2n?1(x) as enumerating polynomials for the height of peaks in alternating permutations of length 2n?1. A divisibility property of the coefficients of these polynomials is proved, which generalizes and explains combinatirially a well-known property of the tangent numbers. Furthermore, a version of the exponential generating function for the H_2n?1(x) is given, leading to a new combinatorial interpretation of Dumont's modified Ghandi-polynomials.     

A recursion formula for k-Schur functions

The Bernstein operators allow one to build recursively the Schur functions. We present a recursion formula for k-Schur functions at t=1 based on combinatorial operators that generalize the Bernstein operators. The recursion leads immediately to a combinatorial interpretation for the expansion coefficients of k-Schur functions at t=1 in terms of homogeneous symmetric functions.     

Lah numbers and Lindström's lemma

We provide a combinatorial interpretation of Lah numbers by means of planar networks. Henceforth, as a consequence of Lindström's lemma, we conclude that the related Lah matrix possesses a remarkable property of total non-negativity.     

A new q-analogue of the sum of cubes

A new q-analogue of the sum of cubes is given with a combinatorial interpretation on the lattice of subspaces.     

Genealized poly-Cauchy and poly-Benoulli numbes by using incomplete $${\vavec{}}$$-Stiling numbes

In this paper we introduce restricted r-Stirling numbers of the first kind. Together with restricted r-Stirling numbers of the second kind and the associated r-Stirling numbers of both kinds, by giving more arithmetical and combinatorial properties, we introduce a new generalization of incomplete poly-Cauchy numbers of both kinds and incomplete poly-Bernoulli numbers.     

Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi

We give the first combinatorial interpretation of the coefficients of the power series of the elliptic Jacobi functions sn, cn and dn. This is done by introducing a new class of permutations enumerated by the Euler numbers and a new index about permutations having the same distribution as the Eulerian numbers.     

The r-Stirling numbers

The r-Stirling numbers of the first and second kind count restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively distinct subsets. The combinatorial and algebraic properties of these numbers, which in most cases generalize similar properties of the regular Stirling numbers, are explored starting from the above definition.     

Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R₂,R₃,… of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.     

The Legendre–Stirling numbers

The Legendre–Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre–Stirling numbers. In this paper, we establish several properties of the Legendre–Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.     

Enumeration of labelled threshold graphs and a theorem of frobenius involving eulerian polynomials

We enumerate labelled threshold graphs by the number of vertices, the number of isolated vertices, and the number of distinct vertex-degrees and we give the exact asymptotics for the number of labelled threshold graphs withn vertices. We obtain the appropriate generating function and point out a combinatorial interpretation relating its coefficients to the Stirling numbers of the second kind. We use these results to derive a new proof of a theorem of Frobenius expressing the Eulerian polynomials in terms of the Stirling numbers.