We prove a formula expressing the Kerov polynomial $Σ_k$ as a weighted sum over the lattice of noncrossing partitions of the set $\{1,...,k+1\}$. In particular, such a formula is related to a partial order $\mirr$ on the Lehner's irreducible noncrossing partitions which can be described in terms of left-to-right minima and maxima, descents and excedances of permutations. This provides a translation of the formula in terms of the Cayley graph of the symmetric group $\frak{S}_k$ and allows us to recover the coefficients of $Σ_k$ by means of the posets $P_k$ and $Q_k$ of pattern-avoiding permutations discovered by Bóna and Simion. We also obtain symmetric functions specializing in the coefficients of $Σ_k$.
Kerov polynomials describe normalized irreducible characters of the symmetric groups in terms of the free cumulants associated with Young diagrams. We suggest well-suited counterparts of the Kerov polynomials in spin (or projective) representation settings. We show that spin analogues of irreducible characters are polynomials in even free cumulants associated with double diagrams of strict partitions. Moreover, we present a conjecture for the positivity of their coefficients.
An iterative formula for the Kostka-Foulkes polynomials is given using the vertex operator realization of the Hall-Littlewood polynomials. The operational formula can handle large Kostka-Foulkes polynomials, and a stability property for the Kostka-Foulkes polynomials is shown. We also use our algorithm to give a formula of $K_{λμ}(t)$ for $μ$ being hook-shaped.
We prove Jacobi-Trudi-type determinantal formulas for skew dual Grothendieck polynomials which are $K$-theoretic deformations of Schur polynomials. We also prove a bialternant-type formula analogous to the classical definition of Schur polynomials.
Representation theory of finite groups portrays a marvelous crossroad of group theory, algebraic combinatorics, and probability. In particular the Plancherel measure is a probability that arises naturally from representation theory, and in this thesis we consider three ramifications of asymptotic questions for random Plancherel distributed representations. First we recall irreducible characters of the symmetric group, which are indexed by integer partitions. We focus on the so called 'dual approach', in which the partition indexing a character is now considered to be the variable. We extend a …
Some transformation formulas associated with Askey-Wilson polynomials and Lassalle's formulas for Macdonald-Koornwinder polynomials
我们提出了代表阿斯基-威尔逊多项式的四重级数展开。为了得到这个结果，我们依次使用了几个基本超几何级数的求和和变换公式，包括Verma的域的q-扩张和Wimp展开式，Andrews的Watson的3F 2和的终止q-模拟，Singh的二次变换.作为应用，我们给出了BCn（n in Z>0）型Koornwinder多项式的一个列图显式公式.在参数特殊化的情况下，我们用一行图恢复了Bn，Cn和Dn型Macdonald多项式的Lassalle公式，从而证明了他的结论.
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam, and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These "quasisymmetric Macdonald polynomials" refine the (symmetric) Macdonald polynomials and specialize to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg.
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These quasisymmetric Macdonald polynomials refine the (symmetric) Macdonald polynomials and specialize at $q=t=0$ to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg. This is an extended abstract.
We prove a combinatorial formula for the alternating descent polynomials of type A and B. Combining with Josuat-Vergès' combinatorial interpretation for Hoffman's derivative polynomials for tangent and secant functions, we obtain a unified combinatorial proof of Ma-Yeh's two formulae linking these two families of polynomials.
We give a new operator formula for Grothendieck polynomials that generalizes Magyar's Demazure operator formula for Schubert polynomials. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar. We apply our formula to prove a necessary divisibility condition for a monomial to appear in a given Grothendieck polynomial.
In this paper, we give a reduced formula of the characteristic polynomial of $k$-uniform hypergraphs with a pendant edge. And the explicit characteristic polynomial and all distinct eigenvalues of $k$-uniform hyperpath are given.
In 2007 Lam and Pylyavskyy found a combinatorial formula for the dual stable Grothendieck polynomials, which are the dual basis of the stable Grothendieck polynomials with respect to the Hall inner product. In 2016 Galashin, Grinberg, and Liu introduced refined dual stable Grothendieck polynomials by putting additional sequence of parameters in the combinatorial formula of Lam and Pylyavskyy. Grinberg conjectured a Jacobi--Trudi type formula for refined dual stable Grothendieck polynomials. In this paper this conjecture is proved by using bijections of Lam and Pylyavskyy.
The Al-Salam-Chihara polynomials are an important family of orthogonal polynomials in one variable $x$ depending on 3 parameters $α$, $β$ and $q$. They are closely connected to a model from statistical mechanics called the partially asymmetric simple exclusion process (PASEP) and they can be obtained as a specialization of the Askey-Wilson polynomials. We give two different combinatorial formulas for the coefficients of the (transformed) Al-Salam-Chihara polynomials. Our formulas make manifest the fact that the coefficients are polynomials in $α$, $β$ and $q$ with positive coefficients.
Given a quiver associated to a cluster algebra and a sequence of vertices, iterative mutation leads to $F$-Polynomials which appear in numerous places in the cluster algebraic literature. The coefficients of the monomials in these $F$-Polynomials are difficult to understand and have been an area of study for many years. In this paper, we present a general closed-form formula for these coefficients in terms of elementary manipulations with $C$-matrices. We then demonstrate the effectiveness of the formula by using it to derive simple explicit formulas for $F$-Polynomials of specific classes of …
We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple combinatorial formula we discovered for these polynomials using skew young tableaux. This supports the conjecture that Kazhdan-Lusztig polynomials for all matroids have non-negative coeffiecients. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.
The Cauchy polynomials with a $q$ parameter were recently defined, and several arithmetical properties were studied. In this paper, we establish explicit formulae for computing the Cauchy polynomials with a $q$ parameter in terms of $r$-Whitney numbers of the first kind. We also obtain several properties and combinatorial identities.
