Related papers: Degrees of stretched Kostka coefficients

We identify a close relationship between stable sheaf cohomology for polynomial functors applied to the cotangent bundle on projective space, and Koszul--Ringel duality on the category of strict polynomial functors as described in the work…

For a given graph $H$ and the graphical properties $P_1, P_2,\ldots,P_k$, a graph $H$ is said to be $(V_1, V_2,\ldots,V_k)$-partitionable if there exists a partition of $V(H)$ into $k$-sets $V_1, V_2\ldots,V_k$, such that for each…

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as…

For $G$ a semisimple, simply-connected complex algebraic group and two dominant integral weights $\lambda, \mu$, we consider the dimensions of weight spaces $V_\lambda(\mu)$ of weight $\mu$ in the irreducible, finite-dimensional highest…

We report about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and ``saturation'' properties of several generalizations of the…

Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations…

In this paper one extends the binomial and trinomial coefficients to the concept of 'k-nomial' coefficients, and one obtains some properties of these. As an application one generalizes Pascal's triangle.

We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka--Foulkes polynomials) share symmetries related to the operations…

We study the space, $R_m$, of $m$-symmetric functions consisting of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\dots$ but have no special symmetry in the variables $x_1,\dots,x_m$. We obtain $m$-symmetric…

Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works…

In this article we study asymptotic behavior of the probability that a random monic polynomial with integer coefficients is irreducible over the integers. We consider the cases where the coefficients grow together with the degree of the…

Let $K(q,t)= \|K_{\la\mu}(q,t)\|_{\la,\mu}$ be the Macdonald q,t-Kostka matrix and $K(t)=K(0,t)$ be the matrix of the Kostka-Foulkes polynomials K_{\la\mu}(t). In this paper we present a new proof of the polynomiality of the q,t-Kostka…

We consider the poset of weighted partitions $\Pi_n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $\Pi_n^w$ provide a generalization of the lattice $\Pi_n$ of…

A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The…

We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rz\k{a}\.zewski, Thomass\'e, and Walczak, that for every graph $H$, there is a polynomial $p$ such that for every positive integer $s$, every graph of average degree at least $p(s)$…

A set partition of $[n] := \{1, 2, \dots, n \}$ is called {\em $r$-Stirling} if the numbers $1, 2, \dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k…

We show that for any polynomial $f$ from the integers to the integers, with positive leading coefficient and irreducible over the rationals, if $x$ is large enough then there is a string of $(\log x)(\log\log x)^{1/835}$ consecutive…

We compute the algebraic K-theory of the non-commutative ring k<x_1,...,x_n>/(m^a) when k is a perfect field of positive characteristic and m=(x_1,...,x_n). We express the answer in terms of the truncation poset Witt vectors developed in…

We study graphs of polynomial growth from the perspective of asymptotic geometry and descriptive set theory. The starting point of our investigation is a theorem of Krauthgamer and Lee who showed that every connected graph of polynomial…

Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics.…