A new recursion for three-column combinatorial Macdonald polynomials

The Hilbert series of the Garsia–Haiman module Mμ can be described combinatorially as the generating function of certain fillings of the Ferrers diagram of μ where μ is an integer partition of n . Since there are n ! fillings that generate , it is desirable to find recursions to reduce the number of fillings that need to be considered when computing combinatorially. In this paper, we present a combinatorial recursion for the case where μ is an n by 3 rectangle. This allows us to reduce the number of fillings under consideration from (3n)! to (3n)!/(3!nn!)

A Bijective Proof of a Factorization Formula for Specialized Macdonald Polynomials

Let μ and ν = (ν 1, . . . , ν k ) be partitions such that μ is obtained from ν by adding m parts of sizer. Descouens and Morita proved algebraically that the modified Macdonald polynomials H~μ(X;q,t) satisfy the identity H~μ=H~νH~(rm) when the parameter t is specialize to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ≤ ν k and r∈{1,2}. This note gives a bijective proof of the formula for all r ≤ ν k

New combinatorial formulations of the shuffle conjecture

The shuffle conjecture (due to Haglund, Haiman, Loehr, Remmel, and Ulyanov) provides a combinatorial formula for the Frobenius series of the diagonal harmonics module DHn, which is the symmetric function∇(en). This formula is a sum over all labeled Dyck paths of terms built from combinatorial statistics called area, dinv, and IDes. We provide three new combinatorial formulations of the shuffle conjecture based on other statistics on labeled paths, parking functions, and related objects. Each such reformulation arises by introducing an appropriate new definition of the inverse descent set. Analogous results are proved for the higher-order shuffle conjecture involving ∇m(en). We also give new versions of some recently proposed combinatorial formulas for ∇(Cα) and ∇(s(k,1(n−k))), which translate expansions based on the dinv statistic into equivalent expansions based on Haglund\u27s bounce statistic

Skew row-strict quasisymmetric Schur functions

Mason and Remmel introduced a basis for quasisymmetric functions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur functions. We introduce a modification known as Young row-strict quasisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric functions and then prove this is equivalent to a combinatorial description. We also provide a decomposition of the skew Young row-strict quasisymmetric Schur functions into a sum of Gessel’s fundamental quasisymmetric functions and prove a multiplication rule for the product of a Young row-strict quasisymmetric Schur function and a Schur function

Quasisymmetric (k; l)-hook Schur functions

We introduce a quasisymmetric generalization of Berele and Regev\u27s (k,l)-hook Schur functions. These quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. The quasisymmetric hook Schur functions can be defined as the generating function for a certain set of composition tableaux on two alphabets. We will look at the combinatorics of the quasisymmetric hook Schur functions, including an analogue of the RSK algorithm and a generalized Cauchy Identity

Standard fillings to parking functions

The Hilbert series of the Garsia-Haiman module can be written as a generating function of standard fillings of Ferrers diagrams. It is conjectured by Haglund and Loehr that the Hilbert series of the diagonal harmonics can be written as a generating function of parking functions. In this paper we present a weight-preserving injection from standard fillings to parking functions for certain cases

Quasisymmetric (k,l)-hook Schur functions

We introduce a quasisymmetric generalization of Berele and Regev's hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. In this paper we examine the combinatorics of the quasisymmetric hook Schur functions, providing analogues of the Robinson-Schensted-Knuth algorithm and a generalized Cauchy Identity