Set partitions, tableaux, and subspace profiles under regular diagonal matrices

TL;DR

This paper introduces a family of polynomials linked to set partitions, tableaux, and subspace profiles under regular diagonal matrices, unifying various combinatorial objects and providing new enumeration formulas.

Contribution

It defines new polynomials that generalize several classical combinatorial counts and connects them to subspace enumeration in finite vector spaces.

Findings

Polynomials count subspaces under diagonal matrices at prime powers.

At specific values, they count set partitions, standard tableaux, and shifted tableaux.

They relate to q-Stirling numbers, Catalan triangles, and chord diagram enumeration.

Abstract

We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This enumeration formula is a combinatorial solution to a problem introduced by Bender, Coley, Robbins and Rumsey. At $1$, they count set partitions with specified block sizes. At $0$, they count standard tableaux of specified shape. At $-1$, they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express $q$-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated to the $q$-Hermite orthogonal polynomial sequence. In particular, when all parts are equal to two, they coincide with the polynomials defined by Touchard that enumerate chord diagrams by the number of crossings.