A cornucopia of quasi-Yamanouchi tableaux

TL;DR

This paper explores various enumerative approaches to quasi-Yamanouchi tableaux, a special subset of semistandard Young tableaux, including q-hit numbers, lattice paths, and symmetric functions, revealing their diverse applications.

Contribution

The paper introduces multiple new enumerative methods for quasi-Yamanouchi tableaux, expanding understanding beyond the limited product formula enumeration.

Findings

Enumerations using q-hit numbers and lattice paths are developed.

Schur generating functions are expanded into fundamental and monomial quasisymmetric functions.

Connections to Jack polynomials, Foulkes characters, and coinvariant algebra are discussed.

Abstract

Quasi-Yamanouchi tableaux are a subset of semistandard Young tableaux and refine standard Young tableaux. They are closely tied to the descent set of standard Young tableaux and were introduced by Assaf and Searles to tighten Gessel's fundamental quasisymmetric expansion of Schur functions. The descent set and descent statistic of standard Young tableaux repeatedly prove themselves useful to consider, and as a result, quasi-Yamanouchi tableaux make appearances in many ways outside of their original purpose. Some examples, which we present in this paper, include the Schur expansion of Jack polynomials, the decomposition of Foulkes characters, and the bigraded Frobenius image of the coinvariant algebra. While it would be nice to have a product formula enumeration of quasi-Yamanouchi tableaux in the way that semistandard and standard Young tableaux do, it has previously been shown by the author that there is little hope on that front. The goal of this paper is to address a handful of the numerous alternative enumerative approaches. In particular, we present enumerations of quasi-Yamanouchi tableaux using $q$-hit numbers, semistandard Young tableaux, weighted lattice paths, and symmetric polynomials, as well as the fundamental quasisymmetric and monomial quasisymmetric expansions of their Schur generating function.