Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers

TL;DR

This paper develops generating functions for counting inverted semistandard Young tableaux with various shapes and contents, relating them to higher-dimensional Dyck paths and generalized Ballot numbers, and provides explicit enumerations for specific shapes.

Contribution

It introduces new generating functions for inverted semistandard Young tableaux and connects their enumeration to higher-dimensional combinatorial objects and generalized Ballot numbers.

Findings

Generated explicit formulas for counting k-inverted semistandard Young tableaux.

Connected tableau enumeration to m-dimensional Dyck paths and generalized Ballot numbers.

Provided exact counts for specific two-row and two-column shapes.

Abstract

An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableau of various shapes $\lambda$ and contents $\mu$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableau that "standardize" to a fixed semistandard Young tableau. For $m$-row shapes $\lambda$ and standard content $\mu$, the total number of $k$-inverted standard Young tableau of shape $\lambda$ are then enumerated by relating such tableaux to $m$-dimensional generalizations of Dyck paths and counting the numbers of "returns to ground" in those paths. In the rectangular specialization of $\lambda = n^m$ this yields a generating function that involves $m$-dimensional analogues of the famed Ballot numbers. Our various results are then used to directly enumerate all $k$-inverted semistandard Young tableaux with arbitrary content and two-row shape $\lambda = a^1 b^1$, as well as all $k$-inverted standard Young tableaux with two-column shape $\lambda=2^n$.