Outside nested decompositions of skew diagrams and Schur function determinants

TL;DR

This paper introduces a new determinantal formula for skew Schur functions using outside nested decompositions, generalizing previous results and enabling enumeration of m-strip tableaux.

Contribution

It presents a novel determinantal formula for skew Schur functions based on outside nested decompositions, extending Hamel and Goulden's theorem.

Findings

Derived a determinantal formula involving thickened strips

Extended the transfer operator approach for counting m-strip tableaux

Generalized previous theorems on skew shape decompositions

Abstract

In this paper we describe the thickened strips and the outside nested decompositions of any skew shape $\lambda/\mu$. For any such decomposition $\Phi=(\Theta_1,\Theta_2,\ldots,\Theta_g)$ of the skew shape $\lambda/\mu$ where $\Theta_i$ is a thickened strip for every $i$, if $r$ is the number of boxes that are contained in any two distinct thickened strips of $\Phi$, we establish a determinantal formula of the function $s_{\lambda/\mu}(X)p_{1^r}(X)$ with the Schur functions of thickened strips as entries, where $s_{\lambda/\mu}(X)$ is the Schur function of the skew shape $\lambda/\mu$ and $p_{1^r}(X)$ is the power sum symmetric function index by the partition $(1^r)$. This generalizes Hamel and Goulden's theorem on the outside decompositions of the skew shape $\lambda/\mu$. As an application of our theorem, we derive the number of $m$-strip tableaux which was first counted by Baryshnikov and Romik via extending the transfer operator approach due to Elkies.