Generalised Stretched Littlewood-Richardson Coefficients

TL;DR

This paper explores how Littlewood-Richardson coefficients, which count certain tableaux, change when shapes and contents are incrementally added, generalizing the concept of stretched LR coefficients.

Contribution

It introduces a generalization of stretched LR coefficients by analyzing the incremental addition of shapes and contents to LR tableaux and their effects on the counts.

Findings

Number of LR tableaux weakly increases with added shapes and contents.

Behavior of LR tableaux counts under repeated shape additions is characterized.

Generalization extends the concept of stretched LR coefficients to broader cases.

Abstract

The Littlewood-Richardson (LR) coefficient counts among many other things the LR tableaux of a given shape and a given content. We prove, that the number of LR tableaux weakly increases if one adds to the shape and the content the shape and the content of another LR tableau. We also investigate the behaviour of the number of LR tableaux, if one repeatedly adds to the shape another shape with either fixed or arbitrary content. This is a generalisation of the stretched LR coefficients, where one repeatedly adds the same shape and content to itself.