Tableaux and plane partitions of truncated shapes

TL;DR

This paper introduces a new class of plane partitions and Young tableaux with truncated shapes, providing formulas for counting standard tableaux in specific cases using generating functions and Schur functions.

Contribution

It develops formulas for counting standard tableaux of truncated shapes, a novel class not previously studied, using generating functions and Schur function techniques.

Findings

Formulas for counting standard tableaux of truncated shapes.

Generating functions derived from restricted Schur functions.

Explicit enumeration results for specific truncated shapes.

Abstract

We consider a new kind of straight and shifted plane partitions/Young tableaux --- ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.