Weight-preserving bijections between integer partitions and a class of alternating sign trapezoids

TL;DR

This paper introduces weight-preserving bijections linking column strict shifted plane partitions with a single row to a specific class of alternating sign trapezoids, including configurations with -1s, enriching the combinatorial understanding.

Contribution

It constructs novel bijections that incorporate -1 configurations, connecting two combinatorial objects and generalizing concepts in plane partitions and alternating sign arrays.

Findings

Bijections preserve weights between the two structures.

Includes configurations with -1s, expanding previous bijections.

Relates the number of -1s to elements in plane partitions.

Abstract

We construct weight-preserving bijections between column strict shifted plane partitions with one row and alternating sign trapezoids with exactly one column in the left half that sums to $1$. Amongst other things, they relate the number of $-1$s in the alternating sign trapezoids to certain elements in the column strict shifted plane partitions that generalise the notion of special parts in descending plane partitions. The advantage of these bijections is that they include configurations with $-1$s, which is a feature that many of the bijections in the realm of alternating sign arrays lack.