A Fourfold Refined Enumeration of Alternating Sign Trapezoids

Abstract

Alternating sign trapezoids have recently been introduced as a generalisation of alternating sign triangles. Fischer established a threefold refined enumeration of alternating sign trapezoids and provided three statistics on column strict shifted plane partitions with the same joint distribution. In this paper, we are able to add a new pair of statistics to these results. More precisely, we consider the number of $-1$s on alternating sign trapezoids and introduce a corresponding statistic on column strict shifted plane partitions that has the same distribution. More generally, we show that the joint distributions of the two quadruples of statistics on alternating sign trapezoids and column strict shifted plane partitions, respectively, coincide. In addition, we provide a closed-form expression for the $2$-enumeration of alternating sign trapezoids.

arXiv comment20 pages, 7 figures. Final revised version to appear in the Electronic Journal of Combinatorics

References in corpus (4)

• On the weighted enumeration of alternating sign matrices and descending plane partitions
• Diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Refined Enumeration of Halved Monotone Triangles and Applications to Vertically Symmetric Alternating Sign Trapezoids

Cited by in corpus (1)

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