On refined enumerations of totally symmetric self-complementary plane partitions II

TL;DR

This paper proves a weak version of a conjecture relating symmetric plane partitions and alternating sign matrices, introduces new classes of domino plane partitions, and provides a determinantal formula for the general case.

Contribution

It establishes a connection between invariant shifted plane partitions and symmetric alternating sign matrices, and introduces two new classes of domino plane partitions.

Findings

Number of invariant shifted plane partitions equals invariant alternating sign matrices.

Introduces two new classes of domino plane partitions with specific symmetries.

Provides a determinantal formula for the general conjecture, though it remains difficult to evaluate.

Abstract

In this paper we settle a weak version of a conjecture (i.e. Conjecture 6) by Mills, Robbins and Rumsey in the paper "Self-complementary totally symmetric plane partitions" (J. Combin. Theory Ser. A 42, 277-292). In other words we show that the number of shifted plane partitions invariant under an involution is equal to the number of alternating sign matrices invariant under the vertical flip. We also give a determinantal formula for the general conjecture (Conjecture 6), but this determinant is still hard to evaluate. In this paper we introduce two new classes of domino plane partitions, one has the same cardinality as the set of half-turn symmetric alternating sign matrices and the other has the same cardinality as the set of vertically symmetric alternating sign matrices.