Monotone triangles and 312 Pattern Avoidance

TL;DR

This paper establishes a bijection between certain classes of alternating sign matrices and plane partitions, generalizes 312 pattern avoidance to Gog words, and estimates their enumeration via p-branchings.

Contribution

It introduces the gapless condition on monotone triangles, connects it to pattern avoidance in permutations, and extends pattern avoidance concepts to Gog words.

Findings

Bijection between gapless ASMs and symmetric plane partitions.

Reduction of gapless ASMs to 312-avoiding permutations.

Enumeration of gapless monotone triangles using p-branchings.

Abstract

We demonstrate a natural bijection between a subclass of alternating sign matrices (ASMs) defined by a condition on the corresponding monotone triangle which we call the gapless condition and a subclass of totally symmetric self-complementary plane partitions defined by a similar condition on the corresponding fundamental domains or Magog triangles. We prove that, when restricted to permutations, this class of ASMs reduces to 312-avoiding permutations. This leads us to generalize pattern avoidance on permutations to a family of words associated to ASMs, which we call Gog words. We translate the gapless condition on monotone trangles into a pattern avoidance-like condition on Gog words associated. We estimate the number of gapless monotone triangles using a bijection with p-branchings.