Forest webs and pattern avoidance

TL;DR

This paper establishes a bijection between certain forest-like $ ext{sl}_r$-webs and permutations avoiding specific patterns, advancing the combinatorial understanding of Springer fiber components.

Contribution

It provides a new bijection linking $ ext{sl}_r$-webs to pattern-avoiding permutations, refining previous enumerative results and addressing Cummings' posed problem.

Findings

Bijection between $ ext{sl}_r$-webs and permutations avoiding {132,4321,3214}

Confirmation that these webs are enumerated by sequence A116731

Enhanced combinatorial understanding of Springer fiber components

Abstract

In a recent preprint, Mike Cummings showed that the smooth components of suitably parametrized Springer fibers are in bijection with contracted, fully reduced Pl\"ucker degree-two $\mathfrak{sl}_r$-webs of standard type and that are forests. He showed these are enumerated by sequence A116731 in the OEIS, which is equinumerous with permutations avoiding the patterns {321,2143,3124}. Cummings posed the problem of strengthening this enumerative result by finding a bijection between these webs and a collection of pattern-avoiding permutations. Here we solve this problem, although notably not with the collection of patterns that Cummings had proposed. Rather, we give a bijection between this class of webs and permutations avoiding the patterns {132,4321,3214}.