Forest webs and pattern avoidance
Jessica Striker, Bridget Eileen Tenner

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 -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}.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Finite Group Theory Research · Geometric and Algebraic Topology
