Forest webs and pattern avoidance

Abstract

In a recent preprint, Mike Cummings showed that the smooth components of suitably parametrized Springer fibers are in bijection with contracted, fully reduced Plücker 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}.

Figures (5)

• Figure 1: Examples of the two types of forest webs and their corresponding tableaux.
• Figure 2: A two-column tableau, its corresponding Dyck path, and the resulting permutation that is constructed by $\pi$.
• Figure 3: A tableau and path corresponding to a web having four short arcs of the form $\{i,i+1\}$.
• Figure 4: A tableau and path corresponding to a web having four short arcs, one of which is $\{2r,1\}$.
• Figure 5: A path with three northwest corners whose corresponding permutation has a $3214$-pattern.

Theorems & Definitions (12)

• Lemma 2.1: GPPSS25
• Lemma 2.2: MC
• Example 3.1
• Example 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Theorem 4.4
• proof