Two examples of unbalanced Wilf-equivalence

Abstract

We prove that the set of patterns {1324, 3416725} is Wilf-equivalent to the pattern 1234 and that the set of patterns {2143, 3142, 246135} is Wilf-equivalent to the set of patterns {2413, 3142}. These are the first known unbalanced Wilf-equivalences for classical patterns between finite sets of patterns.

Tilings

“…which consists of those permutations that can be sorted by two increasing stacks in series (for a visualization of these basis elements, see Figure 11; a proof of this result using frps is given in Bloom and Vatter [42]). Burstein and Pantone [70] later showed that Av(1234) and Av(1324, 3416725) are Wilf-equivalent, along with other unbalanced Wilf-equivalences.…”

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

“…is also the generating function of the sequence that enumerates the permutation classes Av(4321, 4213) [4,18] and Av(2413, 2431, 23145) [39]. In particular, Theorem 3.1 tells us that the permutation classes Av(4321, 4213) and Av(2341, 3241, 45231) form a new example of an unbalanced Wilf equivalence, as defined in [3,17].…”

“…As in the previous two sections, we use the kernel method to find the generating function J(x, 0, z). There is a unique power series Y (17) shows that J(x, 0, z) = C(Y ). Let J(x, z) = xJ(x, 0, z) = xC(Y ).…”

Pattern avoidance of generalized permutations

“…There are some interpretations on Motzkin and Riordan numbers as SYTs and pattern avoidances of permutations. For Motzkin numbers, Zabrocki interpreted M n as the number of SYTs of height ≤ 3, Baril [3] obtained that M n is the number of (n + 1)-length permutations avoiding the pattern 132 and the dotted pattern 231, Burstein and Pantone [5] showed that M n is the number of n-length involutions avoiding patterns 4231 and 5276143. For Riordan numbers, Callan obtained that R n is the number of 321-avoiding permutations on [n] in which each left-to-right maximum is a descent, Chen-Deng-Yang [6] proved that R n is the number of derangements on [n] that avoid both 321 and 3142, Regev [14] found that R n is the number of SYTs of shape (k, k, 1 n−2k ) for all k ≥ 0 which gave an expression of R n through the hook length formula.…”