Inversion sequences avoiding a pair of vincular patterns of type (2, 1)

An inversion sequence of length n is a sequence of integers e = e₀· · · e_nwhich satisfies 0 ≤ e_i≤ i, for all i = 0, 1, ..., n. We say that e avoids a pattern ab-c of type (2, 1) if does not exist i, j such that 0 ≤ i < j − 1 ≤ n − 1 and the subsequence π_i, π_{i +1}, π_jhas the same order isomorphic as abc. For a set of patterns B, let I_n(B) be the set of inversion sequences of length n that avoid all the patterns from B. We say that two sets of patterns B and C are I-Wilf equivalent if |I_n(B)| = |I_n(C)|, for all n ≥ 0. In this paper, we show that the number of I-Wilf equivalences among pairs of patterns of type (2, 1) is 72. In particular, we present connections to Bell numbers, ascent sequences, and permutations avoiding length-4 vincular pattern.