Wilf equivalences between vincular patterns in inversion sequences

Abstract

Inversion sequences are finite sequences of non-negative integers, where the value of each entry is bounded from above by its position. They provide a useful encoding of permutations. Patterns in inversion sequences have been studied by Corteel–Martinez–Savage–Weselcouch and Mansour–Shattuck in the classical case, where patterns can occur in any positions, and by Auli–Elizalde in the consecutive case, where only adjacent entries can form an occurrence of a pattern. These papers classify classical and consecutive patterns of length 3 into Wilf equivalence classes according to the number of inversion sequences avoiding them.In this paper we consider vincular patterns in inversion sequences, which, in analogy to Babson–Steingrímsson patterns in permutations, require only certain entries of an occurrence to be adjacent, and thus generalize both classical and consecutive patterns. Solving three conjectures of Lin and Yan, we provide a complete classification of vincular patterns of length 3 in inversion sequences into Wilf equivalence classes, and into more restrictive classes that consider the number of occurrences of the pattern and the positions of such occurrences. We find the first known instance of patterns in inversion sequences where these two more restrictive classes do not coincide.