Vincular patterns in inversion sequences

Abstract

Inversion sequences have intriguing applications in combinatorics, computer sciences and polyhedral geometry. Ascent sequences, as one of the most important subsets of inversion sequences, were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev to encode the (2+2)-free posets. Pattern avoidance in ascent sequences was first studied by Ducan and Steingrímsson in 2011, while the systematic study of patterns in inversion sequences was initiated only recently by Corteel–Martinez–Savage–Weselcouch and Mansour–Shattuck. In this paper, we investigate systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length 3, where two of the three letters are required to be adjacent. Our results connect restricted inversion sequences and ascent sequences to a number of well-known combinatorial sequences including Bell numbers, Fishburn numbers, Powered Catalan numbers, Semi-Baxter numbers and the number of 3-noncrossing partitions. A variety of combinatorial bijections, known or newly constructed, are applied to achieve these interesting connections.