Séminaire Lotharingien de Combinatoire, 84B.27 (2020), 12 pp.

Juan S. Auli and Sergi Elizalde

Inversion Sequences Avoiding Consecutive Patterns

Abstract. Inversion sequences are integer sequences e₁e₂ ... e_n such that 0 <= e_i < i for each i. The study of classical patterns in inversion sequences was initiated by Corteel-Martinez-Savage-Weselcouch and Mansour-Shattuck. Here we focus on consecutive patterns in inversion sequences, namely patterns whose entries are required to occur in adjacent positions. We enumerate inversion sequences that avoid small consecutive patterns. We also study the notion of Wilf equivalence in this setting, as well as generalizations that consider the positions of the occurrences, and classify patterns of length up to 4 into equivalence classes.

Finally, in analogy to the work of Martinez-Savage in the classical case, we consider consecutive patterns of relations among 3 adjacent entries. Our setting allows us to give a simple bijective proof of a result of Baxter-Shattuck and Kasraoui about vincular permutation patterns, and to prove a conjecture of Martinez-Savage about certain unimodal inversion sequences.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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