Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations

Inversion sequences of length n, In, are integer sequences (e1, . . . , en) with 0 ≤ ei < i for each i. The study of patterns in inversion sequences was initiated recently by Mansour-Shattuck and Corteel-Martinez-Savage-Weselcouch through a systematic study of inversion sequences avoiding words of length 3. We continue this investigation by reframing the notion of a length-3 pattern from a “word of length 3,” w1w2w3, to a “triple of binary relations,” (ρ1, ρ2, ρ3), and consider the set In(ρ1, ρ2, ρ3) consisting of those e ∈ In with no i < j < k such that eiρ1ej , ejρ2ek, and eiρ3ek. We show that “avoiding a triple of relations” can characterize inversion sequences with a variety of monotonicity or unimodality conditions, or with multiplicity constraints on the elements. We uncover several interesting enumeration results and relate pattern avoiding inversion sequences to familiar combinatorial families. We highlight open questions about the relationship between pattern avoiding inversion sequences and a variety of