Journal of Integer Sequences, Vol. 22 (2019), Article 19.7.5 |

Department of Mathematics

Dartmouth College

Hanover, NH 03755

USA

**Abstract:**

Inversion sequences are integer sequences
*e* = *e*_{1} *e*_{2}
··· *e*_{n}
such that 0 ≤ *e*_{i} < *i*
for each *i*. The study of patterns in inversion sequences was
initiated by Corteel-Martinez-Savage-Weselcouch and Mansour-Shattuck in
the classical (non-consecutive) case, and later by Auli-Elizalde in the
consecutive case, where the entries of a pattern are required to occur
in adjacent positions. In this paper we continue this investigation
by considering consecutive patterns of relations, in analogy to the
work of Martinez-Savage in the classical case. Specifically, given
two binary relations *R*_{1},
*R*_{2} ∈ {≤, ≥, <, >, =, ≠}, we study
inversion sequences *e* with no subindex *i* such that
*e*_{i} *R*_{1} *e*_{i+1}
*R*_{2} *e*_{i+2}. By
enumerating such inversion sequences according to their length, we
obtain well-known quantities such as Catalan numbers, Fibonacci numbers
and central polynomial numbers, relating inversion sequences to other
combinatorial structures. We also classify consecutive patterns of
relations into Wilf equivalence classes, according to the number of
inversion sequences avoiding them, and into more restrictive classes
that consider the positions of the occurrences of the patterns.

As a byproduct of our techniques, we obtain a simple bijective proof of a result of Baxter-Shattuck and Kasraoui about Wilf-equivalence of vincular patterns, and we prove a conjecture of Martinez and Savage, as well as related enumeration formulas for inversion sequences satisfying certain unimodality conditions.

(Concerned with sequences A000027 A000045 A000071 A000079 A000085 A000108 A000124 A000325 A000522 A003422 A033321 A040000 A049774 A052169 A071356 A200403 A279561 A328357 A328358 A328429 A328430 A328431 A328432 A328433 A328434 A328435 A328436 A328437 A328438 A328439 A328440 A328441 A328442.)

Received June 17 2019; revised version received October 16 2019.
Published in *Journal of Integer Sequences*,
October 17 2019.

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