Refined restricted inversion sequences

Abstract

Recently, the study of patterns in inversion sequences was initiated by Corteel-Martinez-Savage-Weselcouch and Mansour-Shattuck independently. Motivated by their works and a double Eulerian equidistribution due to Foata (1977), we investigate several classical statistics on restricted inversion sequences that are either known or conjectured to be enumerated by {\em Catalan}, {\em Large Schröder}, {\em Baxter} and {\em Euler} numbers. One of the two highlights of our results is a fascinating bijection between $000$-avoiding inversion sequences and Simsun permutations, which together with Foata's V- and S-codes, provide a proof of a restriced double Eulerian equdistribution. The other one is a refinement of a conjecture due to Martinez and Savage that the cardinality of $\I_n(\geq,\geq,>)$ is the $n$-th Baxter number, which is proved via the so-called {\em obstinate kernel method} developed by Bousquet-Mélou.

25 pages, 6 figures. This is the full version of the extended abstract that appears in FPSAC'17 London

References in corpus (3)

• Patterns in Inversion Sequences I
• Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations
• Tableau sequences, open diagrams, and Baxter families

Cited by in corpus (4)

• Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers
• Consecutive Patterns in Inversion Sequences
• Consecutive patterns in inversion sequences II: avoiding patterns of relations
• $k$-arrangements, statistics and patterns