On Ascent, Repetition and Descent Sequences

TL;DR

This paper explores ascent, repetition, and descent sequences, establishing bijections with well-known combinatorial objects, deriving generating functions, and revealing new enumerative relationships involving Catalan and Bell numbers.

Contribution

It introduces new bijections and generating functions for 021-avoiding ascent sequences and links descent sequences to Dyck paths, expanding understanding of these combinatorial structures.

Findings

Repetition sequences are counted by Bell numbers.

021-avoiding repetition sequences are counted by Catalan numbers.

Derived a 4-variable generating function for 021-avoiding ascent sequences.

Abstract

Ascent sequences have received a lot of attention in recent years in connection with (2 + 2)-free posets and other combinatorial objects. Here, we first show bijectively that analogous repetition sequences are counted by the Bell numbers, and 021-avoiding repetition sequences by the Catalan numbers. Then we adapt a bijection of Chen et al and use it along with the "symbolic" method of Flajolet to find the 4-variable generating function for 021-avoiding ascent sequences by length, number of 0's, number of isolated 0's, and number of runs of 2 or more 0's. We deduce that 021-avoiding ascent sequences that have no consecutive 0's (resp. no isolated 0's) both satisfy a Catalan-like recurrence, differing only in initial conditions, and give a bijective proof for the case of no consecutive 0's. Lastly, we show that 021-avoiding descent sequences are equinumerous with same-size $UUDU$-avoiding Dyck paths.