Patterns of relation triples in inversion and ascent sequences

Abstract

Recently, Martinez and Savage carried out the systematic study of inversion sequences avoiding triples of relations. They reported many nice connections with familiar combinatorial families and posed several enumeration conjectures. All of their conjectures have already been solved except those related to the OEIS sequence A098746. In this paper, we address those remaining conjectures, which completes a picture for all the suspected connections arising in their investigation.As one of the most important subsets of inversion sequences, ascent sequences were introduced by Bousquet-Mélou et al. in bijection with (2+2)-free posets and their pattern avoidance properties have been extensively studied. We investigate some classical Eulerian and Stirling statistics on ascent sequences avoiding triples of relations. This leads us to find two new interpretations of the Catalan numbers and its refinements, and to interpret combinatorially a natural refinement of the binomial transformation of Catalan numbers. The latter discovery answers a challenging open problem posed by Pudwell.