On $\underline{12}0$-avoiding inversion and ascent sequences

TL;DR

This paper addresses two conjectures about counting inversion and ascent sequences that avoid the vincular pattern 120, expanding understanding of their combinatorial properties and connections.

Contribution

It provides proofs for two conjectures related to the enumeration of 120-avoiding inversion and ascent sequences.

Findings

Confirmed enumeration formulas for 120-avoiding sequences

Established new combinatorial connections involving these sequences

Extended previous conjectures with rigorous proofs

Abstract

Recently, Yan and the first named author investigated systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length $3$, where two of the three letters are required to be adjacent. They established many connections with familiar combinatorial families and proposed several interesting conjectures. The objective of this paper is to address two of their conjectures concerning the enumeration of $\underline{12}0$-avoiding inversion or ascent sequences.