Abstract
Recently, Yan and the first 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 main objective of this paper is to prove two of their conjectures concerning the enumeration of 12̲0-avoiding inversion or ascent sequences. Additionally, concerning the last entry of 12̲0-avoiding inversion sequences, we prove two equidistribution results and find a new succession rule for the powered Catalan numbers. Towards a proof of an enumerative conjecture on 23̲14-avoiding permutations, we offer two refined equidistribution conjectures in connection with 110- and 12̲0-avoiding inversion sequences.
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