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Guo-Niu Han [Sémin. Lothar. Comb. 85 (2021) B85c (electronic)] has introduced a new combinatorial object named standard puzzle. We use digraphs to show the relations between numbers in standard puzzles and propose a skeleton model. By this model, we solve the enumeration problem of over fifty thousand standard puzzle sequences. Most of them can be represented by classical numbers, such as Catalan numbers, double factorials, secant numbers and so on. Also, we prove several identities for standard puzzle sequences. [ABSTRACT FROM AUTHOR]

We launch a systematic study of the refined Wilf-equivalences by the statistics and , where and are the number of components and the length of the initial ascending run of a permutation , respectively. As Comtet was the first one to consider the statistic in his book Analyse combinatoire , any statistic equidistributed with over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on -avoiding permutations, and a recent result of the first and third authors that is a Comtet statistic over separable permutations. Some highlights of our results are: ● Bijective proofs of the symmetry of the joint distribution over several Catalan and Schröder classes, preserving the values of the left-to-right maxima. ● A complete classification of - and -Wilf-equivalences for length patterns and pairs of length patterns. Calculations of the generating functions over these pattern avoiding classes and separable permutations. ● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and -avoiding permutations by the Comtet statistic . [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is two-fold. We first prove several new interpretations of a kind of (q , t) -Catalan numbers along with their corresponding γ -expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain (− 1) -phenomenon for each subset of permutations avoiding a single pattern of length three, and discuss their q -analogues utilizing the newly obtained q - γ -expansions, as well as the continued fraction of a quint-variate generating function due to Shin and the fourth author. Moreover, we enumerate the alternating permutations avoiding simultaneously two patterns, namely (2413 , 3142) and (1342 , 2431) , of length four, and consider such (− 1) -phenomenon for these two subsets as well. [ABSTRACT FROM AUTHOR]

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 12 ̲ 0 -avoiding inversion or ascent sequences. Additionally, concerning the last entry of 12 ̲ 0 -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 23 ̲ 14 -avoiding permutations, we offer two refined equidistribution conjectures in connection with 110- and 12 ̲ 0 -avoiding inversion sequences. [ABSTRACT FROM AUTHOR]