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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]