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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]

Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations for the moments of Laguerre polynomials via Flajolet's combinatorial theory of continued fractions. In this paper, we first introduce a reflection-like involution on restricted Laguerre histories. Then, we demonstrate its power by composing this involution with three bijections due to Françon-Viennot, Foata-Zeilberger, and Yan-Zhou-Lin, respectively. A host of equidistribution results involving various (multiset-valued) permutation statistics follow from these applications. As byproducts, seven apparently new Mahonian statistics present themselves; new interpretations of known Mahonian statistics are discovered as well. Finally, in our effort to show the interconnections between these Mahonian statistics, we are naturally led to a new link between the variant Yan-Zhou-Lin bijection and the Kreweras complement. [ABSTRACT FROM AUTHOR]

Weakly increasing tree is a new kind of multiset-labeled tree introduced in a recent work of Lin-Ma-Ma-Zhou [9] , which naturally unifies the classical concepts of plane trees and increasing trees on the set { 1 , 2 , ... , n }. Two bijections defined on weakly increasing trees are introduced. The first map has its roots in a bijection on plane trees due to Deutsch [4]. The second map is a closely related variant and turns out to be an involution. This involution amounts to give a combinatorial proof of certain equidistribution results for a sextuple of tree statistics over weakly increasing trees defined on any multiset M , extending a previous result of Lin-Ma-Ma-Zhou on restricted M. [ABSTRACT FROM AUTHOR]

Recently, Gu, Lai and Liang proved necessary and sufficient conditions for a given sequence of positive integers to be the degree sequence of a line-Hamiltonian multigraph. Our goal in this note is to utilize this result to prove a closed formula for the function , the number of degree sequences with degree sum representable by line-Hamiltonian multigraphs. Indeed, we give a truly elementary proof that where is the number of unrestricted integer partitions of j. [ABSTRACT FROM AUTHOR]

Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank. [ABSTRACT FROM AUTHOR]