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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]

The distribution of certain Mahonian statistic (called BAST) introduced by Babson and Steingrímsson over the set of permutations that avoid vincular pattern 1 32 _ , is shown bijectively to match the distribution of major index over the same set. This new layer of equidistribution is then applied to give alternative interpretations of two related q -Stirling numbers of the second kind, studied by Carlitz and Gould. Moreover, extensions to an Euler-Mahonian statistic over ordered set partitions, and to statistics over ordered multiset partitions present themselves naturally. The latter of which is shown to be related to the Delta Conjecture. During the course, a refined relation between BAST and its reverse complement STAT is derived as well. [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]

Abstract Berkovich–Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage–Sills's new little Göllnitz identities. Noticing the connection between their results and Boulet's earlier four-parameter partition generating functions, we discover a new class of partitions, called k -strict partitions, to generalize their results. By applying both horizontal and vertical dissections of Ferrers' diagrams with appropriate labellings, we provide a unified combinatorial treatment of their results and shed more lights on the intriguing conditions of their companion to Capparelli's identities. [ABSTRACT FROM AUTHOR]

A generalized crank ( k -crank) for k -colored partitions is introduced. Following the work of Andrews–Lewis and Ji–Zhao, we derive two results for this newly defined k -crank. Namely, we first obtain some inequalities between the k -crank counts M k ( r , m , n ) for m = 2 , 3 and 4, then we prove the positivity of symmetrized even k -crank moments weighted by the parity for k = 2 and 3. We conclude with several remarks on furthering the study initiated here. [ABSTRACT FROM AUTHOR]

Abstract: The Rogers–Ramanujan identities have many natural and significant generalizations. The generalization presented in this note was first studied by D. Bressoud, by considering the partitions that he named as “footed partition”. A bijection is described to prove his conjecture and some examples are attached at the end. [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]

Let r (n , k) (resp. s (n , k)) be the number of Schröder paths (resp. little Schröder paths) of length 2 n with k hills, and set r (0 , 0) = s (0 , 0) = 1. We bijectively establish the following recurrence relations: r (n , 0) = ∑ j = 0 n − 1 2 j r (n − 1 , j) , n ≥ 1 , r (n , k) = r (n − 1 , k − 1) + ∑ j = k n − 1 2 j − k r (n − 1 , j) , 1 ≤ k ≤ n , s (n , 0) = ∑ j = 1 n − 1 2 ⋅ 3 j − 1 s (n − 1 , j) , n ≥ 1 , s (n , k) = s (n − 1 , k − 1) + ∑ j = k + 1 n − 1 2 ⋅ 3 j − k − 1 s (n − 1 , j) , 1 ≤ k ≤ n. The infinite lower triangular matrices [ r (n , k) ] n , k ≥ 0 and [ s (n , k) ] n , k ≥ 0 , whose row sums produce the large and little Schröder numbers respectively, are two Riordan arrays of Bell type. Hence the above recurrences can also be deduced from their A - and Z -sequences characterizations. On the other hand, it is well-known that the large Schröder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run, whose distribution on separable permutations is shown to be given by [ r (n , k) ] n , k ≥ 0 as well. [ABSTRACT FROM AUTHOR]

As shown by Bousquet-Mélou–Claesson–Dukes–Kitaev (2010), ascent sequences can be used to encode (2 + 2) -free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series ∑ m = 1 ∞ ∏ i = 1 m (1 − (1 − t) i). In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: • a calculation of the Euler–Stirling distribution on ascent sequences, including the numbers of ascents (asc), repeated entries (rep) , zeros (zero) and maximal entries (max). In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of zero and max. • a far-reaching generalization of the generating function formula for (asc , zero) due to Jelínek. This is accomplished via a bijective proof of the quadruple equidistribution of (asc , rep , zero , max) and (rep , asc , rmin , zero) , where rmin denotes the right-to-left minima statistic of ascent sequences. • an extension of a conjecture posed by Levande, which asserts that the pair (asc , zero) on ascent sequences has the same distribution as the pair (rep , max) on (2 − 1) -avoiding inversion sequences. This is achieved via a decomposition of (2 − 1) -avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples (asc , rep , zero , max) and (rep , asc , max , zero) are equidistributed on ascent sequences. [ABSTRACT FROM AUTHOR]