Search

Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of 0021-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern-restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted permutation case and the $\mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result. [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]

In 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discovered by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on R (w) , the rearrangement class of a given word w. We construct such an involution in this paper, which yields a comparable joint equidistribution between two sextuple statistics over R (w). Our involution builds on Burstein's involution and Foata–Schützenberger's involution that utilizes the celebrated RSK algorithm. [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]

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]

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the γ -coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted. [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]