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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., Comment: 19 pages, comments are welcome

In this paper, we prove six conjectures on the evaluations of permanents, relating them to Genocchi numbers of the first and second kinds as well as (binomial) Euler numbers. For example, we prove that $$\mathrm{per}\left[\left\lfloor\frac{2j-k}{n}\right\rfloor\right]_{1\le j,k\le n}=2(2^{n+1}-1)B_{n+1},$$ where $\lfloor\cdot\rfloor$ is the floor function and $B_0,B_1,\ldots$ are the Bernoulli numbers. This was previously conjectured by the third author. Various permutation statistics are at the heart of our approaches., Comment: 28 pages, Expanded version with new results added

Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schr��der paths (resp. little Schr��der paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: \begin{align*} r(n,0)&=\sum\limits_{j=0}^{n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+\sum\limits_{j=k}^{n-1}2^{j-k}r(n-1,j),\quad 1\le k\le n, s(n,0) &=\sum\limits_{j=1}^{n-1}2\cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+\sum\limits_{j=k+1}^{n-1}2\cdot3^{j-k-1}s(n-1,j),\quad 1\le k\le n. \end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,k\ge 0}$ and $[s(n,k)]_{n,k\ge 0}$, whose row sums produce the large and little Schr��der 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 Schr��der 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 $[r(n,k)]_{n,k\ge 0}$ as well., 20 pages, 6 figures and 2 tables

For an integer $m\ge 2$, a partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ is called $m$-falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $\lambda_i$'s form a nonincreasing sequence. We extend a bijection originally due to the third author to deduce a lecture hall theorem for such $m$-falling partitions. A special case of this result gives rise to a finite version of Pak-Postnikov's $(m,c)$-generalization of Euler's theorem. Our work is partially motivated by a recent extension of Euler's theorem for all moduli, due to Keith and Xiong. We note that their result actually can be refined with one more parameter., Comment: 14 pages, 3 figures, 1 tables

As shown by Bousquet-M\'elou--Claesson--Dukes--Kitaev (2010), ascent sequences can be used to encode $({\bf2+2})$-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series $$\sum_{m=1}^{\infty}\prod_{i=1}^m (1-(1-t)^i).$$ In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: (i) 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 $\max$. (ii) a far-reaching generalization of the generating function formula for $(\asc,\zero)$ due to Jel\'inek. 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. (iii) 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 $({\bf2-1})$-avoiding inversion sequences. This is achieved via a decomposition of $({\bf2-1})$-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., Comment: 25 pages, to appear in Journal of Combinatorial Theory, Series A

The distribution of certain Mahonian statistic (called $\mathrm{BAST}$) introduced by Babson and Steingr\'{i}msson over the set of permutations that avoid vincular pattern $1\underline{32}$, 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 recently proven Delta Conjecture. During the course, a refined relation between $\mathrm{BAST}$ and its reverse complement $\mathrm{STAT}$ is derived as well., Comment: 29 pages, a new extension to ordered multiset partitions added

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 $\gamma$-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 $q$-$\gamma$-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., Comment: 36 pages, 3 figures, 4 tables

Motivated by Andrews' recent work related to Euler's partition theorem, we consider the set of partitions of an integer $n$ where the set of even parts has exactly $j$ elements, versus the set of partitions of $n$ where the set of repeated parts has exactly $j$ elements. These two sets of partitions turn out to be equinumerous, and this naturally encloses Euler's theorem and Andrews' theorem as two special cases. We give two proofs, one using generating function, and the other is a direct bijection that builds on Glaisher's bijection, Comment: 7 pages, 2 Tables; Submitted to Math. Student

Berkovich-Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage-Sills' new little G��llnitz 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., This is the second revision submitted to JCTA in June, comments are welcome