Search

Let $w_{n,k,m}$ be the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. We establish in two ways a new interpretation of the numbers $w_{n,k,m}$ in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of $w_{n,k,m}$ in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, $2$-dominant compositions) is introduced, which amounts to give a combinatorial proof of the $\gamma$-positivity of the Narayana polynomial, as well as the $\gamma$-positivity of the polynomial $W_{2k+1,k}(t):=\sum_{1\le m\le k}w_{2k+1,k,m}t^m$ previously obtained by B\'{o}na et al, with apparently new combinatorial interpretations of their $\gamma$-coefficients., Comment: 19 pages, 3 figures

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. First off, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews' bivariate generating function, and two families of Andrews--Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e., invariant under conjugation) and explore their connections with three third order mock theta functions $\omega(q)$, $\nu(q)$, and $\psi^{(3)}(q)$, introduced by Ramanujan and Watson., Comment: 20 pages, 6 figures

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

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 Fran\ccon-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., Comment: 7 figures and 3 tables

In his recent work, Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions. In this note, we provide bijective proofs as well as refinements of those three theorems of Andrews. Our refinements take into account the numbers of parts in each of the two colors., Comment: 11 pages, 2 figures, comments are welcome

Given a general multiset $\mathcal{M}=\{1^{m_1},2^{m_2},\ldots,n^{m_n}\}$, where $i$ appears $m_i$ times, a multipermutation $\pi$ of $\mathcal{M}$ is called {\em quasi-Stirling}, if it contains no subword of the form $abab$ with $a\neq b$. We designate exactly one entry of $\pi$, say $k\in \mathcal{M}$, which is not the leftmost entry among all entries with the same value, by underlining it in $\pi$, and we refer to the pair $(\pi,k)$ as a quasi-Stirling multipermutation of $\mathcal{M}$ rooted at $k$. By introducing certain vertex and edge labeled trees, we give a new bijective proof of an identity due to Yan, Yang, Huang and Zhu, which links the enumerator of rooted quasi-Stirling multipermutations by the numbers of ascents, descents, and plateaus, with the exponential generating function of the {\em bivariate Eulerian polynomials}. This identity can be viewed as a natural extension of Elizalde's result on $k$-quasi-Stirling permutations, and our bijective approach to proving it enables us to: (1) prove bijectively a Carlitz type identity involving quasi-Stirling polynomials on multisets that was first obtained by Yan and Zhu; (2) confirm a recent partial $\gamma$-positivity conjecture due to Lin, Ma and Zhang, and find a combinatorial interpretation of the $\gamma$-coefficients in terms of two new statistics defined on quasi-Stirling multipermutations called sibling descents and double sibling descents., Comment: 17 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: 29 pages, Expanded version with final remarks

A di-sk tree is a rooted binary tree whose nodes are labeled by $\oplus$ or $\ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees proving the two quintuples $(\LMAX,\LMIN,\DESB,\iar,\comp)$ and $(\LMAX,\LMIN,\DESB,\comp,\iar)$ have the same distribution over separable permutations. Here for a permutation $\pi$, $\LMAX(\pi)/\LMIN(\pi)$ is the set of values of the left-to-right maxima/minima of $\pi$ and $\DESB(\pi)$ is the set of descent bottoms of $\pi$, while $\comp(\pi)$ and $\iar(\pi)$ are respectively the number of components of $\pi$ and the length of initial ascending run of $\pi$. Interestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides (up to the classical {\em Knuth--Richards bijection}) an alternative approach to a result of Rubey (2016) that asserts the two triples $(\LMAX,\iar,\comp)$ and $(\LMAX,\comp,\iar)$ are equidistributed on $321$-avoiding permutations. Rubey's result is a symmetric extension of an equidistribution due to Adin--Bagno--Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive. Some equidistribution results for various statistics concerning tree traversal are presented in the end., Comment: Minor revision: 20 pages, 7 figures

We launch a systematic study of the refined Wilf-equivalences by the statistics $\mathsf{comp}$ and $\mathsf{iar}$, where $\mathsf{comp}(\pi)$ and $\mathsf{iar}(\pi)$ are the number of components and the length of the initial ascending run of a permutation $\pi$, respectively. As Comtet was the first one to consider the statistic $\mathsf{comp}$ in his book {\em Analyse combinatoire}, any statistic equidistributed with $\mathsf{comp}$ over a class of permutations is called by us a {\em Comtet statistic} over such class. This work is motivated by a triple equidistribution result of Rubey on $321$-avoiding permutations, and a recent result of the first and third authors that $\mathsf{iar}$ is a Comtet statistic over separable permutations. Some highlights of our results are: (1) Bijective proofs of the symmetry of the double Comtet distribution $(\mathsf{comp},\mathsf{iar})$ over several Catalan and Schr\"oder classes, preserving the values of the left-to-right maxima. (2) A complete classification of $\mathsf{comp}$- and $\mathsf{iar}$-Wilf-equivalences for length $3$ patterns and pairs of length $3$ patterns. Calculations of the $(\mathsf{des},\mathsf{iar},\mathsf{comp})$ generating functions over these pattern avoiding classes and separable permutations. (3) A further refinement by the Comtet statistic $\mathsf{iar}$, of Wang's recent descent-double descent-Wilf equivalence between separable permutations and $(2413,4213)$-avoiding permutations., Comment: 39 pages, 2 tables, 2 figures. Comments are welcome

The $k$-arrangements are permutations whose fixed points are $k$-colored. We prove enumerative results related to statistics and patterns on $k$-arrangements, confirming several conjectures by Blitvi\'c and Steingr\'imsson. In particular, one of their conjectures regarding the equdistribution of the number of descents over the derangement form and the permutation form of $k$-arrangements is strengthened in two interesting ways. Moreover, as one application of the so-called Decrease Value Theorem, we calculate the generating function for a symmetric pair of Eulerian statistics over permutations arising in our study., Comment: 25 pages, 1 figure and 1 table

Recently, Yan and the first named 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 objective of this paper is to address two of their conjectures concerning the enumeration of $\underline{12}0$-avoiding inversion or ascent sequences., Comment: 13 pp, comments are welcome

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

Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schr\"oder paths (resp. little Schr\"oder 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\"oder 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\"oder 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., Comment: 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

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 Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchy's identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan., Comment: 9 pages, 2 figures

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., Comment: 11 pages, 2 figures, 1 table

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a function on $k$-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results., Comment: 11 pages, 1 table

In 2000, Babson and Steingr\'{i}msson 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 discoverd 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-Sch\"{u}tzenberger's involution that utilizes the celebrated RSK algorithm., Comment: 11 pages

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., Comment: 12 pages,1 table

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

Multiranks and new rank/crank analogs for a variety of partitions are given, so as to imply combinatorially some arithmetic properties enjoyed by these types of partitions. Our methods are elementary relying entirely on the three classical theta functions, and are motivated by the seminal work of Ramanujan, Garvan, Hammond and Lewis., Comment: 17 pages, 3 figures

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's Odd-Distinct partition theorem, derive a generalization in the spirit of Alder's conjecture, as well as a curious analogue of the first Rogers-Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler's pentagonal number theory in this setting, and connect it with the Rogers-Fine identity. We concludes with some congruence properties., Comment: 15 pages, 3 figures, 4 tables

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

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $\gamma$-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., Comment: 16 pages, 4 figures