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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Guo-Niu Han [Sémin. Lothar. Comb. 85 (2021) B85c (electronic)] has introduced a new combinatorial object named standard puzzle. We use digraphs to show the relations between numbers in standard puzzles and propose a skeleton model. By this model, we solve the enumeration problem of over fifty thousand standard puzzle sequences. Most of them can be represented by classical numbers, such as Catalan numbers, double factorials, secant numbers and so on. Also, we prove several identities for standard puzzle sequences. [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]

We derive some new signed Mahonian polynomials over the complex reflection group G (r , 1 , n) = C r ≀ S n , where the "sign" is taken to be any of the 2r 1-dim characters and the "Mahonian" statistics are the lmaj defined by Bagno and the sor defined by Eu et al. Various new signed Mahonian polynomials over Coxeter groups of types B n and D n are obtained as well. We also investigate the signed counting polynomials on G(r, 1, n) for those statistics with the distribution [ r ] q [ 2 r ] q ⋯ [ n r ] q . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For an integer m ≥ 2 , a partition λ = (λ 1 , λ 2 , ...) is called m-falling, a notion introduced by Keith, if the least non-negative residues mod m of λ 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 Xiong and Keith. We note that their result actually can be refined with one more parameter. [ABSTRACT FROM AUTHOR]

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for k-colored partition functions p-k(n) for all k≥2. 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 theorem in this setting, and connect it with the Rogers-Fine identity. We conclude with some congruence properties. [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]

In one of their recent collaborative papers, Andrews and Paule continue their study of partition functions via MacMahon's Partition Analysis by considering partition functions associated with directed graphs which consist of chains of diamond shape. They prove a congruence related to one of these partition functions and conjecture a number of similar congruence results. In this note, we reprove this congruence by constructing an explicit way to group partitions. Then we keep the essence of the method and manage to apply it to a different kind of plane partitions to get more general results and several other congruences. [ABSTRACT FROM AUTHOR]