Related papers: Reciprocity between partitions and compositions
Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle…
We introduce constellation ensembles, in which charged particles on a line (or circle) are linked with charged particles on parallel lines (or concentric circles). We present formulas for the partition functions of these ensembles in terms…
We introduce a new Hopf algebra that operates on pairs of finite interval partitions and permutations of equal length. This algebra captures vincular patterns, which involve specifying both the permutation patterns and the consecutive…
Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in…
In this note we give three identities for partitions with parts separated by parity, which were recently introduced by Andrews.
We study the boundedness of composition operators on the weighted Bergman spaces and the Hardy space over the polydisc. For arbitrary polydisc we prove the rank sufficiency theorem which, in particular, provides us with a simple criterion…
We study three different poset structures on the set of all compositions. In the first case, the covering relation consists of inserting a part of size one to the left or to the right, or increasing the size of some part by one. The…
The emergence of structure in cooperative relation is studied in a game theoretical model. It is proved that specific types of reciprocity norm lead individuals to split into two groups. The condition for the evolutionary stability of the…
The preparation of many composites requires the intermixing of several macromolecular fluids along with the addition of solid filler particles. These fillers are usuallly polydisperse and there is an extensive experimental evidence that…
Ramanujan's celebrated congruences of the partition function $p(n)$ have inspired a vast amount of results on various partition functions. Kwong's work on periodicity of rational polynomial functions yields a general theorem used to…
We obtain a combinatorial proof of a surprising weighted partition equality of Berkovich and Uncu. Our proof naturally leads to a formula for the number of partitions with a given parity of the smallest part, in terms of S(i), the number of…
This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of…
Recently, Andrews considered the partitions with parts separated by parity, in which parts of a given parity are all smaller than those of the other parity. Inspired from the partitions with parts separated by parity, we investigate the…
In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of…
We offer some partition functions related to ternary quadratic forms, and note on their upper bounds and related properties. We offer these results as an application of a simple method related to conjugate Bailey pairs presented in a prior…
The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the…
The reciprocal Pascal matrix has entries $\binom{i+j}{j}^{-1}$. Explicit formullae for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, $q$-analogues are also presented.
We prove a reciprocity formula between Gauss sums that is used in the computation of certain quantum invariants of 3-manifolds. Our proof uses the discriminant construction applied to the tensor product of lattices.
We define a generalized vector partition function and derive an identity for generating series of such functions associated with solutions of basic recurrence relation of combinatorial analysis. As a consequence, we obtain the generating…
In the study of the algebra $\mathrm{NCSym}$ of symmetric functions in noncommutative variables, Bergeron and Zabrocki found a free generating set consisting of power sum symmetric functions indexed by atomic partitions. On the other hand,…