Related papers: Linked partition ideals, directed graphs and $q$-m…
We investigate some weighted integer partitions whose generating functions are double-series. We will establish closed formulas for these $q$-double series and deduce that their coefficients are non-negative. This leads to inequalities…
We construct a $k$-fold $q$-series as a generating function of $k$-regular partitions for each positive integer $k$. The $k=1$ case is one of Euler's $q$-series identities pertaining to the partitions into distinct parts. The construction…
We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us to give a combinatorial interpretation of the Macdonald identities for affine root systems of the seven…
We consider random partitions of the vertex set of a given finite graph that can be sampled by means of loop-erased random walks stopped at a random exponential time of parameter $q>0$. The related random blocks tend to cluster nodes…
Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of…
Two fundamental algorithm-design paradigms are Tree Search and Dynamic Programming. The techniques used therein have been shown to complement one another when solving the complete set partitioning problem, also known as the coalition…
The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements--all in analogy with formulas for finite sets (which are the special case of q=1). A direct-sum decomposition of a…
We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined…
We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us on the one hand to obtain enumerations connecting products of hook lengths and vectors of integers. This…
We study families of partitions with gap conditions that were introduced by Schur and Andrews, and describe their fundamental connections to combinatorial q-series and automorphic forms. In particular, we show that the generating functions…
The Andrews-Bressoud identities are one of many families of $q$-series identities relating an infinite sum to an infinite product. While the original motivation for studying these series relates to partitions, they can also be viewed in…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
Andrews investigated parity conditions in the Rogers-Ramanujan-Gordon theorem. Under the conditions that even parts or odd parts appear an even number of times, Andrews discovered two Rogers-Ramanujan-Gordon type partition theorems and…
We analyse q-functional equations arising from tree-like combinatorial structures, which are counted by size, internal path length, and certain generalisations thereof. The corresponding counting parameters are labelled by a positive…
Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these…
In this paper, we propose several graph-based extensions of the Douglas-Rachford splitting (DRS) method to solve monotone inclusion problems involving the sum of $N$ maximal monotone operators. Our construction is based on a two-layer…
We establish factoriality of $q$-Araki-Woods von Neumann algebras (with the number of generators at least two) in full generality, exploiting the approach via conjugate variables developed recently in the tracial case by Akihiro Miyagawa…
Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A…
We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial…
We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a…