Related papers: Basis partitions and their signature
We define noncrossing partitions of a marked surface without punctures (interior marked points). We show that the natural partial order on noncrossing partitions is a graded lattice and describe its rank function topologically. Lower…
The elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These…
Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored…
In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) $s$-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to $0$ or…
In this paper, we introduce Gram matrices for the signed partition algebras, the algebra of $\mathbb{Z}_2$-relations and the partition algebras. We prove that the Gram matrix is similar to a matrix which is a direct sum of block…
Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…
We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results…
We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrew's in which he considers the generation function for partitions with respect to size, number of odd parts, and number of…
In their seminal work J. Lepowsky and R. L. Wilson gave a vertex-operator theoretic interpretation of Gordon-Andrews-Bressoud's generalization of Rogers-Ramanujan combinatorial identities, by constructing bases of vacuum spaces for the…
We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when…
Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a…
In the representation theory of simple Lie algebras, we consider the problem of constructing a monomial basis in an arbitrary irreducible finite-dimensional highest weight module. We construct a PBW-type basis in every finite-dimensional…
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…
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…
We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by…
Extending the notion of $r$-(class) regular partitions, we define $(r_{1},...,r_{m})$-class regular partitions. A partition identity is presented and described by making use of the Glaisher correspondence.
We focus on writing closed forms of generating functions for the number of partitions with gap conditions as double sums starting from a combinatorial construction. Some examples of the sets of partitions with gap conditions to be discussed…
We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. These permutations are the linear…
We give a direct deduction and proof of two identities in the theory of plane partitions. The first one is known to enumerate the traces of plane partitions. The second one comes without any combinatorial interpretation.
We present a different way to obtain generators of metric spaces having the property that the ``position'' of every element of the space is uniquely determined by the distances from the elements of the generators. Specifically we introduce…