Related papers: The combinatorics of k-marked Durfee symbols
Several authors have recently considered the smallest positive part missing from an integer partition, known as the minimum excludant or mex. In this work, we revisit and extend connections between Dyson's crank statistics, the mex, and…
We generalize the method of combinatorial telescoping to the case of multiple summations. We shall demonstrate this idea by giving combinatorial proofs for two identities of Andrews on parity indices of partitions.
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
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…
A combinatorial interpretation is provided for the moments of characteristic polynomials of random unitary matrices. This leads to a rather unexpected consequence of the Keating and Snaith conjecture: the moments of $\mid\xi(1/2+it)\mid$…
We establish a new simple explicit description of combinatorial wall-crossing for the rational Cherednik algebra applied to the trivial representation. In this way we recover a theorem of P. Dimakis and G. Yue. We also present two…
We prove a family of partition identities which is "dual" to the family of Andrews-Gordon's identities. These identities are inspired by a correspondence between a special type of partitions and "hypergraphs" and their proof uses…
In 1978 Durfee conjectured various inequalities between the signature and the geometric genus of a normal surface singularity. Since then a few counter examples have been found and positive results established in some special cases. We…
We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and…
Hecke algebras are beautiful q-extensions of Coxeter groups. In this paper, we prove several results on their characters, with an emphasis on characters induced from trivial and sign representations of parabolic subalgebras. While most of…
In this series of articles we study connections between combinatorics of multidimensional generalizations of Cauchy identity and continuous objects such as multidimensional Brownian motions and Brownian bridges. In Part I of the series we…
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model…
We prove K-theoretic generalizations of the component formulas of Knutson, Miller, and Shimozono, and deduce that K-theoretic quiver coefficients have alternating signs. We also prove new variants of the factor sequences conjecture, and a…
In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee…
In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's "Lost" Notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by…
In this work, we study the generalized k-th power symbol (a/n)_k and present a comprehensive collection of its algebraic properties. The results are classified according to their dependence on the three main parameters a, n, and k. In…
Several intersection matrices of $s$-subsets vs. $k$-subsets of a $v$-set are introduced in the literature. We study these matrices systematically through counting arguments and generating function techniques. A number of new or known…
A well-studied statistic of an integer partition is the size of its Durfee square. In particular, the number $D_k (n)$ of partitions of $n$ with Durfee square of fixed size $k$ has a well-known simple rational generating function. We study…
The enumeration of diagonally symmetric alternating sign matrices (DSASMs) is studied, and a Pfaffian formula is obtained for the number of DSASMs of any fixed size, where the entries for the Pfaffian are positive integers given by simple…
We present what we call a "motivated proof" of the G\"{o}llnitz-Gordon-Andrews identities. A similar motivated proof of the Rogers-Ramanujan identities was previously given by G. E. Andrews and R. J. Baxter, and was subsequently generalized…