Related papers: $q$-Partition Algebra Combinatorics
The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities:…
A standard bicovariant differential calculus on a quantum matrix space ${\tt Mat}(m,n)_q$ is considered. The principal result of this work is in observing that the $U_q\frak{s}(\frak{gl}_m\times \frak{gl}_n))_q$ is in fact a…
In this paper we complete the derivations of finite volume partition functions for QCD using random matrix theories by calculating the effective low-energy partition function for three-dimensional QCD in the adjoint representation from a…
The class of minimal difference partitions MDP($q$) (with gap $q$) is defined by the condition that successive parts in an integer partition differ from one another by at least $q\ge 0$. In a recent series of papers by A. Comtet and…
The tensor product of a positive and a negative discrete series representation of the quantum algebra U_q(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
We discuss the problem posed by Bender, Coley, Robbins and Rumsey of enumerating the number of subspaces which have a given profile with respect to a linear operator over the finite field $\mathbb{F}_q$. We solve this problem in the case…
Let $\mathbb{F}_q$ be the finite field with $q$ elements, $K$ be an algebraically closed field containing $\mathbb{F}_q$, $K\{\tau\}$ be the Ore ring of $\mathbb{F}_q$-linear polynomials and $\Lambda_n$ be a free $K\{\tau\}$-module of rank…
The number of standard Young tableaux of shape a partition $\lambda$ is called the dimension of the partition and is denoted by $f^{\lambda}$. Partitions with odd dimensions were enumerated by McKay and were further characterized by…
Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the…
Four classes of three dimensional quadratic algebras of the type $\lsb Q_0 , Q_\pm \rsb$ $=$ $\pm Q_\pm$, $\lsb Q_+ , Q_- \rsb$ $=$ $aQ_0^2 + bQ_0 + c$, where $(a,b,c)$ are constants or central elements of the algebra, are constructed using…
We define new generalizations of (q,t)-Catalan numbers applying nabla operator on k-Schur functions indexed by column partitions. In some special cases, we give a combinatorial interpretation of these numbers using configurations of Dyck…
We study divisibility for the $q$-trinomial coefficients $\tau_0(n,m,q)$, $T_0(n,m,q)$ and $T_1(n,m,q)$, which were first introduced by Andrews and Baxter. In particular, we completely determine $\tau_0(an,bn,q)$, $T_0(an,bn,q)$ and…
We give a $q$-analogue of some binomial coefficient identities of Y. Sun [Electron. J. Combin. 17 (2010), #N20] as follows: {align*} \sum_{k=0}^{\lfloor n/2\rfloor}{m+k\brack k}_{q^2}{m+1\brack n-2k}_{q} q^{n-2k\choose 2} &={m+n\brack…
Traditional methods in quantum chemistry rely on Hartree-Fock-based Slater-determinant (SD) representations, whose underlying zeroth-order picture assumes separability by particle. Here, we explore a radically different approach, based on…
We derive orthogonality relations for discrete q-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra su_q(1,1). Spectra and eigenfunctions of these operators are found explicitly. These…
We give combinatorial proofs of two identities from the representation theory of the partition algebra $C A_k(n), n \ge 2k$. The first is $n^k = \sum_\lambda f^\lambda m_k^\lambda$, where the sum is over partitions $\lambda$ of $n$,…
A many variable $q$-calculus is introduced using the formalism of braided covector algebras. Its properties when certain of its deformation parameters are roots of unity are discussed in detail, and related to fractional supersymmetry. The…
In this paper, we show that the r-Stirling numbers of both kinds, the r-Whitney numbers of both kinds, the r-Lah numbers and the r-Whitney-Lah numbers form particular cases of family of polynomials forming a generalization of the partial…
The integer division of a numerator n by a divisor d gives a quotient q and a remainder r. Optimizing compilers accelerate software by replacing the division of n by d with the division of c * n (or c * n + c) by m for convenient integers c…