Related papers: Enumerative Properties of NC^B(p,q)

This article is part of an ongoing investigation of the combinatorics of $q,t$-Catalan numbers $\textrm{Cat}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle…

Let the columns of a $p \times q$ matrix $M$ over any ring be partitioned into $n$ blocks, $M = [M_1, ..., M_n]$. If no $p \times p$ submatrix of $M$ with columns from distinct blocks $M_i$ is invertible, then there is an invertible $p…

We generalize the construction of elliptic stable envelopes to actions of connected reductive groups and give a direct inductive proof of their existence and uniqueness in a rather general situation. We show these have powerful enumerative…

In this paper we investigate the following "polynomial moment problem": for a complex polynomial $P(z)$ and distinct complex numbers $a,b$ to describe polynomials $q(z)$ orthogonal to all integer non-negative powers of $P(z)$ on the segment…

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $D(a):=\{ q\in \mathbb{C}$, $0<|q|\leq a$, $\arg (q)\in…

Let $n,p,k$ be three positive integers. We prove that the rational fractions of $q$: $${n \brack k}_{q} {}_3\phi_{2} [ . {matrix}q^{1-k},q^{-p},q^{p-n} q,q^{1-n} {matrix}| q;q^{k+1}]\quad\textrm{and}\quad q^{(n-p)p}\qbi{n}{k}{q} {}_3\phi_2[…

In this work, we study type B set partitions for a given specific positive integer $k$ defined over $\langle n\rangle=\{-n, -(n-1),\cdots -1,0,1,\cdots n-1,n\}$. We found a few generating functions of type B analogue for some of the set…

An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so…

The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the n'th Catalan number. In this paper, we will generalize…

The $n^{th}$ partial flag incidence algebra of a poset $P$ is the set of functions from $P^n$ to some ring which are zero on non-partial flag vectors. These partial flag incidence algebras for $n>2$ are not commutative, not unitary, and not…

A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close…

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex…

Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values,…

We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative…

In this paper, a de Casteljau algorithm to compute (p,q)-Bernstein Bezier curves based on (p,q)-integers is introduced. We study the nature of degree elevation and degree reduction for (p,q)-Bezier Bernstein functions. The new curves have…

We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure,…

This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these…

An important question arising from the Frobenius Coin Problem is to decide whether or not a given monetary sum S can be obtained from N coin denominations. We develop a new Generating Function G(x), where the coefficient of x^i is equal to…

We consider the problem of counting matrices over a finite field with fixed rank and support contained in a fixed set. The count of such matrices gives a $q$-analogue of the classical rook and hit numbers, known as the $q$-rook and $q$-hit…

For an arbitrary set or multiset $A$ of positive integers, we associate the $A$-partition function $p_A(n)$ (that is the number of partitions of $n$ whose parts belong to $A$). We also consider the analogue of the $k$-colored partition…