Related papers: Asymptotic formulas for general colored partition …

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$.…

Given a graph $G$ of order $n$, the $\sigma$-$polynomial$ of $G$ is the generating function $\sigma(G,x) = \sum a_{i}x^{i}$ where $a_{i}$ is the number of partitions of the vertex set of $G$ into $i$ nonempty independent sets. Such…

We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of numbers, illustrate how different mathematical fields play…

Let $A$ be a nonempty set of positive integers. The restricted partition function $p_A(n)$ denotes the number of partitions of $n$ with parts in $A$. When the elements in $A$ are pairwise relatively prime positive integers, Ehrhart,…

In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the…

The study of the well-known partition function $p(n)$ counting the number of solutions to $n = a_{1} + \dots + a_{\ell}$ with integers $1 \leq a_{1} \leq \dots \leq a_{\ell}$ has a long history in combinatorics. In this paper, we study a…

Let $f:\mathbb{N}\to\{0,\pm 1\}$, for $n \in \mathbb{N}$ let $\Pi[n]$ be the set of partitions of $n$, and for all partitions $\pi = (a_1,a_2,\ldots,a_k) \in \Pi[n]$ let \[ f(\pi) := f(a_1)f(a_2) \cdots f(a_k). \] With this we define the…

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers $p_i$ indexed by its parts…

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…

The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this…

We determine asymptotic formulas for the coefficients of a natural class of negative index and negative weight Jacobi forms. These coefficients can be viewed as a refinement of the numbers $p_k(n)$ of partitions of n into k colors. Part of…

We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions $\{\mathcal{P}_t\}_{t \ge 1}$, where $\mathcal{P}_t$ is a partition of $\{1,2,\dots, Rt\}$. At each time~$t$,~$R$ numbers are…

We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of…

In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks.

In this paper, we investigate $\pi(m,n)$, the number of partitions of the \emph{bipartite number} $(m,n)$ into \emph{steadily decreasing} parts, introduced by L.Carlitz ['A problem in partitions', Duke Math Journal 30 (1963), 203--213]. We…

The $A$-partition function $p_A(n)$ enumerates those partitions of $n$ whose parts belong to a fixed (finite or infinite) set $A$ of positive integers. On the other hand, the extended $A$-partition function…

Very recently, Thejitha, Sellers, and Fathima defined the function $a_{r,s}(n)$, which enumerates the number of multicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors,…

We define a new pair of dual bases that generalize the immaculate and dual immaculate bases to the colored algebras $QSym_A$ and $NSym_A$. The colored dual immaculate functions are defined combinatorially via tableaux, and we present…

We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z|…

The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If…