Related papers: Closed-Form Formula for the Partition Function and…

This paper presents the methods to utilizing the $s$-fold extension of Bailey's lemma to obtain $spt$-type functions related to the symmetrized rank function $\eta_{2k}(n).$ We provide the $k=2$ example, but clearly illustrate how deep…

This article introduces recursive relations allowing the calculation of the number of partitions with constraints on the minimum and/or on the maximum fragment size.

New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that a normalized…

Partition density functional theory is a formally exact procedure for calculating molecular properties from Kohn-Sham calculations on isolated fragments, interacting via a global partition potential that is a functional of the fragment…

The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing…

We consider the Ising model on an $M\times N$ rectangular lattice with an asymmetric self-dual boundary condition, and derive a closed-form expression for its partition function. We show that zeroes of the partition function are given by…

We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the…

Motivated by constraints on the dark energy equation of state from supernova-data, we propose a formalism for the Bayesian inference of functions: Starting at a functional variant of the Kullback-Leibler divergence we construct a functional…

In statistical physics and information theory, although the exponent of the partition function is often of our primary interest, there are cases where one needs more detailed information. In this paper, we present a general framework to…

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \varphi(n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes $ \sum_{p\leq…

Let p(n, k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m, p(n,k) is periodic modulo m. Using this, we…

We propose a new approximate method for counting the number of the solutions for constraint satisfaction problem (CSP). The method derives from the partition function based on introducing the free energy and capturing the relationship of…

We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an…

We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series…

We present several elementary closed-forms that express a non-trivial divisor for every composite integer $n > 1$. Each closed-form consists of a fixed number of elementary arithmetic operations drawn from the set: addition, subtraction,…

We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…

The partition function, $p_A(n)$, is defined to be the number of partitions of $n$ with parts in the set A, where $n$ is a positive integer and $A$ is a set of positive integers. It is well documented that: if A is a finite set with…

Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit…

We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers-Ramanujan…

We prove an asymptotic formula for the number of partitions of $n$ into distinct parts where the largest part is at most $t\sqrt{n}$ for fixed $t \in \mathbb{R}$. Our method follows a probabilistic approach of Romik, who gave a simpler…