Related papers: Gaussian Polynomials and Restricted Partition Func…

We derive closed formulas for the number of $k$-coloured partitions and the number of plane partitions of $n$ in terms of the Bell polynomials.

We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…

We introduce a division formula on a possibly singular projective subvariety $X$ of complex projective space $\Pk^N$, which, e.g., provides explicit representations of solutions to various polynomial division problems on the affine part of…

We consider Gaussian distributions on certain Riemannian symmetric spaces. In contrast to the Euclidean case, it is challenging to compute the normalization factors of such distributions, which we refer to as partition functions. In some…

We narrow in on the number of graphical partitions for which there is no known generating function by manipulating the well known generating function for Frobenius partitions.

It is known that if an algebra of polymorphisms of the constraint language has the Polynomially Generated Powers (PGP) Property then the Quantified CSP can be reduced to the CSP over the same constraint language with constants. The only…

Give a convenient new approach to obtain the algebraic constraints on tau functions of p-reduced KP with the constraints of string equation. The classical results about algebraic constraints on associated tau function are included.

We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In…

In this note, we obtain a formula which leads to a practical and efficient method to calculate the number of partitions of n into parts not divisible by m for given natural numbers n and m.

We prove new formulas and congruences for $p(n,k):=$ the number of partitions of $n$ into $k$ parts and $q(n,k):=$ the number of partitions of $n$ into $k$ distinct parts. Also, we give lower and upper bounds for the density of the set…

We show that explicit forms for certain polynomials~$\psi^{(a)}_m(n)$ with the property \[ \psi^{(a+1)}_m(n) = \sum_{\nu=1}^n \psi_m^{(a)}(\nu) \] can be found (here, $a,m,n\in\mathbb{N}_0$). We use these polynomials as a basis to express…

We continue to investigate combinatorial properties of functions $f_m$ and $c_m$ considered in our previous papers. They depend on an initial arithmetic function $f_0$. In this paper, the values of $f_0$ are the binomial coefficients. We…

Consider the representations of an algebraic group G. In general, polynomial invariant functions may fail to separate orbits. The invariant subring may not be finitely generated, or the number and complexity of the generators may grow…

This paper is a survey of plurisubharmonic theory where the usual polynomial ring is replaced by a polynomial ring $\mathcal P^S(\mathbb C^n)$ where the $m$-th degree polynomials have exponents restricted to $mS$, where $S\subseteq \mathbb…

For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2x2 polynomial differential systems satisfied by the associated orthogonal polynomials is…

The partition functions $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, and $Q(n,m,p)$, the number of integer partitons of $n$ into exactly $m$ distinct parts with each part at most…

Using previous work by Merca, we show the partition function involving parts of k different magnitudes, shifted by the triangular numbers, equals the self convolution of the unrestricted partition function. We also provide a combinatorial…

An approximate formula for the partitions of Goldbach's Conjecture is derived using Prime Number Theorem and a heuristic probabilistic approach. A strong form of Goldbach's conjecture follows in the form of a lower bounding function for the…

For a given set of integers $\mathcal{S}$, let $\mathcal{R}_n^*(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ over $\mathbb{Z}[X]$ with $a_i\in\mathcal{S}$ and $a_0a_n\ne 0$. In this…

We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution.