Related papers: A Chinese Remainder Theorem for Partitions

We derive polynomial identities of arbitrary degree $n$ for syzygies degrees of numerical semigroups S_m=<d_1,...,d_m> and show that for n>=m they contain higher genera G_r=\sum_{s\in Z_>\setminus S_m}s^r of S_m. We find a number…

Let $p_2(n)$ denote the number of cubic partitions. In this paper, we shall present two new congruences modulo $11$ for $p_2(n)$. We also provide an elementary alternative proof of a congruence established by Chan. Furthermore, we will…

The restricted partitions in which the largest part is less than or equal to $N$ and the number of parts is less than or equal to $k$ were investigated by Andrews in \cite{Andrews76}. These partitions were extended recently by the author to…

We derive a formula for $p(n)$ (the number of partitions of $n$) in terms of the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's pentagonal number theorem.

In this paper, we study the $n$-point function of $t$-core partitions. The main tool is the topological vertex, originally developed to study the topological string theory for toric Calabi--Yau 3-folds. By virtue of the topological vertex,…

This paper gives an exposition of well known results on vector partition functions. The exposition is based on works of M. Brion, A. Szenes and M. Vergne and is geared toward explicit computer realizations. In particular, the paper presents…

The polynomial partitioning method of Guth and Katz [arXiv:1011.4105] has numerous applications in discrete and computational geometry. It partitions a given $n$-point set $P\subset\mathbb{R}^d$ using the zero set $Z(f)$ of a suitable…

We introduce the idea of (s,t)-closure and delta-sets and show that (s,t)-closed beta-sets which are contained set-wise in (s,t)-closed delta-sets are also contained partition-wise. This implies the maximal (s,t)-core partition theorem of…

Residue number systems based on pairwise relatively prime moduli are a powerful tool for accelerating integer computations via the Chinese Remainder Theorem. We study a structured family of moduli of the form $2^n - 2^k + 1$, originally…

We prove that for every nonempty set $\Sigma$ of integers bigger than $1$, which has at most three elements, there exists a numerical semigroup $T$ and an element $x$ of $T$ such that a natural number $n$ is the number of atoms in a…

Let A={a_s(mod n_s)}_{s=0}^k be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results concerning system A. In particular, we show that if every integer lies in…

For a fixed permutation $\sigma \in S_k$, let $N_{\sigma}$ denote the function which counts occurrences of $\sigma$ as a pattern in permutations from $S_n$. We study the expected value (and $d$-th moments) of $N_{\sigma}$ on conjugacy…

A generalized Chinese remainder theorem (CRT) for multiple integers from residue sets has been studied recently, where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known. A…

In this paper, we consider different constrained partition problems for weighted trees and cactus graphs. We focus on the (l,u)-partition problem, which is the problem of partitioning a weighted graph into connected clusters such that each…

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…

In the Stable Roommates problem, we seek a stable matching of the agents into pairs, in which no two agents have an incentive to deviate from their assignment. It is well known that a stable matching is unlikely to exist, but a stable…

It is well known that, whenever $k$ divides $n$, the complete $k$-uniform hypergraph on $n$ vertices can be partitioned into disjoint perfect matchings. Equivalently, the set of $k$-subsets of an $n$-set can be partitioned into parallel…

We argue that 6d N=(2,0) theory on S^1 x S^3 x C_2 reduces to the 2d q-deformed Yang-Mills on C_2 at finite area, as a small extension to the result of Gadde, Rastelli, Razamat and Yan. This is done by computing the partition function on…

We present a semiclassical trace formula for the canonical partition function of arbitrary one-dimensional systems. The approximation is obtained via the stationary exponent method applied to the phase-space integration of the density…

A probabilistic characterization of the dominance partial order on the set of partitions is presented. This extends work in "Symmetric polynomials and symmetric mean inequalities". Electron. J. Combin., 20(3): Paper 34, 2013. Let $n$ be a…