Related papers: On the Enumeration and Congruences for m-ary Parti…

In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to…

Let $a$ and $m>0$ be integers. We show that for any integer $b$ relatively prime to $m$, the set $\{a^n+bn:\ n=1,\ldots,m^2\}$ contains a complete system of residues modulo $m$. We also pose several conjectures for further research; for…

For any positive integer $m$, let $\mathbb{Z}_{m}$ be the set of residue classes modulo $m$. For $A\subseteq \mathbb{Z}_{m}$ and $\overline{n}\in \mathbb{Z}_{m}$, let representation function $R_{A}(\overline{n})$ denote the number of…

Beck introduced two partition statistics $NT(r,m,n)$ and $M_{\omega}(r,m,n)$,which denote the total number of parts in the partition of $n$ with rank congruent to $r$ modulo $m$ and the total number of ones in the partition of $n$ with…

Given a sequence A=(a1,...,an) of real numbers, a block B of the A is either a set B={ai,...,aj} where i<=j or the empty set. The size b of a block B is the sum of its elements. We show that when 0<=ai<=1 and k is a positive integer, there…

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$…

For a positive rational $\alpha$, call a set of distinct positive integers $\{a_1, a_2, \ldots, a_r\}$ an $\alpha$-partition of $n$, if the sum of the $a_i$ is equal to $n$ and the sum of the reciprocals of the $a_i$ is equal to $\alpha$.…

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part…

In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer $n$ with even crank and those with odd crank, denoted $C(n)=c_e(n)-c_o(n)$. Inspired by Ramanujan's classical…

The feedback class of a locally Brunovsky linear system is fully determined by the decomposition of state space as direct sum of system invariants [4]. In this paper we attack the problem of enumerating all feedback classes of locally…

For any positive integer $m$, let $\mathbb{Z}_{m}$ be the set of residue classes modulo $m$. For $A\subseteq \mathbb{Z}_{m}$ and $\overline{n}\in \mathbb{Z}_{m}$, let $R_{A}(\overline{n})$ denote the number of solutions of…

Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the…

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove…

A partition on [n] has an m-nesting if there exists i_1 < i_2 < ... < i_m < j_m < j_{m-1} < ... < j_1, where i_l and j_l are in the same block for all 1 <= l <= m. We use generating trees to construct the class of partitions with no…

In this paper, we study partitions of totally positive integral elements $\alpha$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper…

In this paper we present an extension of Stanley's theorem related to partitions of positive integers. Stanley's theorem states a relation between "the sum of the numbers of distinct members in the partitions of a positive integer $n$" and…

We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m.…

Let $M$ be a positive integer and $p(n)$ be the number of partitions of a positive integer $n$. Newman's Conjecture asserts that for each integer $r$, there are infinitely many positive integers $n$ such that \[ p(n)\equiv r \pmod{M}. \]…

Let ${{B}_{3}}(n)$ denote the number of partition triples of $n$ where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for…

Recently, Andrews, Chan, Kim and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture $A_k (n) \geq B_k (n)$ holds for large enough positive integers n, where A_k(n) (resp. B_k(n)) is…