On the number of partitions with distinct even parts

Let ped(n) be the number of partitions of n wherein even parts are distinct (and odd parts are unrestricted). We obtain many congruences for ped(n)mod2 and mod4 by the theory of Hecke eigenforms.     

On the combinatorics of the number of even parts in all partitions with distinct parts

The Ramanujan Journal - Recently, Andrews and Merca obtained two identities concerning the number of even parts in all partitions of n into distinct parts. In this paper, we provide bijective...     

Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts

The Ramanujan Journal - Recently, Andrews and Merca considered the number of even parts in all partitions of n into distinct parts and obtained new combinatorial interpretations for this number....     

Pairs of partitions without repeated odd parts

We prove two identities related to overpartition pairs. One of them gives a generalization of an identity due to Lovejoy, which was used in a joint work by Bringmann and Lovejoy to derive congruences for overpartition pairs. We apply our two identities on pairs of partitions where each partition has no repeated odd parts. We also present three partition statistics that give combinatorial explanations to a congruence modulo 3 satisfied by these partition pairs.     

On the number of partitions of n into k different parts

We study the number of partitions of n into k different parts by constructing a generating function. As an application, we will prove mysterious identities involving convolution of divisor functions and a sum over partitions. By using a congruence property of the overpartition function, we investigate values of a certain convolution sum of two divisor functions modulo 8.     

Identities and inequalities for the $$M_2$$ M 2 -rank of partitions without repeated odd parts modulo 8

The Ramanujan Journal - In 2002, Berkovich and Garvan introduced the $$M_2$$ -rank of partitions without repeated odd parts. Let $$N_2(a, M, n)$$ denote the number of partitions of n without...     

On 3-regular partitions with odd parts distinct

We consider \(\text {pod}_3(n)\), the number of 3-regular partitions with odd parts distinct, whose generating function is

$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3(n)q^n=\frac{(-q;q^2)_\infty (q^6;q^6)_\infty }{(q^2;q^2)_\infty (-q^3;q^3)_\infty }=\frac{\psi (-q^3)}{\psi (-q)}, \end{aligned}$$

where

$$\begin{aligned} \psi (q)=\sum _{n\ge 0}q^{(n^2+n)/2}=\sum _{-\infty }^\infty q^{2n^2+n}. \end{aligned}$$

For each \(\alpha >0\), we obtain the generating function for

$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3\left( 3^{\alpha }n+\delta _\alpha \right) q^n, \end{aligned}$$

where \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha }}\) if \(\alpha \) is even, \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha +1}}\) if \(\alpha \) is odd.

We show that the sequence {\(\text {pod}_3(n)\)} satisfies the internal congruences

$$\begin{aligned} \text {pod}_3(9n+2)\equiv \text {pod}_3(n)\pmod 9, \end{aligned}$$

(0.1)

$$\begin{aligned} \text {pod}_3(27n+20)\equiv \text {pod}_3(3n+2)\pmod {27} \end{aligned}$$

(0.2)

and

$$\begin{aligned} \text {pod}_3(243n+182)\equiv \text {pod}_3(27n+20)\pmod {81}. \end{aligned}$$

(0.3)

     

Arithmetic properties of partitions with even parts distinct

In this work, we consider the function ped(n), the number of partitions of an integer n wherein the even parts are distinct (and the odd parts are unrestricted). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). We prove a number of results for ped(n) including the following: For all n≥0, $$\mathit{ped}(9n+4)\equiv0\pmod{4}$$ and $$\mathit{ped}(9n+7)\equiv0\pmod{12}.$$ Indeed, we compute appropriate generating functions from which we deduce these congruences and find, in particular, the surprising result that $$\sum_{n\geq0}\mathit{ped}(9n+7)q^n=12\frac{ (q^{2};q^{2})_\infty ^{4}(q^{3};q^{3})_\infty ^{6}(q^{4};q^{4})_\infty ^{}}{(q^{};q^{})_\infty ^{11}}.$$ We also show that ped(n) is divisible by 6 at least 1/6 of the time.     

Decomposing complete equipartite graphs into odd square‐length cycles: number of parts odd

In this article, we introduce a new technique for obtaining cycle decompositions of complete equipartite graphs from cycle decompositions of related multigraphs. We use this technique to prove that if n, m and λ are positive integers with n ≥ 3, λ≥ 3 and n and λ both odd, then the complete equipartite graph having n parts of size m admits a decomposition into cycles of length λ² whenever nm ≥ λ² and λ divides m. As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph into cycles of length p², where p is prime. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:401‐414, 2010     

On the number of partitions into primes

There is, apparently, a persistent belief that in the current state of knowledge it is not possible to obtain an asymptotic formula for the number of partitions of a number n into primes when n is large. In this paper such a formula is obtained. Since the distribution of primes can only be described accurately by the use of the logarithmic integral and a sum over zeros of the Riemann zeta-function one cannot expect the main term to involve only elementary functions. However the formula obtained, when n is replaced by a real variable, is in and is readily seen to be monotonic. Research supported by NSA grant, no. MDA904-03-1-0082.     

On the number of even and odd strings along the overpartitions of n

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 the number of odd (resp. even) strings along the overpartitions of n. We introduce m-strings and show how this new combinatorial object is related with another positivity conjecture of Andrews, Chan, Kim, and Osburn. Finally, we confirm that the positivity conjecture is also true for large enough integers.     

关于正整数奇偶分拆数的计算问题

正整数n的分拆是指将正整数n表示成一个或多个正整数的无序和,设O(n,m)表示将正整数n分拆成m个奇数之和的分拆数;e(n,m)表示将正整数n分拆成m个偶数之和的分拆数.本文用初等方法给出了将O(n,m),e(n,m)分别化为有限个O(n,2),e(n,2)的和的计算公式,进而达到计算O(n,m),e(n,m)的值.同时,还讨论了将正整数n分拆成互不相同的奇数或偶数的分拆数的相应的递推计算方法.     

A note on the total number of cycles of even and odd permutations

We prove bijectively that the total number of cycles of all even permutations of [n]={1,2,…,n} and the total number of cycles of all odd permutations of [n] differ by (−1)ⁿ(n−2)!, which was stated as an open problem by Miklós Bóna. We also prove bijectively the following more general identity:

     

Linear inequalities concerning partitions into distinct parts

The Ramanujan Journal - Linear inequalities involving Euler’s partition function p(n) have been the subject of recent studies. In this article, we consider the partition function Q(n)...     

The number of partitions of a natural number n into parts each of which is not less than m

Mathematical Notes - We present recurrence formulas for the number of partitions of a natural number n whose parts must be not less than m. A simple proof of Euler’s formula for the number of...     

Asymptotic formulas related to the $$M_2$$M2-rank of partitions without repeated odd parts

The Ramanujan Journal - We give asymptotic expansions for the moments of the $$M_2$$-rank generating function and for the $$M_2$$-rank generating function at roots of unity. For this we apply the...