Divisibility of certain $$\ell $$ -regular partitions by 2

The Ramanujan Journal - For a positive integer $$\ell $$ , let $$b_{\ell }(n)$$ denote the number of $$\ell $$ -regular partitions of a nonnegative integer n. Motivated by some recent conjectures...     

On hybrid moments of $$\Delta _2(x)$$ Δ 2 ( x ) and $$\Delta _3(x)$$ Δ 3 ( x )

The Ramanujan Journal - Let $$k\geqslant 2$$ be a fixed natural number and $$d_k(n)$$ denote the number of ways n can be written as a product of k positive integers. We use $$\Delta _k(x)$$ to...     

On the structure of partition which the difference of their representation function is a constant

Periodica Mathematica Hungarica - Let $$\mathbb {N}$$ be the set of nonnegative integers. For any set $$A \subset \mathbb {N}$$ , let $$R_1(A, n)$$ , $$R_2(A, n)$$ and $$R_3(A, n)$$ be the number...     

New congruences modulo 2, 4, and 8 for the number of tagged parts over the partitions with designated summands

The Ramanujan Journal - Recently, Lin introduced two new partition functions $$\hbox {PD}_{\mathrm{t}}(n)$$ and $$\hbox {PDO}_{\mathrm{t}}(n)$$, which count the total number of tagged parts over...     

Proof of a conjecture on a congruence modulo 243 for overpartitions

Periodica Mathematica Hungarica - Let $$\bar{p}(n)$$ denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding $$\bar{p}(n)$$ ....     

On a divisor of the central binomial coefficient

The Ramanujan Journal - It is well known that for all $$n\ge 1$$ the number $$n+1$$ is a divisor of the central binomial coefficient $${2n\atopwithdelims ()n}$$ . Since the nth central binomial...     

On a conjecture on shifted primes with large prime factors

Archiv der Mathematik - Let $$\mathcal {P}$$ be the set of all primes and $$\pi (x)$$ be the number of primes up to x. For any $$n\ge 2$$ , let $$P^+(n)$$ be the largest prime factor of n. For...     

On some properties of the number of permutations being products of pairwise disjoint d-cycles

Monatshefte für Mathematik - Let $$d\ge 2$$ be an integer. In this paper we study arithmetic properties of the sequence $$(H_d(n))_{n\in \mathbb {N}}$$, where $$H_{d}(n)$$ is the number of...     

Asymptotics of the Number of Threshold Functions and the Singularity Probability of Random {±1}-Matrices

Doklady Mathematics - Two results concerning the number $$P(2,n)$$ of threshold functions and the singularity probability $${{\mathbb{P}}_{n}}$$ of random ( $$n \times n$$ ) $${\text{\{ }} \pm...     

A New Proof of a Result Concerning a Complete Description of $$ (n,n + 2) $$ -Graphs with Maximum Value of the Hosoya Index

Mathematical Notes - The Hosoya index is an important topological index of graphs defined as the number of their matchings. At present, for any $$ n $$ and $$ k \in \{- 1,0,1,2 \}$$ , all connected...     

An asymptotic formula for the logarithm of generalized partition functions

The Ramanujan Journal - Let $$\Lambda =\{ \lambda _j\}_{j=1}^\infty $$ be a set of positive integers, and let $$p_\Lambda (n)$$ be the number of ways of writing n as a sum of positive integers...     

On the average number of subgroups of the group $${\mathbb {Z}}_m \times {\mathbb {Z}}_n$$ Z m × Z n with k-th power sequence

The Ramanujan Journal - Let $$\mathbb {Z}_{n}$$ be the additive group of residue classes modulo n. Let s(m, n) denote the total number of subgroups of the group $$\mathbb {Z}_{m} \times...     

Congruences modulo 7 for broken 3-diamond partitions

The Ramanujan Journal - In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let $$\Delta _k(n)$$ denote the number of broken k-diamond partitions of n for a fixed...     

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 the Chromatic Numbers of Random Hypergraphs

Doklady Mathematics - The asymptotic behavior of the chromatic number of the binomial random hypergraph $$H(n,k,p)$$ is studied in the case when $$k \geqslant 4$$ is fixed, n tends to infinity, and...     

Congruences for overpartitions with restricted odd differences

The Ramanujan Journal - In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function $$\overline{t}(n)$$ to be the number of overpartitions of weight n where (i) the difference...     

Generalized Sidon sets of perfect powers

The Ramanujan Journal - For $$h \ge 2$$ and an infinite set of positive integers A, let $$R_{A,h}(n)$$ denote the number of representations of the positive integer n as the sum of h distinct terms...     

$$M_2$$M2-ranks of overpartitions modulo 6 and 10

The Ramanujan Journal - In this paper, we obtain inequalities on $$M_2$$-ranks of overpartitions modulo 6. Let $$\overline{N}_2(s,m,n)$$ be the number of overpartitions of n whose $$M_2$$-rank is...