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Acta Arithmetica

### A binomial coefficient identity associated with Beukers' conjecture on Apéry numbers.

The Electronic Journal of Combinatorics [electronic only]

### A combinatorial approach to partitions with parts in the gaps

Acta Arithmetica

Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let ${p}_{k,m}\left(j,n\right)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p{*}_{k,m}\left(j,n\right)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p{*}_{k,m}\left(j,n\right)={p}_{k,m}\left(j,n\right)$.

### A formula for the number of solutions of a restricted linear congruence

Mathematica Bohemica

Consider the linear congruence equation ${x}_{1}+...+{x}_{k}\equiv b\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{n}^{s}\right)$ for $b\in ℤ$, $n,s\in ℕ$. Let ${\left(a,b\right)}_{s}$ denote the generalized gcd of $a$ and $b$ which is the largest ${l}^{s}$ with $l\in ℕ$ dividing $a$ and $b$ simultaneously. Let ${d}_{1},...,{d}_{\tau \left(n\right)}$ be all positive divisors of $n$. For each ${d}_{j}\mid n$, define ${𝒞}_{j,s}\left(n\right)=\left\{1\le x\le {n}^{s}:{\left(x,{n}^{s}\right)}_{s}={d}_{j}^{s}\right\}$. K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on ${x}_{i}$. We generalize their result with generalized gcd restrictions on ${x}_{i}$ and prove that for the above linear congruence, the number of solutions...

Integers

Acta Arithmetica

Integers

### Another smallest part function related to Andrews' spt function

Acta Arithmetica

We offer some relations and congruences for two interesting spt-type functions, which together form a relation to Andrews' spt function.

Integers

Acta Arithmetica

### BG-ranks and 2-cores.

The Electronic Journal of Combinatorics [electronic only]

### Bijections and congruences for generalizations of partition identities of Euler and Guy.

The Electronic Journal of Combinatorics [electronic only]

### Blocks for symmetric groups and their covering groups and quadratic forms.

Beiträge zur Algebra und Geometrie

### Computational proofs of congruences for 2-colored Frobenius partitions.

International Journal of Mathematics and Mathematical Sciences

### Congruence Properties and Density Problems for the Fourier Coefficients of Modular Forms.

Mathematica Scandinavica

Integers

Acta Arithmetica

### Congruences for the Convolution of Divisor sum function

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

### Congruences in ordered pairs of partitions.

International Journal of Mathematics and Mathematical Sciences

### Congruences involving F-partition functions.

International Journal of Mathematics and Mathematical Sciences

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