Related papers: A note on congruences for generalized cubic partit…

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 of Keith and Zanello, we establish infinite families of congruences modulo…

Andrews, Brietzke, R\o dseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$…

In this note, we provide three new, very short proofs of two interesting congruences for Merca's partition function $a(n)$, which enumerates integer partitions where the odd parts have multiplicity at most 2. These modulo 2 congruences were…

Let $a, b, c,$ and $n$ be integers, with $a$ nonzero and $n$ at least two. Necessary and sufficient conditions on these parameters are derived which guarantee that all solutions of the congruence \[ ax^2+bx+c \equiv 0\ \textrm{mod}\ n \]…

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number…

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n$-th Bernoulli number. In this paper, we will…

We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\sum\limits_{n=0}^{p-1}\dbinom{2n}{n} \equiv\eta_{p}\mod p^{2},$$ $$\sum\limits_{n=0}^{rp-1}\dbinom{2n}{n}…

We compute the congruence class modulo 16 of the number of unique path partitions of $n$ (as defined by Olsson), thus generalising previous results by Bessenrodt, Olsson and Sellers [Ann. Combin. 13 (2013), 591-602].

We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for…

In this paper, based on Serre's $p$-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series $G_k$ in the form $$ \sum_{i=1}^n g_i(p)G_{f_i(p)}\equiv g_0(p)\mod p^N, $$ where…

Recent work by Craig, van Ittersum, and Ono constructs explicit expressions in the partition functions of MacMahon that detect the prime numbers. Furthermore, they define generalizations, the MacMahonesque functions, and prove there are…

Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of $n$ with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is…

Recently, Amdeberhan et al. proved congruences for the number of hooks of fixed even length among the set of self-conjugate partitions of an integer $n$, therefore answering positively a conjecture raised by Ballantine et al.. In this…

Let $k$ and $n$ be positive integers. Let $c\phi_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$ and $\mathrm{C}\Phi_k(q)$ be the generating function of $c\phi_{k}(n)$. In this article, we study…

Let $ p \ge 5 $ be a prime and let $ b, c \in \mathbb{Z} $. Denote by $ T_k(b,c) $ the generalized central trinomial coefficient, i.e., the coefficient of $ x^k $ in $ (x^2 + bx + c)^k $. In this paper, we establish congruences modulo $ p^3…

In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Ap\'ery numbers $A_k$. In particular, we prove that, for any $r\in\mathbb{N}$, there…

In this paper, we prove several new infinite families of Ramanujan--like congruences satisfied by the coefficients of the generating function $U_t(a,q)$ which is an extension of MacMahon's generalized sum-of-divisors function. As a…

For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for…

Andrews and Merca [J. Combin. Theory Ser. A 203 (2024), Art. 105849] recently obtained two interesting results on the sum of the parts with the same parity in the partitions of $n$ (the modulo $2$ case), the proof of which relies on…

Suppose that $a_1(n),a_2(n),...,a_s(n),m(n)$ are integer-valued polynomials in $n$ with positive leading coefficients. This paper presents Popoviciu type formulas for the generalized restricted partition function…