Related papers: Arithmetic properties of $(\ell,m)$-regular colore…

Let $c_N(n)$ denotes the number of bipartitions $(\lambda, \mu)$ of a positive integer $n$ subject to the restriction that each part of $\mu$ is divisible by $N$. In this paper, we prove some congruence properties of the function $c_N(n)$…

Let $A_k(n)$ denote the set of $k$-distinct partitions of $n$, and let $B_k(n)$ be the set of $k$-regular partitions of $n$. Glaisher showed that $\# A_k(n) = \# B_k(n)$. For $k=2$, this equality yields the celebrated Euler's partition…

The Bell numbers count the number of different ways to partition a set of $n$ elements while the graphical Bell numbers count the number of non-equivalent partitions of the vertex set of a graph into stable sets. This relation between graph…

Holmsen, Kyn\v{c}l and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset…

In this note, we obtain a formula which leads to a practical and efficient method to calculate the number of partitions of n into parts not divisible by m for given natural numbers n and m.

The partition relation N \to (n)_{\ell}^k means that whenever the k-tuples of an N-element set are \ell-colored, there is a monochromatic set of size n, where a set is called monochromatic if all its k-tuples have the same color. The…

Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function $\overline{R_{\ell,\mu}}(n)$, which enumerates the overpartitions of $n$ where no part is divisible by…

Let A and M be nonempty sets of positive integers. A partition of the positive integer n with parts in A and multiplicities in M is a representation of n in the form n = \sum_{a\in A} m_a a, where m_a is in M U {0} for all a in A, and m_a…

Recently, Nadji, Ahmia and Ram\'{i}rez \cite{Nadji2025} investigate the arithmetic properties of ${\bar B}_{\ell_1,\ell_2}(n)$, the number of overpartitions where no part is divisible by $\ell_1$ or $\ell_2$ with $\gcd(\ell_1,\ell_2)$$=1$…

Recently, using modular forms and Smoot's {\tt Mathematica} implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all $n\geq 0,$ \begin{align*} A(9n+5) & \equiv 0 \pmod{3}, \\…

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a…

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…

Inspired by Andrews' 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial…

It is proved that the number of partitions of n with odd mex and k parts that aren't ones equals the number of partitions of n with nonnegative crank and k parts that aren't ones..

Andrews' $(k, i)$-singular overpartition function $\overline{C}_{k, i}(n)$ counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. In recent times, divisibility…

For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$…

We show that for all $\ell, k, n$ with $\ell \leq k/2$ and $(k-\ell)$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph $\mathcal{K}_n^{(k)}$ on $n$…

The partition functions $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, and $Q(n,m,p)$, the number of integer partitons of $n$ into exactly $m$ distinct parts with each part at most…

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…

We continue the study of the $(a,b,m)$-copartition function $\mathrm{cp}_{a,b,m}(n)$, which arose as a combinatorial generalization of Andrews' partitions with even parts below odd parts. The generating function of $\mathrm{cp}_{a,b,m}(n)$…