Related papers: Alder-type partition inequality at the general lev…

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…

Let $P$ be a set $n$ points in a $d$-dimensional space. Tverberg's theorem says that, if $n$ is at least $(k-1)(d+1)+1$, then $P$ can be partitioned into $k$ sets whose convex hulls intersect. Partitions with this property are called {\em…

We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical Rogers-Ramanujan type identity implies the existence of…

Basis partitions are minimal partitions corresponding to successive rank vectors. We show combinatorially how basis partitions can be generated from primary partitions which are equivalent to the Rogers-Ramanujan partitions. This leads to…

A Goellnitz-Gordon partition is one in which the parts differ by at least 2, and where the inequality is strict if a part is even. Let Q_i(n) denote the number of partitions of n into distinct parts not congruent to i mod 4. By attaching…

Let $\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-4}(n)$ involving the following infinite family of…

Recently, Andrews gave a detailed study of partitions with even parts below odd parts in which only the largest even part appears an odd number of times. In this paper, we provide a combinatorial proof of the generating function identity of…

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*}…

Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal{A} \subseteq \mathcal{B}(n)$ is a non-trivial $t$-intersecting family of set partitions i.e. any two members of $\A$ have at least $t$ blocks in…

Partitions with even (respectively odd) parts distinct and all other parts unrestricted are often referred to as PED (respectively POD) partitions. In this article, we generalize these notions and study sets of partitions in which parts…

For a positive integer $r$, George Andrews proved that the set of partitions of $n$ in which odd multiplicities are at least $2r + 1$ is equinumerous with the set of partitions of $n$ in which odd parts are congruent to $2r + 1$ modulo $4r…

In this paper, we prove that every graph with average degree at least $s+t+2$ has a vertex partition into two parts, such that one part has average degree at least $s$, and the other part has average degree at least $t$. This solves a…

We give manifestly positive Andrews-Gordon type series for the level 3 standard modules of the affine Lie algebra of type $A^{(1)}_2$. We also give corresponding bipartition identities, which have representation theoretic interpretations…

We prove a family of partition identities involving integer partitions in three colors. The conditions imposed on the types of partitions appearing in these identities involve constraints that arise in the Rogers-Ramanujan and…

Gr\"unbaum's equipartition problem asked if for any measure $\mu$ on $\mathbb{R}^d$ there are always $d$ hyperplanes which divide $\mathbb{R}^d$ into $2^d$ $\mu$-equal parts. This problem is known to have a positive answer for $d\le 3$ and…

A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using theta function identities due to Ramanujan, we establish some new infinite…

In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the…

We prove two partition identities which are dual to the Rogers-Ramanujan identities. These identities are inspired by (and proved using) a correspondence between three kinds of objects: a new type of partitions (neighborly partitions),…

Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…

A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Recently, both authors…