Related papers: Glaisher combinatorics of regular partitions

Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition…

We generalize the well known Glaisher partition bijection result. For given positive integers n, d, both greater than 1, we provide a rich family of bijections between the set of partitions of n where at least one part is divisible by d,…

For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not…

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…

Recently, George Andrews has given a Glaisher style proof of a finite version of Euler's partition identity. We generalise this result by giving a finite version of Glaisher's partition identity. Both the generating function and bijective…

For positive integers $k, l \geq 2$, the set of $k$-regular partitions in which parts appear at most $l$ times has attracted a lot of interest in that a composition of Glaisher's mapping can be used to prove the associated partition…

Ferrers graphs and tables of partitions are treated as vectors. Matrix operations are used for simple proofs of identities concerning partitions. Interpreting partitions as vectors gives a possibility to generalize partitions on negative…

A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by Glaisher's bijection. We prove results that suggest surprising usefulness for such a…

We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by $m$, appearing fewer than $m$ times, or differing by less than $m$. We find results on their behavior and…

In this paper, we generalize recent work of Mizuhara, Sellers, and Swisher that gives a method for establishing restricted plane partition congruences based on a bounded number of calculations. Using periodicity for partition functions, our…

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a…

A partition of degree $n$ is a decomposition $n=i_1+i_2+\dots+i_q$, where ${i_1,i_2,\dots,i_q}$ are positive integers called the parts of the partition. Let $\lambda>0$ be an integer. The partition is said to be a $\lambda$--partition if…

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…

Categories of partitions are combinatorial structures arising from the representation theory of certain compact quantum groups and are linked to classical diagram algebras such as the Temperley-Lieb algebra. In this paper, we present…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…

In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) $s$-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to $0$ or…

In this paper, we proof Mizukawa-Yamada's $X-Y=C$ and $m$-tuple version of this.

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by…

In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with $k$-regular partitions. Extending the generating function for $k$-regular partitions…

Euler's classic partition identity states that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We develop a new generalization of this identity, which yields a previous…