Related papers: Counting birthday collisions using partitions

Define a "nuclear partition" to be an integer partition with no part equal to one. In this study we prove a simple formula to compute the partition function $p(n)$ by counting only the nuclear partitions of $n$, a vanishingly small subset…

In this paper, we obtain upper and lower bounds for the partition function $p(n)$ by using an elementary geometric inequality in Euclidean space, and we extend the method to generalizations of the partition function.

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…

This paper considers three different partition relations from partition calculus, two of which are pair relations and one of which is a triple relation. An examination of the first partition relation and the ramification argument used to…

We give a bijection between the set of self-conjugate partitions and that of ordinary partitions. Also, we show the relation between hook lengths of self conjugate partition and corresponding partition via the bijection. As a corollary, we…

Let $A_{t,k}(n)$ denote the number of partition $k$-tuples of $n$ where each partition is $t$-core. In this paper, we establish formulas of $A_{t,k}(n)$ for some values of $t$ and $k$ by employing the method of modular forms, which extends…

We present an involution on set partitions that interchanges two statistics related to relative size of block entries and use it to establish an equidistribution on objects counted by the Bessel numbers.

A formula which only involves a partition number and elementary functions is derived by applying Burnside's Lemma to the set of idempotent maps from a set to itself. One side involves a summation over a set closely related to the partition…

We use the celebrated circle method of Hardy and Ramanujan to develop convergent formulae for counting a restricted class of partitions that arise from the G\"ollnitz--Gordon identities.

We investigate infinite versions of vector and affine space partition results, and thus obtain examples and a counterexample for a partition problem for relational structures. In particular we provide two (related) examples of an age…

In 2003, Maroti showed that one could use the machinery of l-cores and l-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case l=2, using them to give a…

In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…

We describe efficient algorithms to search for cases in which binomial coefficients are equal or almost equal, give a conjecturally complete list of all cases where two binomial coefficients differ by 1, and give some identities for…

Alternative novel measures of the distance between any two partitions of a n-set are proposed and compared, together with a main existing one, namely 'partition-distance' D(.,.). The comparison achieves by checking their restriction to…

We give a direct deduction and proof of two identities in the theory of plane partitions. The first one is known to enumerate the traces of plane partitions. The second one comes without any combinatorial interpretation.

The restricted partition function $p_N(n)$ counts the partitions of the integer $n$ into at most $N$ parts. In the nineteenth century Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and,…

A convenient way to calculate $N$-particle quantum partition functions is by confining the particles in a weak harmonic potential instead of using a finite box or periodic boundary conditions. There is, however, a slightly different…

A scalar integer partition problem asks for a number of nonnegative integer solutions to a linear Diophantine equation with integer positive coefficients. The manuscript discusses an algorithm of derivation of linear relations involving the…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…

Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers have enumerated $(s, m s \pm 1)$-core partitions into distinct parts. We generalize their result in several directions by including the number of parts of these…