Dominant residue classes concerning the summands of partitions

Dominant residue classes concerning the summands of partitions

On a phenomenon of Turán concerning the summands of partitions

Turán [12] proved that for almost all pairs of partitions of an integer, the proportion of common parts is very high, that is greater than 1 2 − ε with ε > 0 arbitrarily small. In this paper we prove that this surprising phenomenon persists when we look only at the summands in a fixed arithmetic progression.

A Theorem concerning Systems of Residue Classes

By system we mean a multi-set whose elements are unordered but may occur repeatedly. FollowingŠ. Znám [8] we use a(n) to denote the residue class { x ∈ Z : x ≡ a (mod n) }. For a system (1) A = {a i (n i)} k i=1 of residue classes, the n i are called its moduli. Definition. An integer T is said to be a covering period of (1) if it is a period of the characteristic function of the set k i=1 a i ...

Pairing conjugate partitions by residue classes

We study integer partitions in which the parts fulfill the same congruence relations with the parts of their conjugates, called conjugate–congruent partitions. The results obtained include uniqueness criteria, weight lower-bounds and enumerating generating functions. © 2007 Elsevier B.V. All rights reserved. MSC: 11P81; 11P83; 05A15

On the enumeration of partitions with summands in arithmetic progression

Enumerating formulae are constructed which count the number of partitions of a positive integer into positive summands in arithmetic progression with common difference D. These enumerating formulae (denoted pD(n)) which are given in terms of elementary divisor functions together with auxiliary arithmetic functions (to be defined) are then used to establish a known characterisation for an intege...