A rank of partitions with overline designated summands

Distribution of integer partitions with large number of summands

The factor c−1 √ n log n is nothing but the asymptotic value of the mean number of summands in a random partition of n, each partition of n being equally likely; cf. [17]. Note that the number of partitions of n with m 1’s is given by p(n −m) − p(n −m − 1) for 0 ≤ m ≤ n; and, consequently, p(n−m) is nothing but the number of partitions of n which have ≥ m 1’s. Thus the factor one plays an impor...

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...

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.

Limit Theorems for the Number of Summands in Integer Partitions

Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cramér-type large deviations and are proved by Mellin transform and the two-dimensional saddle-point method. Applications of these results include partitions into posi...

Dominant residue classes concerning the summands of partitions