Estimates of five restricted partition functions that are quasi polynomials

Abstract

A function f defined on N is said to be a quasi polynomial if, f (αn + r) is a polynomial in n for each r = 0, 1,. .. , α − 1, where α is a positive integer. In this article, we show that the below given restricted partition functions are quasi polynomials: (i) a(n, k)-number of partitions of n with exactly k parts and least part being less than k, (ii) aq(n, k)-number of distinct partitions (partitions with distinct parts) of n with exactly k parts and least part being less than k, (iii) Le(n, k, m)-number of partitions of n with exactly k parts and m least parts, (iv) La(n, k, 1)-number of partitions of n with exactly k parts and one largest part and (v) d(n, k)-number of partitions of n with exactly k parts and difference between least part and largest part exceeds k − 2. Consequently, following estimates were derived: (i) a(n, k) ∼ n k−2 (k − 2)! 2 (ii) aq(n, k) ∼ n k−2 (k − 2)! 2 Communicated by Ari Laptev.

Some general results in combinatorial enumeration

“…In the literature one can find several interesting enumerative results on partitions involving quasipolynomials. We state the results of D. Zeilberger [152] and of G. E. Andrews, M. Beck and N. Robbins [8], for other quasipolynomial results see A. D. Christopher and M. D. Christober [39] and V. Jelínek and M. Klazar [79]. Let t ∈ N 0 with t ≥ 2.…”