Abstract
A function f defined on N is said to be a quasi polynomial if, f ( α n + 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) a q ( n , k ) -number of distinct partitions (partitions with distinct parts) of n with exactly k parts and least part being less than k , (iii) L e ( n , k , m ) -number of partitions of n with exactly k parts and m least parts, (iv) L a ( 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) a q ( n , k ) ∼ n k - 2 ( k - 2 ) ! 2 (iii) L e ( n , k , m ) ∼ ( k - 1 ) ! ( k - m ) ! ( k - m - 2 ) ! n k - m - 1 (iv) L a ( n , k , 1 ) ∼ n k - 1 k ! ( k - 1 ) ! (v) d ( n , k ) ∼ n k - 1 k ! ( k - 1 ) !
Highlights
A function f defined on N is said to be a quasi polynomial if, f is a polynomial in n for each r = 0, 1, . . . , α − 1, where α is a positive integer
The notion of quasi polynomial seems to be subsist from the time of Bell [1], who proved that the partition function, pA(n)-number of partitions of n with parts from a finite set of positive integers A, is a quasi polynomial with each constituent polynomial being of degree at most |A| − 1 and quasi period being a positive common multiple of elements of A
We consider partition functions mentioned in the following definition and we show that they are quasi polynomials
Summary
The notion of quasi polynomial seems to be subsist from the time of Bell [1], who proved that the partition function, pA(n)-number of partitions of n with parts from a finite set of positive integers A, is a quasi polynomial with each constituent polynomial being of degree at most |A| − 1 and quasi period being a positive common multiple of elements of A. We consider partition functions mentioned in the following definition and we show that they are quasi polynomials This characteristic found in defined functions is impetus in deriving its estimates. (ii) The function aq(n, k) is defined to be the number of distinct partitions of n with exactly k parts and least part being less than k, when n
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
