منابع مشابه
On a Multiplicative Partition Function
Let D(s) = ∑∞ m=1 amm −s be the Dirichlet series generated by the infinite product ∏∞ k=2(1− k−s). The value of am for squarefree integers m with n prime factors depends only on the number n, and we let f(n) denote this value. We prove an asymptotic estimate for f(n) which allows us to solve several problems raised in a recent paper by M. V. Subbarao and A. Verma.
متن کاملOn Stanley's Partition Function
Stanley defined a partition function t(n) as the number of partitions λ of n such that the number of odd parts of λ is congruent to the number of odd parts of the conjugate partition λ modulo 4. We show that t(n) equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p(n)− t(n). As a consequen...
متن کاملA Combinatorial Proof on Partition Function Parity
One of the most basic results on the number-theoretic properties of the partition function p(n) is that p(n) takes each value of parity infinitely often. First proved by Kolberg in 1959, this statement was strengthened by Kolberg and Subbarao in 1966 to say that both p(2n) and p(2n + 1) take each value of parity infinitely often. These results have received several other proofs, each relying to...
متن کاملOn a Partition Function of Richard Stanley
In this paper, we examine partitions π classified according to the number r(π) of odd parts in π and s(π) the number of odd parts in π′, the conjugate of π. The generating function for such partitions is obtained when the parts of π are all 5 N . From this a variety of corollaries follow including a Ramanujan type congruence for Stanley’s partition function t(n).
متن کاملcompactifications and function spaces on weighted semigruops
chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main fea...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2000
ISSN: 1077-8926
DOI: 10.37236/1563