Brandt Kronholm

Affiliation: SUNY at Albany

Email: jk174783@albany.edu

Title Of Talk: Generalized Ramanujan Congruence Properties of the Restricted Partition Function p(n,m)

Abstract: The restricted partition function p(n,m) enumerates the number of partitions of n into exactly m parts. The relationship between the unrestricted partition function p(n) and p(n,m) is clear:

p(n) = p(n,1) + p(n,2) + ... + p(n,n).

In 1919 Ramanujan observed proved the following partition congruences:

p(5n+4)== 0 (mod 5) p(7n+5)== 0 (mod7) p(11n+6)== 0 (mod 11) Ono (2000) proved that there are such congruences for p(n) modulo every prime L>3. Ramanujan further conjectured a generalization for a modulus of powers of 5, 7, and 11 which was eventually proved by Atkin in 1967.

In this talk we will discuss a Ramanujan-like congruence relation for p(n,m) where for our choice of any choice of prime power modulus there is no restriction on n. We will highlight a few of the results and techniques in the several papers (Kwong, Nijenhuis, Wilf) preceeding our main result.

If time permits, we will consider future research regarding p(n,m).

Last update made Tue Mar 9 10:04:20 EST 2010.
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