Affiliation: University of Wisconsin
Email: mahlburg@math.wisc.edu
Title Of Talk: A New Approach to Cranks
URL: will be available at http://math.wisc.edu/~mahlburg
Abstract: Most combinatorialists and number theorists are familiar with Ramanujan's celebrated congruences for the partition function - namely, that p(5n+4) mod 5 = 0, p(7n+5) mod 7 = 0, and p(11n+6) mod 11 = 0. There are many different proofs and generalizations of these formulae (including infinite families of related congruences), which follow from methods in q-series, theta functions, and modular forms to list a few. However, most of these methods do not give any combinatorial insight as to why the congruences hold. Dyson's rank function gives a simple statistic on partitions that verifies the congruences modulo 5 and 7, by showing that, for example, $$N(m,5,5n+4) = p(5n+4)/5$$ where $N(m,N,n)$ is the number of partitions of $n$ whose rank is congruent to $m$ modulo $N$, and $m$ ranges through the residues modulo 5. The mod 11 case remained conjectural for fourty years, until Andrews and Garvan found their famous crank function and showed that $$M(m,11,11n+6) = p(11n+6)/11$, where $M(m,N,n)$ is defined analogously to the case of the rank. In this talk, I present a new result that shows that the crank function provides a combinatorial proof of infinitely many congruences. The main theorem states that for any prime $l > 3$, there is an arithmetic progression $An + B$ such that $N(m,l,An+B) \equiv 0 \mod{l}$ for each $0 \leq m < l$. This provides a proof that $p(An+B) \equiv 0 \mod{l}$, as the partitions are grouped into classes whose sizes are all divisible by $l$. The proof uses the theory of modular forms in a manner similar to that found in Ono's seminal work on partition congruences.
Last update made Thu Nov 11 14:46:48 EST 2004.
Please report problems to:
frank@math.ufl.edu