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Probability and Combinatorics

Tuesday, March 29, 2005 - 4:30pm

Karl Mahlberg

University of Wisconsin

Location

University of Pennsylvania

DRL 4N30

In 1944, Freeman Dyson conjectured the existence of an integer-valued crank function for partitions that would provide a combinatorial proof of Ramanujan's congruence $p(11n+6) \mod{11} = 0$ by dividing the partitions of $11n+6$ into $11$ classes of equal size. Forty years later, Andrews and Garvan successfully found such a function, and proved the celebrated result that $M(m,11,11n+6) = p(11n+6)/11,$ where $M(m,N,n)$ is the number of partitions $\lambda$ of $n$ with crank equivalent to $m$ modulo $N$. The main result of this work is that for any prime $l \geq 5$, and any positive integers $j$ and $t$, there are infinitely many arithmetic progressions $An + B$ such that $M(m,l^j,An+B) \equiv 0 \pmod{l^t}$ for each $0 \leq m \leq \l^j -1$. Summing over all $m$ then provides a combinatorial proof that $p(An+B) \equiv 0 \pmod{l^t}$, as the partitions are grouped into classes whose sizes are all divisible by $\ell^j$. Furthermore, these crank congruences appear throughout the framework of known congruences for the partition function as described by Ahlgren and Ono.