Affiliation: University of Wisconsin
Email: anderson@math.wisc.edu
Title Of Talk: An Asymptotic Formula for the $t$-core Partition Function and a Conjecture of Stanton
Abstract: For a positive integer $t$, a partition is said to be a $t$-core if each of the hook numbers from its Ferrers-Young diagram is not a multiple of $t$. In 1996, Granville and Ono proved the $t$-core partition conjecture, that $a_t(n)$, the number of $t$-core partitions of $n$, is positive for every non-negative integer $n$ as long as $t\geq 4$. As part of their proof, they show that if $p\geq 5$ is prime, the generating function for $a_p(n)$ is essentially a multiple of an explicit Eisenstein Series together with a cusp form. This representation of the generating function leads to an asymptotic formula for $a_p(n)$ involving $L$-functions and divisor functions. In 1999, Stanton conjectured that for $t\geq 4$ and $n\geq t+1$, $a_t(n)\leq a_{t+1}(n)$. Here we prove a weaker form of this conjecture, that for $t\geq 4$ and $n$ sufficiently large, $a_t(n)\leq a_{t+1}(n)$. Along the way, we obtain an asymptotic formula for $a_t(n)$ which, in the cases where $t$ is coprime to 6, is a generalization of the formula which follows from the work of Granville and Ono when $t=p\geq 5$ is prime.
Last update made Thu Nov 11 14:45:57 EST 2004.
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