The number of simultaneous core partitions

Abstract

Amdeberhan conjectured that the number of $(t,t+1,t+2)$-core partitions is ${\sum }_{0\le k\le [\frac{t}{2}]}\frac{1}{k+1}\left(\genfrac{}{}{0ex}{}{t}{2k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)$. In this paper, we obtain the generating function of the numbers $f_t$ of $(t,t+1,...,t+p)$-core partitions. In particular, this verifies that Amdeberhan's conjecture is true. We also prove that the number of $(t_1,t_2,...,t_m)$-core partitions is finite if and only if gcd$(t_1,t_2,...,t_m)=1,$ which extends Anderson's result on the finiteness of the number of $(t_1,t_2)$-core partitions for coprime positive integers $t_1$ and $t_2$ and thus rediscover a result of Keith and Nath with a different proof.

9 pages

References in corpus (1)

• Partitions with prescribed hooksets

Cited by in corpus (1)

• On the largest size of $(t,t+1,...,t+p)$-core partitions