The number of simultaneous core partitions

2014/09/24 by Huan Xiong, Xiong, Huan
Mathematics · #05A17 #11P81 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A17 #msc:11P81

paper · pdf · doi:10.48550/arxiv.1409.7038

9 pages

arxiv created 2014/10/12 · arxiv updated 2014/10/14

Abstract

Amdeberhan conjectured that the number of (t,t+1, t+2)-core partitions is ∑_{0≤ k≤ [(t)/(2)]}(1)/(k+1)\binomt2k\binom2kk. 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 (t₁,t₂,..., t_m)-core partitions is finite if and only if gcd(t₁,t₂,..., t_m)=1, which extends Anderson's result on the finiteness of the number of (t₁,t₂)-core partitions for coprime positive integers t₁ and t₂ and thus rediscover a result of Keith and Nath with a different proof.

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