Simultaneous core partitions have attracted much attention since Anderson's work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on simultaneous core partitions with distinct parts. The generating function of $t$-core partitions with distinct parts is obtained. We also prove results on the number, the largest size and the average size of $(t, t + 1)$-core partitions with distinct parts. This gives a complete answer to a conjecture of Amdeberhan, which is partly and independently proved by Straub, Nath and Sellers, and Zaleski recently.
In this paper we prove that Amdeberhan's conjecture on the largest size of (t, t+1, t+2)-core partitions is true. We also show that the number of (t, t + 1, t + 2)-core partitions with the largest size is 1 or 2 based on the parity of t. More generally, the largest size of (t, t + 1, . . . , t + p)-core partitions and the number of such partitions with the largest size are determined.1991 Mathematics Subject Classification. 05A17, 11P81.
Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the 2k-th power sum of hook lengths of partitions with size n is always a polynomial of n for any k ∈ N. This conjecture was generalized and proved by Stanley (Ramanujan J., 23 (1-3) : 91-105, 2010). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators D and D − defined on functions of partitions. Even though the calculations for higher orders of D are extremely complex, we prove that several well-known families of functions of partitions are annihilated by a power of the difference operator D. As an application, our results lead to several generalizations of classic results on partitions, including the marked hook formula, Stanley Theorem, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula. We insist that the Okada constants Kr arise directly from the computation for a single partition λ, without the summation ranging over all partitions of size n.
We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the hook length and content statistics. As an application, several new hook-content formulas for strict partitions are derived.h 2k
Inspired by the works of Dewar, Murty and Kotěšovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylindric partitions. We prove that the order of the asymptotic formula for the skew plane partitions of fixed width depends only on the width of the region, not on the profile (the skew zone) itself, while this is not true for cylindric partitions.
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