Difference Operators for Partitions and Some Applications

Abstract

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.

Polynomiality of Plancherel averages of hook-content summations for strict, doubled distinct and self-conjugate partitions

“…by Lemma 2.2 in [10]. Summing the above equalities, we get Then P (µ, g; n + 1) − P (µ, g; n) = P (µ, D t g; n).…”

“…For a usual partition λ, the outer corners (see [10,2]) are the boxes which can be removed to get a new partition. Let (α 1 , β 1 ), .…”

Polynomiality of Certain Average Weights for Oscillating Tableaux

“…For each box in the Young diagram of the partition λ, let h and c be its hook length and content respectively (see [18,26]). In a preparing paper, by applying results from the study of difference operators on functions of partitions [7,11,12], we will establish the following two explicit formulas with very complicated proofs for the average weights related to hook lengths and contents: (c 2 − j 2 ) = (2r)! (r + 1)!…”

Difference operators for partitions under the Littlewood decomposition

“…is always a polynomial of n for any k ∈ N. This conjecture was generalized and proved by Stanley [22] (see also [1,4,10,20]), and later generalized in [11,12,13]. Let Q be a symmetric function in infinitely many variables and E be a finite set with n elements.…”

“…Let g be a function of partitions and λ a partition. The difference operator D for partitions was defined in [11] as…”