Difference Operators for Partitions and Some Applications

Guo-Niu Han (1) , Huan Xiong (1)

1 IRMA - Institut de Recherche Mathématique Avancée

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.

Domaines: Mathématiques [math]; Combinatoire [math.CO]

Guo-Niu Han, Huan Xiong. Difference Operators for Partitions and Some Applications. Annals of Combinatorics, 2018, 22 (2), pp.317-346. ⟨10.1007/s00026-018-0385-1⟩. ⟨hal-02125572⟩