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

Abstract

In 2009, the first author proved the Nekrasov-Okounkov formula on hook lengths for integer partitions by using an identity of Macdonald in the framework of type A affine root systems, and conjectured that some summations over the set of all partitions of size n are always polynomials in n. This conjecture was generalized and proved by Stanley. Recently, Pétréolle derived two Nekrasov-Okounkov type formulas for C and Cˇwhich involve doubled distinct and self-conjugate partitions. Inspired by all those previous works, we establish the polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions.Date: December 25, 2015. 2010 Mathematics Subject Classification. 05A15, 05A17, 05A19, 05E05, 05E10, 11P81.

Multiplication theorems for self-conjugate partitions

“…This subset of partitions has been of particular interest within the works of Pétréolle [Pét15b,Pét15a] where two Nekrasov-Okounkov type formulas for C and Cˇare derived. See also the work of Han-Xiong [HX19] or Cho-Huh-Sohn [CHS20]. The already mentioned Littlewood decomposition, when restricted to SC, also has interesting properties and can be stated as follows (see for instance [GKS90,Pét15b]):…”

Multiplication theorems for self-conjugate partitions

“…This subset of partitions has been of particular interest within the works of Pétréolle [23,24] where two Nekrasov-Okounkov type formulas for C and Cˇare derived. See also the work of Han-Xiong [16] or Cho-Huh-Sohn [7]. The already mentioned Littlewood decomposition, when restricted to SC, also has interesting properties and can be stated as follows (see for instance [11,23]):…”

On certain unimodal sequences and strict partitions

“…Example 2.6. Assume a = 6, b = 5, take ∆ = (7,8,9,10,11,11,8,7,5,5,4, 3, 1, 1) ∈ S 6,5 , then λ = φ 6 (∆) = (12, 10, 9, 6, 4, 3, 1) ∈ P 6,5 and |λ| = |∆|/2 = 45. Proof.…”

“…In view of this, it is natural to consider a generalization of (4.9) to involve BG-rank as (4.10) does. Although at this moment we are unclear how this could be done, since the hook length of strict partition is essentially defined as the usual hook length of the corresponding doubled distinct partition [7], while the BG-rank of all doubled distinct partitions are zero.…”