Polynomiality of some hook-content summations for 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 $\tilde{A}$ 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 $\tilde{C}$ and $\tilde{C}\stackrel{ˇ}{}$ which 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.

References in corpus (3)

• Difference operators for partitions and some applications
• A Nekrasov-Okounkov Type formula for $\tilde{C}$
• Difference operators for partitions under the Littlewood decomposition

Cited by in corpus (4)

• Difference operators for partitions under the Littlewood decomposition
• Modular Nekrasov-Okounkov formulas
• On certain unimodal sequences and strict partitions
• New hook-content formulas for strict partitions