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In 2009, the first author proved the Nekrasov-Okounkov formula on hooklengths for integer partitions by using an identity of Macdonald in theframework of type $\widetilde A$ affine root systems, and conjectured that somesummations over the set of all partitions of size $n$ are always polynomials in$n$. This conjecture was generalized and proved by Stanley. Recently,P\'etr\'eolle derived two Nekrasov-Okounkov type formulas for $\widetilde C$and $\widetilde C\,\check{}$ which involve doubled distinct and self-conjugatepartitions. Inspired by all those previous works, we establish thepolynomiality of some hook-content summations for doubled distinct andself-conjugate partitions.

Polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions