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

The polynomiality of shifted Plancherel averages for summations of contents of strict partitions were established by the authors and Matsumoto independently in 2015, which is the key to determining the limit shape of random shifted Young diagram, as explained by Matsumoto. In this paper, we prove the polynomiality of shifted Plancherel averages for summations of hook lengths of strict partitions, which is an analog of the authors and Matsumoto's results on contents of strict partitions. 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 the Plancherel averages of 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 C˜ˇ which involve doubled distinct and self-conjugate partitions. Inspired by all those previous works, we establish the polynomiality of t-Plancherel averages of some hook-content summations for doubled distinct and self-conjugate partitions.