  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 $k\in
\mathbb{N}$. 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 $K_r$ arise directly from the computation for a single
partition $\lambda$, without the summation ranging over all partitions of
size~$n$.
