Abstract
The main result of this paper is a generalization of a conjecture of Guoniu Han, originally inspired by an identity of Nekrasov and Okounkov. Our result states that if F is any symmetric function (say over ℚ) and if
$$\Phi_n(F)=\frac{1}{n!}\sum_{\lambda\vdash n}f_\lambda^2F(h_u^2:u\in\lambda),$$
where h
u
denotes the hook length of the square u of the partition λ of n and f
λ
is the number of standard Young tableaux of shape λ, then Φ
n
(F) is a polynomial function of n. A similar result is obtained when F(h
u
2
:u∈λ) is replaced with a function that is symmetric separately in the contents c
u
of λ and the shifted parts λ
i
+n−i of λ.
