Let $C_{\lambda,\mu}$ be the coefficients defined as 
$$ s_\lambda\left(\frac{x_1}{1-x_1},...,\frac{x_N}{1-x_N}\right)=\sum_{\mu\supset \lambda}C_{\lambda\mu}s_\mu(x_1,...,x_N),$$
where $s$ are the Schur polynomials.

These coefficients have an explicit form in terms of a determinant of binomial coefficients,
$$ C_{\lambda\mu}=\det\left[\binom{\mu_j-j}{\lambda_i-i}\right].$$

Given a partition $\lambda$, the set $\{j-i,1\le j\le \lambda_i,1\le i\le \ell(\lambda)\}$ is its content set. Let $t_\lambda$ be the product of all non-zero contents of $\lambda$. 

I have met the following identity which I believe to be true:
$$ \sum_{\mu\supset\lambda}C_{\lambda\mu}\frac{t_\mu^2}{s_\mu(1^N)}=\frac{t_\lambda^2}{s_{\lambda'}(1^N)},$$
where $\lambda'$ is the conjugate partition to $\lambda$.

Anyone seen this before?