# Binomial theorem for content polynomials of partitions

Let $$\lambda$$ be a partition, represented by a usual Young diagram in which $$1\le i\le \ell(\lambda)$$ labels the rows and, for each $$i$$, $$1\le j\le \lambda_i$$ labels the columns. For each box $$\square$$ in the diagram, $$c(\square)=j-i$$ is its content. The polynomial $$P_\lambda(x)=\prod_{\square\in\lambda}(x+c(\square))=\prod_{(i,j)\in\lambda}(x+j-i)$$ is the content polynomial.

I would like to know some family of coefficients $$b^\lambda_{\mu\nu}$$, as a function of three partitions, such that $$P_\lambda(x+y)=\sum_{\mu,\nu\subset \lambda}b^\lambda_{\mu\nu}P_\mu(x)P_\nu(y).$$

(Notice that content polynomials are not linearly independent, so this equation alone is not enough to uniquely determine the coefficients. I ask for some family)

If you lift this to the level of symmetric functions then the structure constants are uniquely determined. Suppose $$x$$ denotes a set of $$m$$ variables and $$y$$ denotes a set of $$n$$ variables. Then you can start with the identity of schur polynomials $$s_{\lambda}(x,y)=\sum_{\mu,\nu}c_{\mu,\nu}^{\lambda}s_{\mu}(x)s_{\nu}(y)$$ where $$c_{\mu,\nu}^{\lambda}$$ are the Littlewood-Richardson coefficients. Setting all the variables to $$1$$ gives $$P_{\lambda}(m+n)=\sum_{\mu,\nu}b_{\mu,\nu}^{\lambda}P_{\mu}(m)P_{\nu}(n)$$ where $$b_{\mu,\nu}^{\lambda}=c_{\mu,\nu}^{\lambda}\frac{\prod_{\square\in\lambda}h_{\square}}{\prod_{\square\in\mu}h_{\square}\prod_{\square\in\nu}h_{\square}}$$ and the $$h_{\square}$$ represent the hook lengths of each partition.