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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)

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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.

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    $\begingroup$ It could be mentioned that the first identity is what defines the coproduct structure for the Hopf algebra of symmetric functions. $\endgroup$ – Sam Hopkins Jul 14 at 15:38

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