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)



Binomial theorem for content polynomials of partitions
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.


