Coloring summands of given n-partition with given weights of colors

Question

Let $\lambda$ and $\sigma$ be partitions of $n$: $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$

Let $M_{\lambda \sigma}$ be the number of ways to colour the parts of lengths $\lambda_1,\lambda_2,\cdots,\lambda_l$ with $s$ colours such that

1) each part is coloured with exactly one colour

2) the sum of the lengths of the parts coloured with colour $i$ is $\sigma_i$.

I am intersted in

1) are these numbers known, for example related to the Kostka numbers?

2) is there any other information about these numbers $M_{\lambda \sigma}$ available? (reccurences, gf's)

These numbers are needed to find the generating function of $n$-ary commutative rooted trees.

In case the definition is not clear, here is an example.

For $\lambda=(2,2,1)$ and $\sigma=(3,2)$ we have $M_{\lambda \sigma}=2$.

This means that there are two ways to colour the parts of lengths $2$, $2$ and $1$ with two colours such that the sum of the lengths of the parts with first colour is $3$ and the sum of the lengths of the parts with the second colour is $2$.

1st way: 1st part and 3rd part are coloured with the 1st colour, 2nd part with the 2nd color.

2nd way: 2nd part and 3rd part are coloured with the 1st colour, 1st part with the 2nd colour.

Answer 1

Your number $M_{\lambda\sigma}$ seems to be the same as $L_{\lambda\mu}$, defined in (6.9) page 103 in Macdonalds book on symmetric functions (second edition) if I am not mistaken.

These numbers are coefficients in the transition matrix between power sum polynomials $p_\lambda$ and monomial symmetrix polynomials, $m_\lambda$.

The Kostka numbers give the matrix between $m_\lambda$ and $s_\lambda$, the Schur polynomials. See figure on page 104.