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

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.

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.
• thank you very much, my colleague also found that these numbers are from transition matrix between $p_λ$ and $m_λ$ by calculating on sage. As i understand, these numbers do not have explicit formula as Kostka numbers? Is there any more information? Thanks. – Radmir Sep 3 '14 at 4:47
• yes, exactly, sorry for my ignorance. as I know, there is algorithm for finding meaning of $K_{\lambda\sigma}$, if $\lambda$ and $\sigma$ are given. is there smthng like this for $L_{\lambda\sigma}$? thanks. – Radmir Sep 4 '14 at 5:37