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.



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


