6
$\begingroup$

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.

$\endgroup$
6
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ 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. $\endgroup$ – Radmir Sep 3 '14 at 4:47
  • 1
    $\begingroup$ What do you mean by explicit formulas? To my knowledge, there are no closed-form formulas for the Kostka numbers. $\endgroup$ – Per Alexandersson Sep 3 '14 at 6:29
  • 1
    $\begingroup$ 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. $\endgroup$ – Radmir Sep 4 '14 at 5:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.