How big are Kostka-Numbers

Question

Let $n\in\mathbf{N}$ and $\lambda=(\lambda_1,\ldots,\lambda_\ell)$ be integers such that $\sum_{i=1}^\ell\lambda_i=n$. To this partition consider the Schur-Polynomial $s_\lambda$. When expressed in terms of monomial symmetric function, we have $s_\lambda=\sum_{\mu\vdash n} K_{\lambda,\mu} m_\mu$, where $\mu$ runs through all partitions of $n$. The coefficients $K_{\lambda,\mu}$ are the so called Kostka numbers. My questions is: How big can these numbers be?

Answer 1

They grow very fast, except for very restrictive shapes. For example, if $\mu=(1,\cdots,1)$ then the Kostka numbers reduce to numbers of Standard Young tableau of shape $\lambda$. Keeping $\mu=(1,1,\cdots,1)$, if $\lambda=(n,0,\cdots,0)$, the Kostka number is always always 1.

Using hook formulas, and Stirling's formula you can get for example that the number of standard staircase shaped Young tableau with shape $\lambda=(n-1,n-2,\cdots,1)$ grows something like $\exp(n^2\log(n)/2)\approx n^{n^2/2}$, meaning that past around $n=10$ the number exceeds the estimated number of atoms in the known universe ($\approx 10^{50}$).

Answer 2

FYI you can get the Kostka numbers with the R package jack:

> KostkaNumbers(6)
              (6) (5,1) (4,2) (4,1,1) (3,3) (3,2,1) (3,1,1,1) (2,2,2) (2,2,1,1) (2,1,1,1,1) (1,1,1,1,1,1)
(6)             1     1     1       1     1       1         1       1         1           1             1
(5,1)           0     1     1       2     1       2         3       2         3           4             5
(4,2)           0     0     1       1     1       2         3       3         4           6             9
(4,1,1)         0     0     0       1     0       1         3       1         3           6            10
(3,3)           0     0     0       0     1       1         1       1         2           3             5
(3,2,1)         0     0     0       0     0       1         2       2         4           8            16
(3,1,1,1)       0     0     0       0     0       0         1       0         1           4            10
(2,2,2)         0     0     0       0     0       0         0       1         1           2             5
(2,2,1,1)       0     0     0       0     0       0         0       0         1           3             9
(2,1,1,1,1)     0     0     0       0     0       0         0       0         0           1             5
(1,1,1,1,1,1)   0     0     0       0     0       0         0       0         0           0             1