I want to show the following.
If $\mu, \nu$ are compositions with the same parts (only rearranged) then for any $\lambda$ we have that $K_{\lambda\mu}=K_{\lambda\nu}$.
I know that the Kostka number $K_{\lambda\mu}$ gives the number of semi standard Young tableaux with shape $\lambda$ and content $\mu$ where the content is the decomposition $\mu=(\mu_1, \mu_2, \dots)$ where $\mu_i$ is the number of $i$'s in the given tableau.
Sagan recommends to look at the case where $\mu$ and $\nu$ differ by an adjacent transposition of parts.
So if $\mu=(\mu_1, \dots, \mu_k, \mu_{k+1}, \dots, \mu_{\ell})$ and $\nu=(\mu_1, \dots, \mu_{k+1}, \mu_{k}, \dots, \mu_{\ell})$ we want to show that $K_{\lambda\mu}$=$K_{\lambda\nu}$. I can see how this should be enough since if we have a set of $\mu_i$ we can use transpositions to permute to whatever $\nu_i$ is. However, I don't know how to prove the hint.
Thanks in advance for any hints or help!
Recall that $K_{\lambda, \mu}$ is the coefficient of the monomial $\mathbf{x}^{\mu} = x_1^{\mu_1} x_2^{\mu_2} x_3^{\mu_3} \cdots$ in the Schur function $s_\lambda$, whereas $K_{\lambda, \nu}$ is the coefficient of the monomial $\mathbf{x}^\nu$ in this Schur function. Thus, proving $K_{\lambda, \mu} = K_{\lambda, \nu}$ whenever $\mu$ and $\nu$ are permutations of each other is tantamount to proving that the Schur function $s_\lambda$ is symmetric. This is done in most texts on symmetric functions. For example: