I want to know the name of or any references for a matrix in the book "The representation theory of the symmetric groups" by Gordon James $\DeclareMathOperator{\Ind}{\operatorname{Ind}}$I'm reading "The representation theory of the symmetric groups" written by Gordon James.
I found the matrix $B$ in the chapter 6 ("The character table of $S_{n}$") useful for my research.
So I want to find recent results on $B$ if there are any.
But as I don't know the name of the matrix I can't search for it directly. So I ask for the name of the matrix or any papers on $B$.
For those who have not read above book, let me explain the matrix $B$.
It is a natural object.
The irreducible representations of $S_{n}$ over $\mathbb{Q}$ are represented by the Specht modules associated to the partitions of $n$. (In the book, for a partition $\lambda$, the Specht module associated to $\lambda$ is denoted $S_{\mathbb{Q}}^{\lambda}$).
There is a partial order (dominance order) on the set of partitions of $n$.
$\lambda \lhd \mu$ if and only if for all $i \geq 1$, $\sum_{j=1}^{i} \lambda_{j} \leq \sum_{j=1}^{i}\mu_{j}$.
For a partition $\lambda$, there is a Young subgroup $G_{\lambda}$ and the induced character $\Ind_{G_{\lambda}}^{S_{n}}1$ is the regular representation on the $\lambda$-tabloids. It is denoted by $M_{\mathbb{Q}}^{\lambda}$ in the book.
In the book, we order the partitions of $n$ in the dictionary order(which is a total order) increasingly and define $B$ by
$B = ( b_{\lambda\mu} )$, where $b_{\lambda\mu}=|G_{\mu}| (\chi^{\lambda}, \Ind_{G_{\mu}}^{S_{n}}1).$
Here $\chi^{\lambda}$ is the character of $S_{\mathbb{Q}}^{\lambda}$.
It is a lower triangular matrix with 1's down the diagonal because there is a theorem that
$S_{\mathbb{Q}}^{\lambda}$ is a constituent of $M_{\mathbb{Q}}^{\mu}$ only if $\mu \lhd \lambda$. And the multiplicity of $S_{\mathbb{Q}}^{\lambda}$ in $M_{\mathbb{Q}}^{\lambda}$ is always 1.
Any references on $(\chi^{\lambda}, \Ind_{G_{\mu}}^{S_{n}}1)$ would be helpful to me.
Thank you in advance.

 A: The $B$-matrix is the Kostka matrix, but where each entry has been multiplied by a certain factor.
In Mathematica code, (using my symmetric functions package) the matrix is given by
Table[
KostkaCoefficient[lam,mu]Times@@(mu!)
,{lam,IntegerPartitions@5}
,{mu,IntegerPartitions@5}]//MatrixForm

which produces
$$
\left(
\begin{array}{ccccccc}
 120 & 24 & 12 & 6 & 4 & 2 & 1 \\
 0 & 24 & 12 & 12 & 8 & 6 & 4 \\
 0 & 0 & 12 & 6 & 8 & 6 & 5 \\
 0 & 0 & 0 & 6 & 4 & 6 & 6 \\
 0 & 0 & 0 & 0 & 4 & 4 & 5 \\
 0 & 0 & 0 & 0 & 0 & 2 & 4 \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
That is, $B_{\lambda \mu} = K_{\lambda\mu}\cdot (\mu_1! \mu_2! \dotsb \mu_\ell !)$.
Similarly, the $A$-matrix in the question is given by
Table[
  Coefficient[
    ToPowerSumBasis[CompleteHSymbol@lam]
    , PowerSumSymbol@mu] Times @@ (lam!)
  , {lam, IntegerPartitions@5}
  , {mu, IntegerPartitions@5}] // MatrixForm

so $A_{\lambda\mu} = [p_\mu]h_\lambda \cdot (\lambda!)$
, where $[p_\mu]h_\lambda $ denotes the coefficient of $p_\mu$
in $h_\lambda$. The coefficient $[p_\mu]h_\lambda $
has a combinatorial formula, using ordered brick tabloids,
see

Ömer Eğecioğlu and Jeffrey B. Remmel. Brick tabloids and the
connection matrices between bases of symmetric functions. Discrete
Applied Mathematics, 34(1-3):107–120, November 1991.

Since $K_{\lambda\mu} = [s_\mu] h_\lambda$, we can
use the matrices above to compute $[s_\mu] p_\lambda$,
which is the character value $\chi_{\lambda,\mu}$, the matrix $C$
in the same chapter.
