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Working with the following problem Expansion in Schur function of negative binomial exponent

I need to find the expansion of $$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$ in terms of schur function. In Macdonald's book, page 47 problem 10 gives an expression for $$s_{\lambda}(x_1 + 1 , x_2 +1, \ldots, x_n +1) = \sum_{\mu\subset \lambda}d_{\lambda \mu}s_{\mu}(x_1, x_2, \ldots x_n) $$ where $d_{\lambda \mu} = det\left(\binom{\lambda_i+n-i}{\mu_i +i-j} \right)_{1\leq i,j\leq 1}$. So can we get an expression in we replace 1 by $y$? Also I wonder about the remark 3 made in page 74 facts that true in $\lambda$ ring if $S^{\lambda \setminus \mu} = \sum_{\nu}c^{\lambda}_{\mu \nu}S^{\nu}$ then $$ S^{\lambda}(x+y) = \sum_{\lambda} S^{\lambda \setminus \mu}(x) S^{\mu}(y)\tag{**} $$

So I am wondering if it's true in the case of the Schur function in multivariable. We have an expression for the skew Schur function in terms of the Little Richardson coefficient that is $s_{\lambda \setminus \mu} = \sum_{\nu}c^{\lambda}_{\mu \nu}s_{\nu}$. What is the relation between $s_{\lambda \setminus \mu} $ and $d_{\lambda \mu}$ ?

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    $\begingroup$ The formula for $s_\lambda(x_1+y,\dots,x_n+y)$ is obtained by "homogenizing" the one for $s_\lambda(x_1+1,\dots,x_n+1)$, i.e., $$ s_\lambda(x_1+y,\dots,x_n+y)=\sum_{\mu\subset \lambda}d_{\lambda\mu} y^{|\lambda/\mu|}s_\mu(x_1,\dots,x_n).$$ For a simplification of this formula, see Problem 87 at klein.mit.edu/~rstan/ec/ch7supp.pdf. For your question about $S^\lambda(x+y)$, perhaps you don't understand the notation. Here $x$ is a set $x_1,x_2,\dots$ of variables, and similarly for $y$, and $S^\lambda(x+y)$ means $s_\lambda(x_1,x_2,\dots, y_1,y_2,\dots)$. $\endgroup$
    – Richard Stanley
    Commented Apr 19, 2022 at 1:28
  • $\begingroup$ I see thanks for the comment, so the problem 87 the map means you are shifting the variable $x_i$ by $x_i +1$ so if I have finitely many variables say $x_1 , \ldots x_n$ then $t$ is $n$. Hence the corresponding schur function formula $\endgroup$
    – GGT
    Commented Apr 19, 2022 at 6:50
  • $\begingroup$ So if this true then I think I can write a schur function expansion of $ \prod_{i=1}^{m}\prod_{j= 1}^{n}(1-z(x_i + y_i))^{-w} \tag{*}$ $\endgroup$
    – GGT
    Commented Apr 19, 2022 at 6:52
  • $\begingroup$ Can you please give me some easy reference for hook content formula for skew Schur function also for $f^{\lambda \ \mu}$ $\endgroup$
    – GGT
    Commented Apr 20, 2022 at 5:58
  • $\begingroup$ I am looking for the definition of a content of a skew young diagram. In problem 87 you mentioned $u\in \lambda \setminus \mu$ $c(u)$. $\endgroup$
    – GGT
    Commented Apr 20, 2022 at 6:06

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