I am trying to use the Garsia–Goupil formula. Fundamentally, the character polynomial satisfies $$ \chi^{(n-|\mu|, \mu)}_{1^{a_1} 2^{a_2} \cdots} = q_\mu(a_1, a_2, \ldots) \equiv q_\mu(1^{a_1} 2^{a_2} \cdots). $$ Given this notation, I can only presume that $q_\mu$ does not depend on $n$? I guess it only really makes sense to look at $\chi^\lambda_\sigma$ if both have parts summing to $n$? (Here is a precursor to my confusion.)
I'm doing it for $\ell$-cycles, ie $a_1 = n-\ell$, $a_\ell = 1$ and $a_i = 0$ for $i \notin \{1, \ell\}$. Then, the main result of that paper says that $$ q_\mu(1^{n-\ell} \ell^1) = q_\mu(1^{n-\ell}) + \sum_{\xi \in BS(\ell, \mu)} q_{\mu \setminus \xi}(1^{n-\ell}), $$ where $BS(\ell, \mu)$ is the set of border-strip tableaux in $\mu$ of length $\ell$. Note that if $k = |\mu| < \ell$, then $BS(\ell, \mu) = \emptyset$. In their notation, we only have $j = 1$ as $x_\ell = 1$; this somewhat simplifies the inner sum.
Main question: how do I evaluate these character polynomials on the right-hand side, probably in terms of dimensions $d_{\lambda'}$ for some $\lambda'$?
I'm rather confused as how to do this. I do have a citation for it (see below), but I'm a little unsure on the legitimacy of it. I want to make sure I'm doing everything completely above board!
I'm not really sure what it means to say that $$ q_\mu(1^{n-\ell}) = \chi^{(n-\ell-|\mu|, \mu)}_{(1^{n-\ell})}. $$ We assume that $\mu_1 \le n - |\mu|$, ie $\mu$ is genuinely a shape $\lambda$ without its first row. But what if $n-\ell-|\mu| < \mu_1$? Or, even worse, what if $n - \ell - |\mu| < 0$?
Garsia–Goupil do mention what to do when the 'exponent' partition isn't correctly ordered. They say about rearranging parts and changing the sign. I am only interested in the absolute value of these---I'm going to apply the triangle inequality across the whole sum to bound $|q_\mu(1^{n-\ell} \ell^1)|$. So I don't mind if the sign changes.
The paper by Bernstein which, on close analysis, turns out to have a number of mistakes, simply claims (p13) that $$ \textstyle q_\mu(1^{n-\ell}) = d_{(n-\ell-|\mu|, \mu)} \le \binom{n-\ell}{|\mu|} d_\mu. $$ I don't really see how to justify this, or what it really even means in the cases I just outlined. The fact that her paper has a number of other mistakes, ranging from minor to actually fairly major, makes me hesitant to just apply it. Moreover, can I just do the same when replacing $\mu$ with $\mu \setminus \xi$, ie $\mu$ without its $\xi$-border-strip?
