It is well-known that the Littlewood-Richardson coefficient $c^{\nu}\_{\lambda \mu}$ is the number of times the irreducible representation $V\_\lambda \bigotimes V\_\mu$ of the product of symmetric groups $S\_{|\lambda|} × S\_{|\mu|}$ appears in the restriction of the representation $V\_{\nu}$ of $S\_{|\nu|}$ to $S\_{|\lambda|} × S\_{|\mu|}$. 


It is also known that the irreducible representations of the hyperoctahedral group $H\_t := (S\_2 \wr S\_t) \leq S\_{2t}$ are indexed by pairs of partitions $(\alpha,\beta)$ such that $|\alpha| + |\beta| = t$.


Let $\nu \vdash 2n$, $\lambda \vdash 2(n-t)$, and let $c^{\nu}\_{\lambda(\alpha,\beta)}$ be the number of times the irreducible representation $V\_\lambda \bigotimes V\_{(\alpha,\beta)}$ of the $S\_{|\lambda|} × H\_t$ appears in the restriction of the representation $V\_{\nu}$ of $S\_{2n}$ to $S\_{|\lambda|} \times H\_t$. My question is the following:


Is there a branching/Littlewood-Richardson-type rule for computing $c^{\nu}\_{\lambda(\alpha,\beta)}$ in the literature?


By the Littlewood-Richardson rule, transitivity of induction, and Frobenius reciprocity, a branching rule for restricting from $S\_{2(n-t)} \times S\_{2t}$ to $S\_{2(n-t)} \times H\_t$ would also do the trick. This is quite similar to an [older post](https://mathoverflow.net/questions/48532/branching-rule-from-symmetric-group-s-2n-to-hyperoctahedral-group-h-n) of Steven Sam, but unfortunately I do not have access to the reference that answers his question. 


I might add that I am really only interested in whether the trivial representation of 
$S\_{|\lambda|} \times H\_t$ appears when an irreducible representation $V\_{\nu}$ of $S\_{2n}$ is restricted to $S\_{|\lambda|} \times H\_t$. I'm not a rep. theorist/algebraist, so if there is some easier way of determining this, I would be happy to know.



Littlewood-Richardson-Type Rule for Restriction from $S_{2n}$ to $S_{2(n-t)} \times (S_2 \wr S_t)$
For the trivial representation the problem is much simplified. Let $1\_G$ denote the trivial character of a group $G$. It is well known that $1\_{S\_2 \wr S\_t}\!\!\uparrow^{S\_{2t}} = \sum\_{\mu \vdash t} \chi^{2\mu}$ where $2\mu$ is the partition obtained from $\mu$ by doubling the length of each part. 


By two applications of Frobenius reciprocity, we have


$$ \begin{align\*} \bigl\langle \chi^\nu \!\!\downarrow\_{S\_{2(n-t)} \times S\_2 \wr S\_t}, 1\_{S\_{2(n-t)}} \times 1\_{S\_2 \wr S\_t} \bigr\rangle &= \bigl\langle \chi^\nu \!\!\downarrow\_{S\_{2(n-t)} \times S\_{2t}}, 1\_{S\_{2(n-t)}} \times 1\_{S\_2 \wr S\_t}\!\!\uparrow^{S\_{2t}} \bigr\rangle \\
&= \bigl\langle \chi^\nu, \sum\_{\mu \vdash t} \bigl( 1\_{S\_{2(n-t)}} \times \chi^{2\mu} \bigr)\!\!\uparrow^{S\_{2n}} \bigr\rangle \end{align\*}
$$


Hence by Young's rule (or the more general Littlewood—Richardson rule), the multiplicity of the trivial character in $\chi^\nu \downarrow\_{S\_{2(n-t)} \times S\_2 \wr S\_t}$ is equal to the number of partitions $\mu$ of $t$ such that the Young diagram of $\nu$ can be obtained from the Young diagram of $2\mu$ by adding $2(n-t)$ boxes, no two in the same column.


