Recently, during the research, I came across a sum, denoted by $H(n,L)$, involving irreducible characters of the symmetric group, \begin{equation} H(n,L)\colon=\sum_{Y_{i,j,w}} \frac{\chi^{Y_{i,j,w}([2^n])} \chi^{Y_{i,j,w}}(\tau)}{\chi^{Y_{i,j,w}}([1^{2n}])} s_{Y_{i,j,w}}(1,\ldots,1). \end{equation} Notations:

Let $S_{2n}$ denote the symmetric group of all permutations over $2n$ objects. Let $\chi^{Y}$ denote the irreducible character of $S_{2n}$ labeled by Ferrers diagram (or shape) $Y$.

Let $Y_{i,j,w}$ denote a special type of Ferrers diagram having one row of $j+w+1$ boxes, one row of $j+1$ boxes, $i-1$ rows of two boxes and $w$ rows of one boxes, where $i\geq 1$, $j\geq 1$ and $i+j+w=n$.

Let $[2^n]$ and $[1^{2n}]$ denote the conjugacy classes of $S_{2n}$ having the cycle types $[2^n]$ and $[1^{2n}]$, respectively. Set $\tau = (1,\ldots,n) \circ (n+1,\ldots,2n)$, a permutation of cylce type $[n^2]$.

Let $s_{Y_{i,j,w}}(x_1,\ldots,x_L)$ denote the Schur-polynomial of $Y_{i,j,w}$ over $L\geq 2n$ indeterminants.

The sum $H(n,L)$ runs over all the Ferrers diagrams $Y_{i,j,w}$ such that $i\geq 1$, $j\geq 1$ and $i+j+w=n$.

Using the Murnaghan-Nakayama Rule, we obtain $\chi^{Y_{i,j,w}}(\tau)=2 (-1)^w$. The Schur-polynomial can be experssed as $s_{Y_{i,j,w}}(1,\ldots,1)= \prod_{(p,q)\in Y_{i,j,w}}\frac{(L-p+q)}{h_{pq}} $, where $h_{pq}$ denote the hook length at position $(p,q)$ of $ Y_{i,j,w}$. Utilizing the hook length formula, we have $\chi^{Y_{i,j,w}}([1^{2n}]) =\frac{(2n)!}{\prod_{(p,q)\in Y_{i,j,w}}h_{pq} }$. Therefore, we can simplify the sum $H(n,L)$ \begin{equation} H(n,L)=\frac{2}{(2n)!} \sum_{Y_{i,j,w}} (-1)^w \chi^{Y_{i,j,w}}([2^n]) \prod_{(p,q)\in Y_{i,j,w}} (L-p+q) \end{equation} We can also use the Murnaghan-Nakayama Rule to compute $\chi^{Y_{i,j,w}}([2^n])$. But I can't find a formula! We can also show that the character values $\chi^{Y_{i,j,w}}([2^n])$ have the property: when $n$ is odd, $\chi^{Y_{i,j,w}}([2^n])=-\chi^{Y_{j,i,w}}([2^n])$, and when $n$ is even, $\chi^{Y_{i,j,w}}([2^n])=\chi^{Y_{j,i,w}}([2^n])$. For specific $n$, the sum $H(n,L)$ is a polynomial of $L$. A few examples are listed below \begin{equation} H(2,L) = \frac{1}{6} L^2 (L^2-1) \end{equation} \begin{equation} H(3,L) = \frac{1}{20} L^3 -\frac{1}{20} L^5 \end{equation} \begin{equation} H(4,L) = \frac{1}{140} L^2 -\frac{131}{5040} L^4+\frac{47}{2520}L^6+\frac{1}{5040} L^8 \end{equation}

Questions: I am wondering if there exists a formula for these character values $\chi^{Y_{i,j,w}}([2^n])$. Are there any other ways to compute this sum $H(n,L)$ smartly? Is there a formula for the sum $H(n,L)$? Any things about these character values $\chi^{Y_{i,j,w}}([2^n])$ and sums $H(n,L)$ would also be appreciated.