Eigenvalues and Eigenvectors of the Matrix of Permutation Counts

Abstract

Define a $(n^4+n^2)/2\times (n^4+n^2)/2$ symmetric $B$. $(ij)(kl)$ is an index where $i,j,k,l\in [n]$, $(ab)$ is an unordered pair and $(kl)$ is an ordered pair when $i\neq j$, otherwise it is also an unordered pair. $B((ij)(kl),(ab)(xy))$ is equal to the number of permutations of S_n in which $\min\{i,j\}$ maps to $k$, $\max\{i,j\}$ maps to $l$, $\min\{a,b\}$ maps to $x$ and $\max\{a,b\}$ maps to $y$. We will show that $B$ has four distinct eigenvalues: $(3/2)n!$, $n(n-3)!$, $(n-1)!/(n-3)$, $2n(n-2)!$ and the corresponding eigenspace dimensions are 1, ${{n-1}\choose{2}}^2$, $({{n-1}\choose{2}}-1)^2$, $(n-1)^2$ respectively.