Pensieve header: The Cayley distance kernel following arXiv://2105.0287 by Corfield, Sati, and Schreiber.

n=3;
S=PermutationCycles/@Permutations@Range@n

Clear[d];
d[p_]:=d[p]=Total[Length[#]-1&/@p[[1]]];
d[p1_,p2_]:=d[PermutationProduct[p1,InversePermutation[p2]]];
d/@S

A=Table[(T^(-d[p1,p2])),{p1,S},{p2,S}];
A // MatrixForm

n=6;
S=PermutationCycles/@Permutations@Range@n;
Clear[d];
d[p_]:=d[p]=Total[Length[#]-1&/@p[[1]]];
d[p1_,p2_]:=d[PermutationProduct[p1,InversePermutation[p2]]];
A=Table[(T^(-d[p1,p2])),{p1,S},{p2,S}];
Plot[(T^(n-1))Min[Eigenvalues[A]], {T,0.5,n}]