Combinatorial Theory

Let \(I = (i_1, \dots, i_k)\) and \(J = (j_1, \dots, j_k)\) be two length \(k\) sequences drawn from \(\{1, \dots, n \}\). We have the group algebra element \([I,J] := \sum_{w(I) = J} w \in {\mathbb {C}}[{\mathfrak{S}}_n]\) where the sum is over permutations \(w \in {\mathfrak{S}}_n\) which satisfy \(w(i_p) = j_p\) for \(p = 1, \dots, k\). We give an algorithm for evaluating irreducible characters \(\chi^\lambda: {\mathbb {C}}[{\mathfrak{S}}_n] \to {\mathbb {C}}\) of the symmetric group on the elements \([I,J]\). This algorithm is a hybrid of the classical Murnaghan-Nakayama rule and a new path Murnaghan-Nakayama rule which reflects the decomposition of a partial permutation into paths and cycles.

These results first appeared in arXiv:2206.06567, which is no longer intended for publication. We originally used the character theoretic results in this paper to prove asymptotic results on moments of certain permutation statistics restricted to conjugacy classes. A referee generously shared a combinatorial argument which is strong enough to prove these results without recourse to character theory. These results now appear in our companion paper "Moments of permutation statistics by cycle type" (to appear in Forum Math. Sigma). However, the approach in this paper are more explicit, as we demonstrate with several examples.

Mathematics Subject Classifications: 05E05, 05E10

Keywords: Symmetric group, character, partial permutation