Murnaghan–Nakayama rule

In mathematics, the Murnaghan–Nakayama rule is a combinatorial method to compute irreducible character values of the symmetric group.^[1] There are several generalizations of this rule.

The Murnaghan–Nakayama is a combinatorial rule for computing the integers χλ
ρ. Here, λ and ρ are both integer partitions of some number k.

Theorem:

$\chi _{\rho }^{{\lambda }}=\sum _{T}(-1)^{{ht(T)}}$

where the sum is taken over all border-strip tableaux of shape λ, and type ρ. That is, each tableau T is a tableau such that

• every row and column is weakly increasing
• the integer i appears ρ_i times
• the set of squares with the number i form a border strip, that is, it is a connected skew-shape with no 2×2-square.

The height, ht(T), is the sum of the heights of the border strips in T. The height of a border strip is one less than the number of rows it touches.

References

1. Richard Stanley, Enumerative Combinatorics, Vol. 2