Murnaghan–Nakayama rule

In mathematics, the Murnaghan–Nakayama rule is a combinatorial method to compute irreducible character values of the symmetric group. 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:

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

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.

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