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:
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 ρi 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
- ↑ Richard Stanley, Enumerative Combinatorics, Vol. 2
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