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Identifier

St001874: Permutations ⟶ ℤ

Values

[1] => 0

[1,2] => 0

[2,1] => 1

[1,2,3] => 0

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 3

[1,2,3,4] => 0

[1,2,4,3] => 1

[1,3,2,4] => 1

[1,3,4,2] => 1

[1,4,2,3] => 1

[1,4,3,2] => 3

[2,1,3,4] => 1

[2,1,4,3] => 2

[2,3,1,4] => 1

[2,3,4,1] => 1

[2,4,1,3] => 2

[2,4,3,1] => 3

[3,1,2,4] => 1

[3,1,4,2] => 2

[3,2,1,4] => 3

[3,2,4,1] => 3

[3,4,1,2] => 2

[3,4,2,1] => 3

[4,1,2,3] => 1

[4,1,3,2] => 3

[4,2,1,3] => 3

[4,2,3,1] => 3

[4,3,1,2] => 3

[4,3,2,1] => 6

[1,2,3,4,5] => 0

[1,2,3,5,4] => 1

[1,2,4,3,5] => 1

[1,2,4,5,3] => 1

[1,2,5,3,4] => 1

[1,2,5,4,3] => 3

[1,3,2,4,5] => 1

[1,3,2,5,4] => 2

[1,3,4,2,5] => 1

[1,3,4,5,2] => 1

[1,3,5,2,4] => 2

[1,3,5,4,2] => 3

[1,4,2,3,5] => 1

[1,4,2,5,3] => 2

[1,4,3,2,5] => 3

[1,4,3,5,2] => 3

[1,4,5,2,3] => 2

[1,4,5,3,2] => 3

[1,5,2,3,4] => 1

[1,5,2,4,3] => 3

[1,5,3,2,4] => 3

[1,5,3,4,2] => 3

[1,5,4,2,3] => 3

[1,5,4,3,2] => 6

[2,1,3,4,5] => 1

[2,1,3,5,4] => 2

[2,1,4,3,5] => 2

[2,1,4,5,3] => 2

[2,1,5,3,4] => 2

[2,1,5,4,3] => 4

[2,3,1,4,5] => 1

[2,3,1,5,4] => 2

[2,3,4,1,5] => 1

[2,3,4,5,1] => 1

[2,3,5,1,4] => 2

[2,3,5,4,1] => 3

[2,4,1,3,5] => 2

[2,4,1,5,3] => 2

[2,4,3,1,5] => 3

[2,4,3,5,1] => 3

[2,4,5,1,3] => 2

[2,4,5,3,1] => 3

[2,5,1,3,4] => 2

[2,5,1,4,3] => 4

[2,5,3,1,4] => 3

[2,5,3,4,1] => 3

[2,5,4,1,3] => 4

[2,5,4,3,1] => 6

[3,1,2,4,5] => 1

[3,1,2,5,4] => 2

[3,1,4,2,5] => 2

[3,1,4,5,2] => 2

[3,1,5,2,4] => 2

[3,1,5,4,2] => 4

[3,2,1,4,5] => 3

[3,2,1,5,4] => 4

[3,2,4,1,5] => 3

[3,2,4,5,1] => 3

[3,2,5,1,4] => 4

[3,2,5,4,1] => 4

[3,4,1,2,5] => 2

[3,4,1,5,2] => 2

[3,4,2,1,5] => 3

[3,4,2,5,1] => 3

[3,4,5,1,2] => 2

[3,4,5,2,1] => 3

[3,5,1,2,4] => 2

[3,5,1,4,2] => 4

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Description

Lusztig's a-function for the symmetric group.
Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$
by the Robinson-Schensted correspondence and
$\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$
where $Q(x)'$ is the transposed tableau.
Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$.
See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.

References

[1] Björner, A., Brenti, F. Combinatorics of Coxeter groups. zbMATH:1110.05001

Code

def statistic(x):
    Q = x.robinson_schensted()[1]
    return sum(binomial(c, 2) for c in Q.conjugate().shape())

Created

Aug 14, 2020 at 20:27 by Rene Marczinzik

Updated

Apr 26, 2023 at 14:16 by Nadia Lafreniere