Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St001874
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001874: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [3,1,2] => [3,2,1] => 3
[3,2,1] => [3,2,1] => [2,3,1] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 3
[2,4,3,1] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 3
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 1
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 3
[3,4,2,1] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [4,3,2,1] => 6
[4,1,3,2] => [4,1,3,2] => [3,1,4,2] => [4,2,1,3] => 3
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => [4,3,1,2] => 3
[4,2,3,1] => [4,1,3,2] => [3,1,4,2] => [4,2,1,3] => 3
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 3
[4,3,2,1] => [4,3,2,1] => [3,4,1,2] => [3,1,4,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => 3
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 3
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => 3
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.