- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001911: Permutations ⟶ ℤ

Values

[1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,5,2,3,4] => [1,2,5,4,3] => 7

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

[1,5,3,2,4] => [1,2,5,4,3] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,5,1,3,4] => [1,2,5,4,3] => 7

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

[2,5,3,1,4] => [1,2,5,4,3] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,5,1,2,4] => [1,3,2,5,4] => 8

[3,5,1,4,2] => [1,3,2,5,4] => 8

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Description

A descent variant minus the number of inversions.
This statistic is defined for general finite crystallographic root system $\Phi$ with Weyl group $W$ as follows: Let $2\rho = \sum_{\beta \in \Phi^+} \beta = \sum_{\alpha\in\Delta}b_\alpha \alpha$ be the sum of the positive roots expressed in the simple roots.
For $w \in W$ this statistic is then
$$\operatorname{stat}(w) = \sum_{\alpha\in\Delta\,:\,w(\alpha) \in \Phi^-}b_\alpha - \ell(w)\,,$$
where the sum ranges over all descents of $w$ and $\ell(w)$ is the Coxeter length.
It was shown in [1], that for irreducible groups, it holds that
$$\sum_{w\in W} q^{\operatorname{stat}(w)} = f\prod_{\alpha \in \Delta} \frac{1-q^{b_\alpha}}{1-q^{e_\alpha}}\,,$$
where $\{ e_\alpha \mid \alpha \in \Delta\}$ are the exponents of the group and $f$ is its index of connection, i.e., the index of the root lattice inside the weight lattice.
For a permutation $\sigma \in S_n$, this becomes
$$\operatorname{stat}(\sigma) = \sum_{i \in \operatorname{Des}(\sigma)}i\cdot(n-i) - \operatorname{inv}(\sigma)\,.$$

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.