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Matching statistic: St000461
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(load all 17 compositions to match this statistic)
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 0
[1,2,3] => 3
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 4
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 0
[2,1,3,4] => 2
[2,1,4,3] => 1
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 5
[1,2,3,5,4] => 1
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 0
[1,3,2,4,5] => 3
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 0
[1,4,2,3,5] => 3
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
[1,4,5,3,2] => 0
Description
The rix statistic of a permutation.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
Matching statistic: St000873
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,2] => 2
[2,1] => [2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 3
[1,3,2] => [1,3,2] => [2,3,1] => [2,1,3] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [1,3,2] => [2,3,1] => [2,1,3] => 1
[3,1,2] => [3,1,2] => [2,1,3] => [2,3,1] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 0
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 1
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 0
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 1
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 1
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 1
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 0
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [2,3,4,1] => 0
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 0
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [2,4,3,1] => 0
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 0
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [3,4,2,1] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => 2
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 1
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => 2
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 2
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => 3
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 1
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 2
[1,4,5,3,2] => [1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 0
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,2,3,5,6,7,4] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 2
[1,2,3,5,7,6,4] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 0
[1,2,3,6,4,7,5] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 0
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [2,3,4,1,5,6,7] => ? = 3
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 0
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 0
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 1
[1,2,4,3,6,7,5] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 0
[1,2,4,5,3,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,2,4,5,6,7,3] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[1,2,4,5,7,3,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 2
[1,2,4,5,7,6,3] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 0
[1,2,4,6,3,7,5] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,2,4,6,5,7,3] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,2,4,6,7,3,5] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 2
[1,2,4,6,7,5,3] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 0
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [2,3,5,1,4,6,7] => ? = 3
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [3,2,5,1,4,6,7] => ? = 1
[1,2,4,7,5,3,6] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 1
[1,2,4,7,5,6,3] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 1
[1,2,4,7,6,3,5] => [1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [3,5,2,1,4,6,7] => ? = 0
[1,2,4,7,6,5,3] => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 0
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 1
[1,2,5,3,6,7,4] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 1
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [2,4,1,5,3,6,7] => ? = 2
[1,2,5,3,7,6,4] => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 0
[1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 1
[1,2,5,4,6,7,3] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 1
[1,2,5,4,7,3,6] => [1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [2,5,1,4,3,6,7] => ? = 2
[1,2,5,4,7,6,3] => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 0
[1,2,5,6,3,7,4] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,2,5,6,4,7,3] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 1
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 2
[1,2,5,6,7,4,3] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 0
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [2,4,5,1,3,6,7] => ? = 3
[1,2,5,7,3,6,4] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [3,2,5,1,4,6,7] => ? = 1
Description
The aix statistic of a permutation.
According to [1], this statistic on finite strings $\pi$ of integers is given as follows: let $m$ be the leftmost occurrence of the minimal entry and let $\pi = \alpha\ m\ \beta$. Then
$$
\operatorname{aix}\pi = \begin{cases}
\operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\
1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\
0 & \text{ if } \beta = \emptyset
\end{cases}\ .
$$
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