Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000541
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 0
[2,3,1] => [1,3,2] => [2,3,1] => [3,1,2] => 1
[3,1,2] => [3,1,2] => [2,1,3] => [3,2,1] => 2
[3,2,1] => [3,2,1] => [1,2,3] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 0
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [4,3,1,2] => 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 1
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 0
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 2
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 1
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,4,3,2] => 0
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,3,4,2] => 0
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 2
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 1
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [4,3,2,1] => 3
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [4,2,3,1] => 2
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 2
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [3,2,4,1] => 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 0
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [5,4,1,2,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,5,3,4] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 1
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 0
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 2
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,5,4,2,3] => 0
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,4,5,2,3] => 0
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 1
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 2
[1,4,5,3,2] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000542
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[3,1,2] => [3,1,2] => [2,1,3] => [3,2,1] => 3 = 2 + 1
[3,2,1] => [3,2,1] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1 = 0 + 1
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [4,3,1,2] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2 = 1 + 1
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1 = 0 + 1
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 3 = 2 + 1
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,4,3,2] => 1 = 0 + 1
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,3,4,2] => 1 = 0 + 1
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2 = 1 + 1
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 3 = 2 + 1
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [4,3,2,1] => 4 = 3 + 1
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 3 = 2 + 1
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [4,2,3,1] => 3 = 2 + 1
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 3 = 2 + 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [3,2,4,1] => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [5,4,1,2,3] => 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,5,3,4] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 2 = 1 + 1
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 3 = 2 + 1
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,5,4,2,3] => 1 = 0 + 1
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,4,5,2,3] => 1 = 0 + 1
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 2 = 1 + 1
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 3 = 2 + 1
[1,4,5,3,2] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 2 = 1 + 1
Description
The number of left-to-right-minima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,3,1] => [1,2,3] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[3,2,1] => [3,2,1] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [4,2,1,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1,4] => 1 = 0 + 1
[1,3,4,2] => [1,2,4,3] => [4,3,1,2] => [4,2,1,3] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [4,3,1,2] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => [3,1,2,4] => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,2,3] => 2 = 1 + 1
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1,4] => 1 = 0 + 1
[2,3,4,1] => [1,2,4,3] => [4,3,1,2] => [4,2,1,3] => 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 3 = 2 + 1
[2,4,3,1] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [2,4,3,1] => [1,3,2,4] => 1 = 0 + 1
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [4,1,2,3] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [4,1,2,3] => 2 = 1 + 1
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 3 = 2 + 1
[3,4,2,1] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => [3,4,2,1] => 3 = 2 + 1
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => [4,2,3,1] => 3 = 2 + 1
[4,2,3,1] => [4,1,3,2] => [1,4,2,3] => [3,4,2,1] => 3 = 2 + 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [2,4,3,1] => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,3,2,1,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [3,4,2,1,5] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,5,4] => [5,4,3,1,2] => [5,3,2,1,4] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [5,4,2,1,3] => 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [4,5,2,1,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [4,2,3,1,5] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [5,2,3,1,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [3,4,2,1,5] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,5,4] => [5,4,3,1,2] => [5,3,2,1,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [5,2,4,1,3] => 3 = 2 + 1
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => [4,5,2,1,3] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [2,4,3,1,5] => 1 = 0 + 1
[1,4,2,5,3] => [1,3,2,5,4] => [5,3,4,1,2] => [5,2,3,1,4] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1,5] => 1 = 0 + 1
[1,4,3,5,2] => [1,3,2,5,4] => [5,3,4,1,2] => [5,2,3,1,4] => 2 = 1 + 1
[1,4,5,2,3] => [1,3,5,2,4] => [5,3,1,4,2] => [5,2,4,1,3] => 3 = 2 + 1
[1,4,5,3,2] => [1,2,5,4,3] => [5,4,1,2,3] => [4,5,2,1,3] => 2 = 1 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [6,4,5,3,2,1,7] => ? = 0 + 1
[1,2,4,5,3,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [6,4,5,3,2,1,7] => ? = 0 + 1
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => [6,5,4,2,3,1,7] => ? = 0 + 1
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0 + 1
[1,3,4,5,2,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [6,4,5,3,2,1,7] => ? = 0 + 1
[1,3,5,2,4,6,7] => [1,3,5,2,4,6,7] => [7,5,3,6,4,2,1] => [6,3,5,2,4,1,7] => ? = 0 + 1
[1,4,2,3,5,6,7] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => [6,5,2,4,3,1,7] => ? = 0 + 1
[1,4,2,5,3,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0 + 1
[1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [7,4,5,6,3,2,1] => [6,5,2,3,4,1,7] => ? = 0 + 1
[1,4,3,5,2,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0 + 1
[1,4,5,2,3,6,7] => [1,3,5,2,4,6,7] => [7,5,3,6,4,2,1] => [6,3,5,2,4,1,7] => ? = 0 + 1
[1,5,3,2,4,6,7] => [1,5,3,2,4,6,7] => [7,3,5,6,4,2,1] => [6,2,5,3,4,1,7] => ? = 0 + 1
[1,5,4,2,3,6,7] => [1,5,4,2,3,6,7] => [7,3,4,6,5,2,1] => [6,2,3,5,4,1,7] => ? = 0 + 1
[1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [7,3,4,5,6,2,1] => [6,2,3,4,5,1,7] => ? = 0 + 1
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => [6,5,4,3,1,2,7] => ? = 0 + 1
[2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [6,7,5,3,4,2,1] => [6,4,5,3,1,2,7] => ? = 0 + 1
[2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [6,7,4,5,3,2,1] => [6,5,3,4,1,2,7] => ? = 0 + 1
[2,1,4,5,3,6,7] => [2,1,3,5,4,6,7] => [6,7,5,3,4,2,1] => [6,4,5,3,1,2,7] => ? = 0 + 1
[2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => [6,7,3,5,4,2,1] => [6,3,5,4,1,2,7] => ? = 0 + 1
[2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => [6,7,3,4,5,2,1] => [6,3,4,5,1,2,7] => ? = 0 + 1
[2,3,1,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => [6,5,4,2,3,1,7] => ? = 0 + 1
[2,3,1,5,4,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0 + 1
[2,3,4,5,1,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [6,4,5,3,2,1,7] => ? = 0 + 1
[2,3,5,1,4,6,7] => [1,3,5,2,4,6,7] => [7,5,3,6,4,2,1] => [6,3,5,2,4,1,7] => ? = 0 + 1
[2,4,1,3,5,6,7] => [2,4,1,3,5,6,7] => [6,4,7,5,3,2,1] => [6,5,2,4,1,3,7] => ? = 0 + 1
[2,4,1,5,3,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0 + 1
[2,4,3,1,5,6,7] => [1,4,3,2,5,6,7] => [7,4,5,6,3,2,1] => [6,5,2,3,4,1,7] => ? = 0 + 1
[2,4,3,5,1,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0 + 1
[2,4,5,1,3,6,7] => [1,3,5,2,4,6,7] => [7,5,3,6,4,2,1] => [6,3,5,2,4,1,7] => ? = 0 + 1
[2,5,1,3,4,6,7] => [2,5,1,3,4,6,7] => [6,3,7,5,4,2,1] => [6,2,5,4,1,3,7] => ? = 0 + 1
[2,5,1,4,3,6,7] => [2,5,1,4,3,6,7] => [6,3,7,4,5,2,1] => [6,2,4,5,1,3,7] => ? = 0 + 1
[2,5,3,1,4,6,7] => [1,5,3,2,4,6,7] => [7,3,5,6,4,2,1] => [6,2,5,3,4,1,7] => ? = 0 + 1
[2,5,4,1,3,6,7] => [2,5,4,1,3,6,7] => [6,3,4,7,5,2,1] => [6,2,3,5,1,4,7] => ? = 0 + 1
[2,5,4,3,1,6,7] => [1,5,4,3,2,6,7] => [7,3,4,5,6,2,1] => [6,2,3,4,5,1,7] => ? = 0 + 1
[3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => [6,5,4,1,3,2,7] => ? = 0 + 1
[3,1,2,5,4,6,7] => [3,1,2,5,4,6,7] => [5,7,6,3,4,2,1] => [6,4,5,1,3,2,7] => ? = 0 + 1
[3,1,4,2,5,6,7] => [2,1,4,3,5,6,7] => [6,7,4,5,3,2,1] => [6,5,3,4,1,2,7] => ? = 0 + 1
[3,1,4,5,2,6,7] => [2,1,3,5,4,6,7] => [6,7,5,3,4,2,1] => [6,4,5,3,1,2,7] => ? = 0 + 1
[3,1,5,2,4,6,7] => [3,1,5,2,4,6,7] => [5,7,3,6,4,2,1] => [6,3,5,1,4,2,7] => ? = 0 + 1
[3,1,5,4,2,6,7] => [2,1,5,4,3,6,7] => [6,7,3,4,5,2,1] => [6,3,4,5,1,2,7] => ? = 0 + 1
[3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => [6,5,4,1,2,3,7] => ? = 0 + 1
[3,2,1,5,4,6,7] => [3,2,1,5,4,6,7] => [5,6,7,3,4,2,1] => [6,4,5,1,2,3,7] => ? = 0 + 1
[3,2,4,1,5,6,7] => [2,1,4,3,5,6,7] => [6,7,4,5,3,2,1] => [6,5,3,4,1,2,7] => ? = 0 + 1
[3,2,4,5,1,6,7] => [2,1,3,5,4,6,7] => [6,7,5,3,4,2,1] => [6,4,5,3,1,2,7] => ? = 0 + 1
[3,2,5,1,4,6,7] => [3,2,5,1,4,6,7] => [5,6,3,7,4,2,1] => [6,3,5,1,2,4,7] => ? = 0 + 1
[3,2,5,4,1,6,7] => [2,1,5,4,3,6,7] => [6,7,3,4,5,2,1] => [6,3,4,5,1,2,7] => ? = 0 + 1
[3,4,1,2,5,6,7] => [2,4,1,3,5,6,7] => [6,4,7,5,3,2,1] => [6,5,2,4,1,3,7] => ? = 0 + 1
[3,4,1,5,2,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0 + 1
[3,4,2,1,5,6,7] => [1,4,3,2,5,6,7] => [7,4,5,6,3,2,1] => [6,5,2,3,4,1,7] => ? = 0 + 1
[3,4,2,5,1,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [6,4,5,2,3,1,7] => ? = 0 + 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St001390
(load all 3 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[2,1,3] => [3,1,2] => [1,3,2] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [2,1,3] => 2 = 1 + 1
[3,1,2] => [2,1,3] => [3,2,1] => 3 = 2 + 1
[3,2,1] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 2 = 1 + 1
[1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 2 = 1 + 1
[1,4,2,3] => [3,2,4,1] => [4,3,1,2] => 3 = 2 + 1
[1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 2 = 1 + 1
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2 = 1 + 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => 1 = 0 + 1
[2,3,4,1] => [1,4,3,2] => [2,1,3,4] => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 3 = 2 + 1
[2,4,3,1] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [1,4,3,2] => 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => [4,1,2,3] => [1,3,4,2] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [2,1,4,3] => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [3,2,1,4] => 3 = 2 + 1
[3,4,2,1] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[4,1,2,3] => [3,2,1,4] => [4,3,2,1] => 4 = 3 + 1
[4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 3 = 2 + 1
[4,2,1,3] => [3,1,2,4] => [4,2,3,1] => 3 = 2 + 1
[4,2,3,1] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[4,3,1,2] => [2,1,3,4] => [3,2,4,1] => 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 2 = 1 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [4,1,2,3,5] => 2 = 1 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,1,2,3] => 3 = 2 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 2 = 1 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,5,3,4] => 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 2 = 1 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => 1 = 0 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [3,1,2,4,5] => 2 = 1 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 3 = 2 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,5,1,2,4] => 2 = 1 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,5,4,2,3] => 1 = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [4,1,5,2,3] => 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,4,5,2,3] => 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [3,1,5,2,4] => 2 = 1 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,1,2,5] => 3 = 2 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [3,4,1,2,5] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 0 + 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,7,3,4,5,6] => ? = 0 + 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,7,4,5,6] => ? = 0 + 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,6,3,4,5,7] => ? = 0 + 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,7,6,3,4,5] => ? = 0 + 1
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [1,2,6,7,3,4,5] => ? = 0 + 1
[1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [1,2,3,4,7,5,6] => ? = 0 + 1
[1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => [1,2,7,3,6,4,5] => ? = 0 + 1
[1,3,4,2,5,6,7] => [7,6,5,2,4,3,1] => [1,2,3,6,4,5,7] => ? = 0 + 1
[1,3,4,5,2,6,7] => [7,6,2,5,4,3,1] => [1,2,5,3,4,6,7] => ? = 0 + 1
[1,3,5,2,4,6,7] => [7,6,4,2,5,3,1] => [1,2,7,5,3,4,6] => ? = 0 + 1
[1,3,5,4,2,6,7] => [7,6,2,4,5,3,1] => [1,2,5,7,3,4,6] => ? = 0 + 1
[1,4,2,3,5,6,7] => [7,6,5,3,2,4,1] => [1,2,3,7,6,4,5] => ? = 0 + 1
[1,4,2,5,3,6,7] => [7,6,3,5,2,4,1] => [1,2,6,3,7,4,5] => ? = 0 + 1
[1,4,3,2,5,6,7] => [7,6,5,2,3,4,1] => [1,2,3,6,7,4,5] => ? = 0 + 1
[1,4,3,5,2,6,7] => [7,6,2,5,3,4,1] => [1,2,5,3,7,4,6] => ? = 0 + 1
[1,4,5,2,3,6,7] => [7,6,3,2,5,4,1] => [1,2,6,5,3,4,7] => ? = 0 + 1
[1,4,5,3,2,6,7] => [7,6,2,3,5,4,1] => [1,2,5,6,3,4,7] => ? = 0 + 1
[1,5,2,3,4,6,7] => [7,6,4,3,2,5,1] => [1,2,7,6,5,3,4] => ? = 0 + 1
[1,5,2,4,3,6,7] => [7,6,3,4,2,5,1] => [1,2,6,7,5,3,4] => ? = 0 + 1
[1,5,3,2,4,6,7] => [7,6,4,2,3,5,1] => [1,2,7,5,6,3,4] => ? = 0 + 1
[1,5,3,4,2,6,7] => [7,6,2,4,3,5,1] => [1,2,5,7,6,3,4] => ? = 0 + 1
[1,5,4,2,3,6,7] => [7,6,3,2,4,5,1] => [1,2,6,5,7,3,4] => ? = 0 + 1
[1,5,4,3,2,6,7] => [7,6,2,3,4,5,1] => [1,2,5,6,7,3,4] => ? = 0 + 1
[1,6,5,4,3,2,7] => [7,2,3,4,5,6,1] => [1,4,5,6,7,2,3] => ? = 0 + 1
[2,1,3,4,5,6,7] => [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => ? = 0 + 1
[2,1,3,5,4,6,7] => [7,6,4,5,3,1,2] => [1,2,7,3,4,6,5] => ? = 0 + 1
[2,1,4,3,5,6,7] => [7,6,5,3,4,1,2] => [1,2,3,7,4,6,5] => ? = 0 + 1
[2,1,4,5,3,6,7] => [7,6,3,5,4,1,2] => [1,2,6,3,4,7,5] => ? = 0 + 1
[2,1,5,3,4,6,7] => [7,6,4,3,5,1,2] => [1,2,7,6,3,5,4] => ? = 0 + 1
[2,1,5,4,3,6,7] => [7,6,3,4,5,1,2] => [1,2,6,7,3,5,4] => ? = 0 + 1
[2,3,1,4,5,6,7] => [7,6,5,4,1,3,2] => [1,2,3,4,6,5,7] => ? = 0 + 1
[2,3,1,5,4,6,7] => [7,6,4,5,1,3,2] => [1,2,7,3,5,4,6] => ? = 0 + 1
[2,3,4,1,5,6,7] => [7,6,5,1,4,3,2] => [1,2,3,5,4,6,7] => ? = 0 + 1
[2,3,4,5,1,6,7] => [7,6,1,5,4,3,2] => [1,2,4,3,5,6,7] => ? = 0 + 1
[2,3,5,1,4,6,7] => [7,6,4,1,5,3,2] => [1,2,7,4,3,5,6] => ? = 0 + 1
[2,3,5,4,1,6,7] => [7,6,1,4,5,3,2] => [1,2,4,7,3,5,6] => ? = 0 + 1
[2,4,1,3,5,6,7] => [7,6,5,3,1,4,2] => [1,2,3,7,5,4,6] => ? = 0 + 1
[2,4,1,5,3,6,7] => [7,6,3,5,1,4,2] => [1,2,6,3,5,4,7] => ? = 0 + 1
[2,4,3,1,5,6,7] => [7,6,5,1,3,4,2] => [1,2,3,5,7,4,6] => ? = 0 + 1
[2,4,3,5,1,6,7] => [7,6,1,5,3,4,2] => [1,2,4,3,7,5,6] => ? = 0 + 1
[2,4,5,1,3,6,7] => [7,6,3,1,5,4,2] => [1,2,6,4,3,5,7] => ? = 0 + 1
[2,4,5,3,1,6,7] => [7,6,1,3,5,4,2] => [1,2,4,6,3,5,7] => ? = 0 + 1
[2,5,1,3,4,6,7] => [7,6,4,3,1,5,2] => [1,2,7,6,4,3,5] => ? = 0 + 1
[2,5,1,4,3,6,7] => [7,6,3,4,1,5,2] => [1,2,6,7,4,3,5] => ? = 0 + 1
[2,5,3,1,4,6,7] => [7,6,4,1,3,5,2] => [1,2,7,4,6,3,5] => ? = 0 + 1
[2,5,3,4,1,6,7] => [7,6,1,4,3,5,2] => [1,2,4,7,6,3,5] => ? = 0 + 1
[2,5,4,1,3,6,7] => [7,6,3,1,4,5,2] => [1,2,6,4,7,3,5] => ? = 0 + 1
[2,5,4,3,1,6,7] => [7,6,1,3,4,5,2] => [1,2,4,6,7,3,5] => ? = 0 + 1
[2,6,5,4,3,1,7] => [7,1,3,4,5,6,2] => [1,3,5,6,7,2,4] => ? = 0 + 1
Description
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. For a given permutation $\pi$, this is the index of the row containing $\pi^{-1}(1)$ of the recording tableau of $\pi$ (obtained by [[Mp00070]]).
Matching statistic: St000991
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => [2,1,3] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => [3,2,1] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1,3] => 2 = 1 + 1
[3,1,2] => [3,1,2] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,3,2] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 1 = 0 + 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [2,1,3,4] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2 = 1 + 1
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 1 = 0 + 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [3,1,2,4] => 3 = 2 + 1
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [4,2,3,1] => 1 = 0 + 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 1 = 0 + 1
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2 = 1 + 1
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [3,1,2,4] => 3 = 2 + 1
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => [1,2,4,3] => 3 = 2 + 1
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => [1,3,2,4] => 3 = 2 + 1
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => [1,2,4,3] => 3 = 2 + 1
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [1,4,2,3] => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [2,3,1,4,5] => 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 1 + 1
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [2,4,1,3,5] => 3 = 2 + 1
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [2,5,3,4,1] => 1 = 0 + 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 1 = 0 + 1
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => [2,4,1,3,5] => 3 = 2 + 1
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => ? = 0 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => ? = 0 + 1
[1,2,4,5,3,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => ? = 0 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [2,3,6,4,5,7,1] => ? = 0 + 1
[1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [2,3,6,5,4,7,1] => ? = 0 + 1
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [2,4,3,5,6,7,1] => ? = 0 + 1
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [2,4,3,6,5,7,1] => ? = 0 + 1
[1,3,4,2,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => ? = 0 + 1
[1,3,4,5,2,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => ? = 0 + 1
[1,3,5,2,4,6,7] => [1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [2,4,6,3,5,7,1] => ? = 0 + 1
[1,3,5,4,2,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [2,3,6,5,4,7,1] => ? = 0 + 1
[1,4,2,3,5,6,7] => [1,4,2,3,5,6,7] => [1,3,4,2,5,6,7] => [2,5,3,4,6,7,1] => ? = 0 + 1
[1,4,2,5,3,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [2,4,3,6,5,7,1] => ? = 0 + 1
[1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [2,5,4,3,6,7,1] => ? = 0 + 1
[1,4,3,5,2,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [2,4,3,6,5,7,1] => ? = 0 + 1
[1,4,5,2,3,6,7] => [1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [2,4,6,3,5,7,1] => ? = 0 + 1
[1,4,5,3,2,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [2,3,6,5,4,7,1] => ? = 0 + 1
[1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => [1,3,4,5,2,6,7] => [2,6,3,4,5,7,1] => ? = 0 + 1
[1,5,2,4,3,6,7] => [1,5,2,4,3,6,7] => [1,3,5,4,2,6,7] => [2,6,3,5,4,7,1] => ? = 0 + 1
[1,5,3,2,4,6,7] => [1,5,3,2,4,6,7] => [1,4,3,5,2,6,7] => [2,6,4,3,5,7,1] => ? = 0 + 1
[1,5,3,4,2,6,7] => [1,5,2,4,3,6,7] => [1,3,5,4,2,6,7] => [2,6,3,5,4,7,1] => ? = 0 + 1
[1,5,4,2,3,6,7] => [1,5,4,2,3,6,7] => [1,4,5,3,2,6,7] => [2,6,5,3,4,7,1] => ? = 0 + 1
[1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [2,6,5,4,3,7,1] => ? = 0 + 1
[1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [2,7,6,5,4,3,1] => ? = 0 + 1
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [3,2,4,5,6,7,1] => ? = 0 + 1
[2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [3,2,4,6,5,7,1] => ? = 0 + 1
[2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [3,2,5,4,6,7,1] => ? = 0 + 1
[2,1,4,5,3,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [3,2,4,6,5,7,1] => ? = 0 + 1
[2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => [2,1,4,5,3,6,7] => [3,2,6,4,5,7,1] => ? = 0 + 1
[2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => [3,2,6,5,4,7,1] => ? = 0 + 1
[2,3,1,4,5,6,7] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [2,4,3,5,6,7,1] => ? = 0 + 1
[2,3,1,5,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [2,4,3,6,5,7,1] => ? = 0 + 1
[2,3,4,1,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,3,5,4,6,7,1] => ? = 0 + 1
[2,3,4,5,1,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => ? = 0 + 1
[2,3,5,1,4,6,7] => [1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [2,4,6,3,5,7,1] => ? = 0 + 1
[2,3,5,4,1,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [2,3,6,5,4,7,1] => ? = 0 + 1
[2,4,1,3,5,6,7] => [2,4,1,3,5,6,7] => [3,1,4,2,5,6,7] => [3,5,2,4,6,7,1] => ? = 0 + 1
[2,4,1,5,3,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [2,4,3,6,5,7,1] => ? = 0 + 1
[2,4,3,1,5,6,7] => [1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [2,5,4,3,6,7,1] => ? = 0 + 1
[2,4,3,5,1,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [2,4,3,6,5,7,1] => ? = 0 + 1
[2,4,5,1,3,6,7] => [1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [2,4,6,3,5,7,1] => ? = 0 + 1
[2,4,5,3,1,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [2,3,6,5,4,7,1] => ? = 0 + 1
[2,5,1,3,4,6,7] => [2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => [3,6,2,4,5,7,1] => ? = 0 + 1
[2,5,1,4,3,6,7] => [2,5,1,4,3,6,7] => [3,1,5,4,2,6,7] => [3,6,2,5,4,7,1] => ? = 0 + 1
[2,5,3,1,4,6,7] => [1,5,3,2,4,6,7] => [1,4,3,5,2,6,7] => [2,6,4,3,5,7,1] => ? = 0 + 1
[2,5,3,4,1,6,7] => [1,5,2,4,3,6,7] => [1,3,5,4,2,6,7] => [2,6,3,5,4,7,1] => ? = 0 + 1
[2,5,4,1,3,6,7] => [2,5,4,1,3,6,7] => [4,1,5,3,2,6,7] => [3,6,5,2,4,7,1] => ? = 0 + 1
[2,5,4,3,1,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [2,6,5,4,3,7,1] => ? = 0 + 1
[2,6,5,4,3,1,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [2,7,6,5,4,3,1] => ? = 0 + 1
Description
The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000260
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 29%●distinct values known / distinct values provided: 17%
Values
[1,2] => 1 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => 0 => [2] => ([],2)
 => ? = 1 + 1
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
 => ? = 1 + 1
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,3,1] => 00 => [3] => ([],3)
 => ? = 1 + 1
[3,1,2] => 00 => [3] => ([],3)
 => ? = 2 + 1
[3,2,1] => 00 => [3] => ([],3)
 => ? = 1 + 1
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
 => ? = 1 + 1
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
 => ? = 2 + 1
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
 => ? = 1 + 1
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,4,1] => 000 => [4] => ([],4)
 => ? = 1 + 1
[2,4,1,3] => 000 => [4] => ([],4)
 => ? = 2 + 1
[2,4,3,1] => 000 => [4] => ([],4)
 => ? = 1 + 1
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,4,2] => 000 => [4] => ([],4)
 => ? = 1 + 1
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,4,1] => 000 => [4] => ([],4)
 => ? = 1 + 1
[3,4,1,2] => 000 => [4] => ([],4)
 => ? = 2 + 1
[3,4,2,1] => 000 => [4] => ([],4)
 => ? = 1 + 1
[4,1,2,3] => 000 => [4] => ([],4)
 => ? = 3 + 1
[4,1,3,2] => 000 => [4] => ([],4)
 => ? = 2 + 1
[4,2,1,3] => 000 => [4] => ([],4)
 => ? = 2 + 1
[4,2,3,1] => 000 => [4] => ([],4)
 => ? = 2 + 1
[4,3,1,2] => 000 => [4] => ([],4)
 => ? = 2 + 1
[4,3,2,1] => 000 => [4] => ([],4)
 => ? = 1 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 1 + 1
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 1 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 1 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 1 + 1
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,4,5,3,2] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 1 + 1
[1,5,2,3,4] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 3 + 1
[1,5,2,4,3] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,5,3,2,4] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,5,3,4,2] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,5,4,2,3] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,5,4,3,2] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 1 + 1
[2,1,3,4,5] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,1,3,5,4] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,1,4,3,5] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,1,4,5,3] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1 + 1
[2,1,5,3,4] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[2,1,5,4,3] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1 + 1
[2,3,1,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,3,1,5,4] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,3,4,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,3,4,5,1] => 0000 => [5] => ([],5)
 => ? = 1 + 1
[2,3,5,1,4] => 0000 => [5] => ([],5)
 => ? = 2 + 1
[2,3,5,4,1] => 0000 => [5] => ([],5)
 => ? = 1 + 1
[2,4,1,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,4,1,5,3] => 0000 => [5] => ([],5)
 => ? = 1 + 1
[2,4,3,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1,2,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1,4,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,2,1,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,2,4,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,4,1,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,4,2,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,2,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,3,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,1,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,3,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,3,1,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,3,2,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,4,5,6] => 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,5,4,6] => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,3,5,6] => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,5,3,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,3,4,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,4,3,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,3,2,4,5,6] => 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,3,2,5,4,6] => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,3,4,2,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,3,4,5,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,3,5,2,4,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,3,5,4,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,4,2,3,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,4,2,5,3,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,4,3,2,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,4,3,5,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,4,5,2,3,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St001491
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 17%
Values
[1,2] => [1,2] => [2,1] => 0 => ? = 0
[2,1] => [2,1] => [1,2] => 1 => 1
[1,2,3] => [1,2,3] => [2,3,1] => 00 => ? = 0
[1,3,2] => [1,3,2] => [2,1,3] => 01 => 1
[2,1,3] => [2,1,3] => [3,2,1] => 00 => ? = 0
[2,3,1] => [1,3,2] => [2,1,3] => 01 => 1
[3,1,2] => [3,1,2] => [3,1,2] => 00 => ? = 2
[3,2,1] => [3,2,1] => [1,3,2] => 10 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 000 => ? = 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 000 => ? = 0
[1,3,4,2] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => 000 => ? = 2
[1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 010 => 1
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 000 => ? = 0
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 001 => 1
[2,3,1,4] => [1,3,2,4] => [2,4,3,1] => 000 => ? = 0
[2,3,4,1] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 000 => ? = 2
[2,4,3,1] => [1,4,3,2] => [2,1,4,3] => 010 => 1
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 000 => ? = 0
[3,1,4,2] => [2,1,4,3] => [3,2,1,4] => 001 => 1
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 000 => ? = 0
[3,2,4,1] => [2,1,4,3] => [3,2,1,4] => 001 => 1
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 000 => ? = 2
[3,4,2,1] => [1,4,3,2] => [2,1,4,3] => 010 => 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 000 => ? = 3
[4,1,3,2] => [4,1,3,2] => [3,1,4,2] => 000 => ? = 2
[4,2,1,3] => [4,2,1,3] => [4,3,1,2] => 000 => ? = 2
[4,2,3,1] => [4,1,3,2] => [3,1,4,2] => 000 => ? = 2
[4,3,1,2] => [4,3,1,2] => [4,1,3,2] => 000 => ? = 2
[4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 100 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0000 => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0000 => ? = 0
[1,2,4,5,3] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => 0000 => ? = 2
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 0000 => ? = 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[1,3,4,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0000 => ? = 0
[1,3,4,5,2] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => 0000 => ? = 2
[1,3,5,4,2] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => 0000 => ? = 0
[1,4,2,5,3] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 0000 => ? = 0
[1,4,3,5,2] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[1,4,5,2,3] => [1,3,5,2,4] => [2,5,3,1,4] => 0000 => ? = 2
[1,4,5,3,2] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1
[1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => 0000 => ? = 3
[1,5,2,4,3] => [1,5,2,4,3] => [2,4,1,5,3] => 0000 => ? = 2
[1,5,3,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => 0000 => ? = 2
[1,5,3,4,2] => [1,5,2,4,3] => [2,4,1,5,3] => 0000 => ? = 2
[1,5,4,2,3] => [1,5,4,2,3] => [2,5,1,4,3] => 0000 => ? = 2
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,5,4,3] => 0100 => 1
[2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => 0000 => ? = 0
[2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1
[2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 0000 => ? = 0
[2,1,4,5,3] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1
[2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => 0000 => ? = 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1
[2,3,1,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 0000 => ? = 0
[2,3,1,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[2,3,4,1,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0000 => ? = 0
[2,3,4,5,1] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[2,3,5,1,4] => [1,3,5,2,4] => [2,5,3,1,4] => 0000 => ? = 2
[2,3,5,4,1] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1
[2,4,1,3,5] => [2,4,1,3,5] => [4,2,5,3,1] => 0000 => ? = 0
[2,4,1,5,3] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[2,4,3,1,5] => [1,4,3,2,5] => [2,5,4,3,1] => 0000 => ? = 0
[2,4,3,5,1] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[2,4,5,1,3] => [1,3,5,2,4] => [2,5,3,1,4] => 0000 => ? = 2
[2,4,5,3,1] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1
[2,5,1,3,4] => [2,5,1,3,4] => [4,2,5,1,3] => 0000 => ? = 3
[2,5,1,4,3] => [2,5,1,4,3] => [4,2,1,5,3] => 0000 => ? = 2
[2,5,3,1,4] => [1,5,3,2,4] => [2,5,4,1,3] => 0000 => ? = 2
[2,5,3,4,1] => [1,5,2,4,3] => [2,4,1,5,3] => 0000 => ? = 2
[2,5,4,1,3] => [2,5,4,1,3] => [5,2,1,4,3] => 0000 => ? = 2
[2,5,4,3,1] => [1,5,4,3,2] => [2,1,5,4,3] => 0100 => 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,1] => 0000 => ? = 0
[3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,1,5] => 0001 => 1
[3,1,4,2,5] => [2,1,4,3,5] => [3,2,5,4,1] => 0000 => ? = 0
[3,1,4,5,2] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1
[3,1,5,2,4] => [3,1,5,2,4] => [3,5,2,1,4] => 0000 => ? = 2
[3,1,5,4,2] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1
[3,2,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => 0000 => ? = 0
[3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => 0001 => 1
[3,2,4,5,1] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1
[3,2,5,4,1] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1
[3,4,1,5,2] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[3,4,2,5,1] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1
[3,4,5,2,1] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1
[3,5,4,2,1] => [1,5,4,3,2] => [2,1,5,4,3] => 0100 => 1
[4,1,2,5,3] => [3,1,2,5,4] => [3,4,2,1,5] => 0001 => 1
[4,1,3,5,2] => [3,1,2,5,4] => [3,4,2,1,5] => 0001 => 1
[4,1,5,3,2] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1
[4,2,1,5,3] => [3,2,1,5,4] => [4,3,2,1,5] => 0001 => 1
[4,2,3,5,1] => [3,1,2,5,4] => [3,4,2,1,5] => 0001 => 1
[4,2,5,3,1] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.