Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [2,1] => [1,2] => 1
[2,1] => [1,2] => [2,1] => 2
[1,2,3] => [3,2,1] => [1,2,3] => 1
[1,3,2] => [2,3,1] => [3,1,2] => 2
[2,1,3] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [1,3,2] => [2,1,3] => 2
[3,1,2] => [2,1,3] => [3,2,1] => 3
[3,2,1] => [1,2,3] => [2,3,1] => 2
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 2
[1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 2
[1,4,2,3] => [3,2,4,1] => [4,3,1,2] => 3
[1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 2
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => 1
[2,3,4,1] => [1,4,3,2] => [2,1,3,4] => 2
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 3
[2,4,3,1] => [1,3,4,2] => [2,4,1,3] => 2
[3,1,2,4] => [4,2,1,3] => [1,4,3,2] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,2,1,4] => [4,1,2,3] => [1,3,4,2] => 1
[3,2,4,1] => [1,4,2,3] => [2,1,4,3] => 2
[3,4,1,2] => [2,1,4,3] => [3,2,1,4] => 3
[3,4,2,1] => [1,2,4,3] => [2,3,1,4] => 2
[4,1,2,3] => [3,2,1,4] => [4,3,2,1] => 4
[4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 3
[4,2,1,3] => [3,1,2,4] => [4,2,3,1] => 3
[4,2,3,1] => [1,3,2,4] => [2,4,3,1] => 3
[4,3,1,2] => [2,1,3,4] => [3,2,4,1] => 3
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [4,1,2,3,5] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,1,2,3] => 3
[1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,5,3,4] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [3,1,2,4,5] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 3
[1,3,5,4,2] => [2,4,5,3,1] => [3,5,1,2,4] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,5,4,2,3] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [4,1,5,2,3] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,4,5,2,3] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [3,1,5,2,4] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,1,2,5] => 3
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: St000007
(load all 2 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[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] => 2
[3,1,2] => [3,1,2] => [2,3,1] => [3,2,1] => 3
[3,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[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] => 2
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[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] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 3
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[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] => [3,2,1,4] => [2,3,1,4] => 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] => [3,1,4,2] => [4,3,1,2] => 3
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [4,3,2,1] => 4
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => [3,4,2,1] => 3
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => [2,4,3,1] => 3
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => [3,4,2,1] => 3
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [4,2,3,1] => 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[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] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 2
[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] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [1,5,4,2,3] => 3
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[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] => [1,4,3,2,5] => [1,3,4,2,5] => 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] => [1,4,2,5,3] => [1,5,4,2,3] => 3
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: 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] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [1,2] => 1
[2,1] => [2,1] => [1,2] => [2,1] => 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [1,3,2] => [2,3,1] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [2,1,3] => [3,2,1] => 3
[3,2,1] => [3,2,1] => [1,2,3] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 2
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [4,3,1,2] => 3
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 3
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 2
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,4,3,2] => 1
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,3,4,2] => 1
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 3
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 2
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [4,3,2,1] => 4
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 3
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [4,2,3,1] => 3
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 3
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [3,2,4,1] => 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 1
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [5,4,1,2,3] => 3
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,5,3,4] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 2
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 1
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 3
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,5,4,2,3] => 1
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,4,5,2,3] => 1
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 2
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 3
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: St000991
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 1
[1,3,2] => [1,3,2] => [1,3,2] => [2,1,3] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [3,2,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1,3] => 2
[3,1,2] => [3,1,2] => [2,3,1] => [1,2,3] => 3
[3,2,1] => [3,2,1] => [3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [2,1,3,4] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [3,1,2,4] => 3
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [4,2,3,1] => 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 1
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [3,1,2,4] => 3
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 4
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => [1,2,4,3] => 3
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => [1,3,2,4] => 3
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => [1,2,4,3] => 3
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [1,4,2,3] => 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [2,3,1,4,5] => 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [2,4,1,3,5] => 3
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [2,5,3,4,1] => 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 1
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => [2,4,1,3,5] => 3
Description
The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].
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] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [1,2] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 0 = 1 - 1
[2,3,1] => [1,3,2] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[3,1,2] => [3,1,2] => [2,1,3] => [3,2,1] => 2 = 3 - 1
[3,2,1] => [3,2,1] => [1,2,3] => [2,3,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 0 = 1 - 1
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [4,3,1,2] => 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1 = 2 - 1
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 0 = 1 - 1
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [4,1,2,3] => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 2 = 3 - 1
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 1 = 2 - 1
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,4,3,2] => 0 = 1 - 1
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,3,4,2] => 0 = 1 - 1
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1 = 2 - 1
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 2 = 3 - 1
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [3,4,1,2] => 1 = 2 - 1
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [4,3,2,1] => 3 = 4 - 1
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 2 = 3 - 1
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [4,2,3,1] => 2 = 3 - 1
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [3,4,2,1] => 2 = 3 - 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [3,2,4,1] => 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [5,4,1,2,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,5,3,4] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 1 = 2 - 1
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,3,4] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 2 = 3 - 1
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,5,4,2,3] => 0 = 1 - 1
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,4,5,2,3] => 0 = 1 - 1
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,2,4] => 2 = 3 - 1
[1,4,5,3,2] => [1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,2,3] => 1 = 2 - 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: St001491
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1] => => ? = 1 - 1
[1,2] => [1,2] => [2,1] => 0 => ? = 1 - 1
[2,1] => [2,1] => [1,2] => 1 => 1 = 2 - 1
[1,2,3] => [1,2,3] => [2,3,1] => 00 => ? = 1 - 1
[1,3,2] => [1,3,2] => [2,1,3] => 01 => 1 = 2 - 1
[2,1,3] => [2,1,3] => [3,2,1] => 00 => ? = 1 - 1
[2,3,1] => [1,3,2] => [2,1,3] => 01 => 1 = 2 - 1
[3,1,2] => [3,1,2] => [3,1,2] => 00 => ? = 3 - 1
[3,2,1] => [3,2,1] => [1,3,2] => 10 => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 000 => ? = 1 - 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 000 => ? = 1 - 1
[1,3,4,2] => [1,2,4,3] => [2,3,1,4] => 001 => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => 000 => ? = 3 - 1
[1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 010 => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 000 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 001 => 1 = 2 - 1
[2,3,1,4] => [1,3,2,4] => [2,4,3,1] => 000 => ? = 1 - 1
[2,3,4,1] => [1,2,4,3] => [2,3,1,4] => 001 => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 000 => ? = 3 - 1
[2,4,3,1] => [1,4,3,2] => [2,1,4,3] => 010 => 1 = 2 - 1
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 000 => ? = 1 - 1
[3,1,4,2] => [2,1,4,3] => [3,2,1,4] => 001 => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 000 => ? = 1 - 1
[3,2,4,1] => [2,1,4,3] => [3,2,1,4] => 001 => 1 = 2 - 1
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 000 => ? = 3 - 1
[3,4,2,1] => [1,4,3,2] => [2,1,4,3] => 010 => 1 = 2 - 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 000 => ? = 4 - 1
[4,1,3,2] => [4,1,3,2] => [3,1,4,2] => 000 => ? = 3 - 1
[4,2,1,3] => [4,2,1,3] => [4,3,1,2] => 000 => ? = 3 - 1
[4,2,3,1] => [4,1,3,2] => [3,1,4,2] => 000 => ? = 3 - 1
[4,3,1,2] => [4,3,1,2] => [4,1,3,2] => 000 => ? = 3 - 1
[4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 100 => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0000 => ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0000 => ? = 1 - 1
[1,2,4,5,3] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => 0000 => ? = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 0000 => ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[1,3,4,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0000 => ? = 1 - 1
[1,3,4,5,2] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => 0000 => ? = 3 - 1
[1,3,5,4,2] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => 0000 => ? = 1 - 1
[1,4,2,5,3] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 0000 => ? = 1 - 1
[1,4,3,5,2] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[1,4,5,2,3] => [1,3,5,2,4] => [2,5,3,1,4] => 0000 => ? = 3 - 1
[1,4,5,3,2] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => 0000 => ? = 4 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [2,4,1,5,3] => 0000 => ? = 3 - 1
[1,5,3,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => 0000 => ? = 3 - 1
[1,5,3,4,2] => [1,5,2,4,3] => [2,4,1,5,3] => 0000 => ? = 3 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [2,5,1,4,3] => 0000 => ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,5,4,3] => 0100 => 1 = 2 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => 0000 => ? = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1 = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 0000 => ? = 1 - 1
[2,1,4,5,3] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1 = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => 0000 => ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1 = 2 - 1
[2,3,1,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 0000 => ? = 1 - 1
[2,3,1,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[2,3,4,1,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0000 => ? = 1 - 1
[2,3,4,5,1] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1 = 2 - 1
[2,3,5,1,4] => [1,3,5,2,4] => [2,5,3,1,4] => 0000 => ? = 3 - 1
[2,3,5,4,1] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1 = 2 - 1
[2,4,1,3,5] => [2,4,1,3,5] => [4,2,5,3,1] => 0000 => ? = 1 - 1
[2,4,1,5,3] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[2,4,3,1,5] => [1,4,3,2,5] => [2,5,4,3,1] => 0000 => ? = 1 - 1
[2,4,3,5,1] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[2,4,5,1,3] => [1,3,5,2,4] => [2,5,3,1,4] => 0000 => ? = 3 - 1
[2,4,5,3,1] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1 = 2 - 1
[2,5,1,3,4] => [2,5,1,3,4] => [4,2,5,1,3] => 0000 => ? = 4 - 1
[2,5,1,4,3] => [2,5,1,4,3] => [4,2,1,5,3] => 0000 => ? = 3 - 1
[2,5,3,1,4] => [1,5,3,2,4] => [2,5,4,1,3] => 0000 => ? = 3 - 1
[2,5,3,4,1] => [1,5,2,4,3] => [2,4,1,5,3] => 0000 => ? = 3 - 1
[2,5,4,1,3] => [2,5,4,1,3] => [5,2,1,4,3] => 0000 => ? = 3 - 1
[2,5,4,3,1] => [1,5,4,3,2] => [2,1,5,4,3] => 0100 => 1 = 2 - 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,1] => 0000 => ? = 1 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,1,5] => 0001 => 1 = 2 - 1
[3,1,4,2,5] => [2,1,4,3,5] => [3,2,5,4,1] => 0000 => ? = 1 - 1
[3,1,4,5,2] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1 = 2 - 1
[3,1,5,2,4] => [3,1,5,2,4] => [3,5,2,1,4] => 0000 => ? = 3 - 1
[3,1,5,4,2] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1 = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => 0001 => 1 = 2 - 1
[3,2,4,5,1] => [2,1,3,5,4] => [3,2,4,1,5] => 0001 => 1 = 2 - 1
[3,2,5,4,1] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1 = 2 - 1
[3,4,1,5,2] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[3,4,2,5,1] => [1,3,2,5,4] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[3,4,5,2,1] => [1,2,5,4,3] => [2,3,1,5,4] => 0010 => 1 = 2 - 1
[3,5,4,2,1] => [1,5,4,3,2] => [2,1,5,4,3] => 0100 => 1 = 2 - 1
[4,1,2,5,3] => [3,1,2,5,4] => [3,4,2,1,5] => 0001 => 1 = 2 - 1
[4,1,3,5,2] => [3,1,2,5,4] => [3,4,2,1,5] => 0001 => 1 = 2 - 1
[4,1,5,3,2] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1 = 2 - 1
[4,2,1,5,3] => [3,2,1,5,4] => [4,3,2,1,5] => 0001 => 1 = 2 - 1
[4,2,3,5,1] => [3,1,2,5,4] => [3,4,2,1,5] => 0001 => 1 = 2 - 1
[4,2,5,3,1] => [2,1,5,4,3] => [3,2,1,5,4] => 0010 => 1 = 2 - 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.