Your data matches 26 different statistics following compositions of up to 3 maps.
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Matching statistic: St000795
(load all 3 compositions to match this statistic)
Mapping
St000795: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 3
[3,1,2] => 2
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 3
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 5
[2,4,1,3] => 4
[2,4,3,1] => 4
[3,1,2,4] => 2
[3,1,4,2] => 5
[3,2,1,4] => 2
[3,2,4,1] => 6
[3,4,1,2] => 4
[3,4,2,1] => 4
[4,1,2,3] => 3
[4,1,3,2] => 4
[4,2,1,3] => 3
[4,2,3,1] => 5
[4,3,1,2] => 3
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 3
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 3
[1,3,4,5,2] => 5
[1,3,5,2,4] => 4
[1,3,5,4,2] => 4
[1,4,2,3,5] => 2
[1,4,2,5,3] => 5
[1,4,3,2,5] => 2
[1,4,3,5,2] => 6
[1,4,5,2,3] => 4
[1,4,5,3,2] => 4
Description
The mad of a permutation. According to [1], this is the sum of twice the number of occurrences of the vincular pattern of $(2\underline{31})$ plus the number of occurrences of the vincular patterns $(\underline{31}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000008
(load all 2 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2] => 0
[2,1] => [2,1] => [2,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [1,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,2] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [1,1,1] => 3
[3,1,2] => [3,1,2] => [2,3,1] => [2,1] => 2
[3,2,1] => [2,3,1] => [1,3,2] => [2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [1,1,2] => 3
[1,4,2,3] => [1,4,2,3] => [3,4,1,2] => [2,2] => 2
[1,4,3,2] => [1,3,4,2] => [2,4,1,3] => [2,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => [2,2] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => [2,1,1] => 5
[2,4,1,3] => [4,2,1,3] => [3,2,4,1] => [1,2,1] => 4
[2,4,3,1] => [3,2,4,1] => [2,1,4,3] => [1,2,1] => 4
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [2,2] => 2
[3,1,4,2] => [4,3,1,2] => [3,4,2,1] => [2,1,1] => 5
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => [2,2] => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[3,4,1,2] => [4,1,3,2] => [4,2,3,1] => [1,2,1] => 4
[3,4,2,1] => [2,4,3,1] => [4,1,3,2] => [1,2,1] => 4
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [3,1] => 3
[4,1,3,2] => [3,4,1,2] => [3,1,4,2] => [1,2,1] => 4
[4,2,1,3] => [2,4,1,3] => [1,3,4,2] => [3,1] => 3
[4,2,3,1] => [3,4,2,1] => [1,4,3,2] => [2,1,1] => 5
[4,3,1,2] => [3,1,4,2] => [4,2,1,3] => [1,1,2] => 3
[4,3,2,1] => [2,3,4,1] => [1,2,4,3] => [3,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,4] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [4,5,1,2,3] => [2,3] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,1,2,4] => [2,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,4] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,5,1,3,4] => [2,3] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [1,1,3] => 3
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => [2,1,2] => 5
[1,3,5,2,4] => [1,5,3,2,4] => [4,3,5,1,2] => [1,2,2] => 4
[1,3,5,4,2] => [1,4,3,5,2] => [3,2,5,1,4] => [1,2,2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => [2,3] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [4,5,3,1,2] => [2,1,2] => 5
[1,4,3,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => [2,3] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,2] => 6
[1,4,5,2,3] => [1,5,2,4,3] => [5,3,4,1,2] => [1,2,2] => 4
[1,4,5,3,2] => [1,3,5,4,2] => [5,2,4,1,3] => [1,2,2] => 4
Description
The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000330
(load all 2 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [[1,2]]
 => 0
[2,1] => [2,1] => [2,1] => [[1],[2]]
 => 1
[1,2,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[1,3,2] => [1,3,2] => [3,1,2] => [[1,3],[2]]
 => 1
[2,1,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 1
[2,3,1] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 3
[3,1,2] => [3,1,2] => [1,3,2] => [[1,2],[3]]
 => 2
[3,2,1] => [2,3,1] => [2,3,1] => [[1,2],[3]]
 => 2
[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] => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
 => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
 => 2
[1,4,3,2] => [1,3,4,2] => [3,4,1,2] => [[1,2],[3,4]]
 => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => [[1,2],[3,4]]
 => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 3
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
 => 5
[2,4,1,3] => [4,2,1,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 4
[2,4,3,1] => [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
 => 4
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 2
[3,1,4,2] => [4,3,1,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 5
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
 => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 6
[3,4,1,2] => [4,1,3,2] => [4,1,3,2] => [[1,3],[2],[4]]
 => 4
[3,4,2,1] => [2,4,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
 => 4
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[4,1,3,2] => [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
 => 4
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
 => 3
[4,2,3,1] => [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
 => 5
[4,3,1,2] => [3,1,4,2] => [1,3,4,2] => [[1,2,3],[4]]
 => 3
[4,3,2,1] => [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
 => 3
[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] => [5,1,2,3,4] => [[1,3,4,5],[2]]
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
 => 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
 => 3
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [[1,2,4,5],[3]]
 => 2
[1,2,5,4,3] => [1,2,4,5,3] => [4,5,1,2,3] => [[1,2,5],[3,4]]
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 2
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
 => 3
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => [[1,2],[3,5],[4]]
 => 5
[1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,2,4] => [[1,3,5],[2],[4]]
 => 4
[1,3,5,4,2] => [1,4,3,5,2] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
 => 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
 => 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
 => 5
[1,4,3,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
 => 6
[1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => [[1,3,5],[2],[4]]
 => 4
[1,4,5,3,2] => [1,3,5,4,2] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
 => 4
Description
The (standard) major index of a standard tableau. A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000391
(load all 2 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0 => 0
[2,1] => [2,1] => [2,1] => 1 => 1
[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => [3,1,2] => 10 => 1
[2,1,3] => [2,1,3] => [2,1,3] => 10 => 1
[2,3,1] => [3,2,1] => [3,2,1] => 11 => 3
[3,1,2] => [3,1,2] => [2,3,1] => 01 => 2
[3,2,1] => [2,3,1] => [1,3,2] => 01 => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 100 => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 110 => 3
[1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 010 => 2
[1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 010 => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 100 => 1
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => 010 => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 110 => 3
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => 011 => 5
[2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 101 => 4
[2,4,3,1] => [3,2,4,1] => [2,1,4,3] => 101 => 4
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 010 => 2
[3,1,4,2] => [4,3,1,2] => [3,4,2,1] => 011 => 5
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => 010 => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 111 => 6
[3,4,1,2] => [4,1,3,2] => [4,2,3,1] => 101 => 4
[3,4,2,1] => [2,4,3,1] => [4,1,3,2] => 101 => 4
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 001 => 3
[4,1,3,2] => [3,4,1,2] => [3,1,4,2] => 101 => 4
[4,2,1,3] => [2,4,1,3] => [1,3,4,2] => 001 => 3
[4,2,3,1] => [3,4,2,1] => [1,4,3,2] => 011 => 5
[4,3,1,2] => [3,1,4,2] => [4,2,1,3] => 110 => 3
[4,3,2,1] => [2,3,4,1] => [1,2,4,3] => 001 => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 1000 => 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1100 => 3
[1,2,5,3,4] => [1,2,5,3,4] => [4,5,1,2,3] => 0100 => 2
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,1,2,4] => 0100 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 1000 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,5,1,3,4] => 0100 => 2
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => 1100 => 3
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => 0110 => 5
[1,3,5,2,4] => [1,5,3,2,4] => [4,3,5,1,2] => 1010 => 4
[1,3,5,4,2] => [1,4,3,5,2] => [3,2,5,1,4] => 1010 => 4
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => 0100 => 2
[1,4,2,5,3] => [1,5,4,2,3] => [4,5,3,1,2] => 0110 => 5
[1,4,3,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => 0100 => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => 1110 => 6
[1,4,5,2,3] => [1,5,2,4,3] => [5,3,4,1,2] => 1010 => 4
[1,4,5,3,2] => [1,3,5,4,2] => [5,2,4,1,3] => 1010 => 4
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St001397
(load all 2 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => ([(0,1)],2)
 => 0
[2,1] => [2,1] => ([],2)
 => 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 1
[2,3,1] => [3,2,1] => ([],3)
 => 3
[3,1,2] => [3,1,2] => ([(1,2)],3)
 => 2
[3,2,1] => [2,3,1] => ([(1,2)],3)
 => 2
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
 => 5
[2,4,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 4
[2,4,3,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 4
[3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,1,4,2] => [4,3,1,2] => ([(2,3)],4)
 => 5
[3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,2,4,1] => [4,3,2,1] => ([],4)
 => 6
[3,4,1,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 4
[3,4,2,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 4
[4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[4,1,3,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 3
[4,2,3,1] => [3,4,2,1] => ([(2,3)],4)
 => 5
[4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 3
[4,3,2,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 3
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 5
[1,3,5,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,4,2,5,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 5
[1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 6
[1,4,5,2,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 4
[1,4,5,3,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 4
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
 => ? = 3
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => ? = 2
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ? = 2
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => ([(0,5),(4,3),(5,6),(6,1),(6,2),(6,4)],7)
 => ? = 5
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => ([(0,5),(4,3),(5,6),(6,1),(6,2),(6,4)],7)
 => ? = 5
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => ? = 2
[1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => ([(0,4),(4,6),(5,2),(5,3),(6,1),(6,5)],7)
 => ? = 4
[1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => ([(0,4),(4,6),(5,2),(5,3),(6,1),(6,5)],7)
 => ? = 4
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
 => ? = 3
[1,2,3,7,4,6,5] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7)
 => ? = 4
[1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ? = 3
[1,2,3,7,5,6,4] => [1,2,3,6,7,5,4] => ([(0,5),(4,3),(5,6),(6,1),(6,2),(6,4)],7)
 => ? = 5
[1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ? = 3
[1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
 => ? = 3
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ? = 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
 => ? = 4
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ? = 3
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ? = 3
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 5
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => ([(0,6),(4,5),(5,3),(6,1),(6,2),(6,4)],7)
 => ? = 7
[1,2,4,5,7,3,6] => [1,2,7,4,5,3,6] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 6
[1,2,4,5,7,6,3] => [1,2,6,4,5,7,3] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 6
[1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ? = 4
[1,2,4,6,3,7,5] => [1,2,7,4,6,3,5] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
 => ? = 7
[1,2,4,6,5,3,7] => [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ? = 4
[1,2,4,6,5,7,3] => [1,2,7,4,6,5,3] => ([(0,6),(5,3),(5,4),(6,1),(6,2),(6,5)],7)
 => ? = 8
[1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ? = 5
[1,2,4,7,3,6,5] => [1,2,6,4,7,3,5] => ([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3)],7)
 => ? = 6
[1,2,4,7,5,6,3] => [1,2,6,4,7,5,3] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
 => ? = 7
[1,2,4,7,6,5,3] => [1,2,5,4,6,7,3] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ? = 5
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ? = 2
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ? = 3
[1,2,5,3,6,4,7] => [1,2,6,5,3,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 5
[1,2,5,3,6,7,4] => [1,2,7,5,3,6,4] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
 => ? = 7
[1,2,5,3,7,4,6] => [1,2,7,5,3,4,6] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 6
[1,2,5,3,7,6,4] => [1,2,6,5,3,7,4] => ([(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3),(5,4)],7)
 => ? = 6
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ? = 2
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ? = 3
[1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ? = 4
[1,2,5,6,3,7,4] => [1,2,7,6,5,3,4] => ([(0,6),(5,4),(6,1),(6,2),(6,3),(6,5)],7)
 => ? = 9
[1,2,5,6,4,3,7] => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ? = 4
[1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => ([(0,6),(4,3),(5,2),(5,4),(6,1),(6,5)],7)
 => ? = 6
Description
Number of pairs of incomparable elements in a finite poset. For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
Matching statistic: St000081
(load all 2 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 3
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ? = 2
[1,2,3,7,4,6,5] => [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 4
[1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ? = 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => ([(2,3),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => ([(2,3),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,2,4,6,3,7,5] => [1,2,7,4,6,3,5] => ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 6
[1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 6
[1,2,4,7,3,6,5] => [1,2,6,4,7,3,5] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,2,4,7,5,3,6] => [1,2,5,4,7,3,6] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5
[1,2,4,7,5,6,3] => [1,2,6,4,7,5,3] => ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,5,3,6,7,4] => [1,2,7,5,3,6,4] => ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,2,5,3,7,6,4] => [1,2,6,5,3,7,4] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,5,7,3,6,4] => [1,2,6,7,5,3,4] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,7,4,3,6] => [1,2,4,7,5,3,6] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,5,7,6,3,4] => [1,2,6,3,5,7,4] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,6,3,5,4,7] => [1,2,5,6,3,4,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 4
[1,2,6,3,7,5,4] => [1,2,5,7,6,3,4] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 7
[1,2,6,4,3,5,7] => [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,6,4,3,7,5] => [1,2,4,7,6,3,5] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,2,6,5,3,4,7] => [1,2,5,3,6,4,7] => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,6,5,3,7,4] => [1,2,7,5,6,3,4] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,6,7,3,5,4] => [1,2,5,7,3,6,4] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,2,6,7,4,3,5] => [1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5
[1,2,6,7,4,5,3] => [1,2,5,7,4,6,3] => ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
[1,2,6,7,5,3,4] => [1,2,5,3,7,6,4] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5
[1,2,7,3,4,6,5] => [1,2,6,3,7,4,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,7,3,5,4,6] => [1,2,5,7,3,4,6] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,7,3,5,6,4] => [1,2,6,7,3,5,4] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 7
[1,2,7,3,6,4,5] => [1,2,6,7,3,4,5] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 6
[1,2,7,3,6,5,4] => [1,2,5,6,7,3,4] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 6
[1,2,7,4,3,5,6] => [1,2,4,7,3,5,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,7,4,3,6,5] => [1,2,4,6,7,3,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,7,4,5,3,6] => [1,2,5,7,4,3,6] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,2,7,4,5,6,3] => [1,2,6,7,4,5,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,7,4,6,3,5] => [1,2,6,7,4,3,5] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 7
[1,2,7,5,3,4,6] => [1,2,5,3,7,4,6] => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,7,5,3,6,4] => [1,2,6,5,7,3,4] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 7
[1,2,7,5,4,3,6] => [1,2,4,5,7,3,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,7,5,6,3,4] => [1,2,6,3,7,5,4] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,2,7,6,3,4,5] => [1,2,6,3,4,7,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,7,6,3,5,4] => [1,2,5,6,3,7,4] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
Description
The number of edges of a graph.
Matching statistic: St000833
(load all 2 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[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] => [3,1,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[3,1,2] => [3,1,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [2,3,1] => [2,3,1] => [2,1,3] => 2
[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] => [4,1,2,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,3,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,3,4,1] => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => [2,4,1,3] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,4,2,1] => 3
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => [4,2,1,3] => 5
[2,4,1,3] => [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,4,3,1] => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 4
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [4,3,1,2] => [1,4,3,2] => [4,3,1,2] => 5
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[3,4,1,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 4
[3,4,2,1] => [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 4
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => [4,1,2,3] => 3
[4,1,3,2] => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 4
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 3
[4,2,3,1] => [3,4,2,1] => [3,4,2,1] => [3,2,1,4] => 5
[4,3,1,2] => [3,1,4,2] => [1,3,4,2] => [3,1,2,4] => 3
[4,3,2,1] => [2,3,4,1] => [2,3,4,1] => [2,1,3,4] => 3
[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] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,5,3] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,4,5,2,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,3,4,5,2] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [1,3,5,2,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [1,4,5,3,2] => 3
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => [3,5,4,1,2] => 5
[1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,2,4] => [1,5,3,4,2] => 4
[1,3,5,4,2] => [1,4,3,5,2] => [4,3,5,1,2] => [4,5,1,3,2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => 5
[1,4,3,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [3,4,5,1,2] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,5,4,3,2] => 6
[1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => [1,5,2,4,3] => 4
[1,4,5,3,2] => [1,3,5,4,2] => [5,3,4,1,2] => [3,5,1,4,2] => 4
[1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => [1,5,6,7,2,3,4] => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [4,6,7,1,2,3,5] => [1,4,6,7,2,3,5] => ? = 3
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => [1,4,6,7,5,2,3] => ? = 5
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => [4,5,7,6,1,2,3] => ? = 7
[1,2,4,5,7,3,6] => [1,2,7,4,5,3,6] => [1,7,4,5,2,3,6] => [1,4,7,2,5,3,6] => ? = 6
[1,2,4,5,7,6,3] => [1,2,6,4,5,7,3] => [4,6,5,7,1,2,3] => [4,6,7,1,5,2,3] => ? = 6
[1,2,4,6,5,3,7] => [1,2,5,4,6,3,7] => [5,4,6,1,2,3,7] => [1,5,6,7,2,4,3] => ? = 4
[1,2,4,6,5,7,3] => [1,2,7,4,6,5,3] => [7,4,6,5,1,2,3] => [1,4,7,6,2,5,3] => ? = 8
[1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => [1,7,4,6,2,3,5] => [1,4,7,2,6,3,5] => ? = 6
[1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => [5,7,4,6,1,2,3] => [1,5,7,4,6,2,3] => ? = 6
[1,2,4,7,5,6,3] => [1,2,6,4,7,5,3] => [4,6,7,5,1,2,3] => [4,6,7,5,1,2,3] => ? = 7
[1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => [6,1,4,7,2,3,5] => [1,6,7,2,4,5,3] => ? = 5
[1,2,4,7,6,5,3] => [1,2,5,4,6,7,3] => [5,4,6,7,1,2,3] => [5,6,7,1,2,4,3] => ? = 5
[1,2,5,3,6,7,4] => [1,2,7,5,3,6,4] => [1,5,7,6,2,3,4] => [1,5,7,6,2,3,4] => ? = 7
[1,2,5,3,7,6,4] => [1,2,6,5,3,7,4] => [1,6,5,7,2,3,4] => [1,6,7,2,5,3,4] => ? = 6
[1,2,5,4,6,7,3] => [1,2,7,5,4,6,3] => [5,4,7,6,1,2,3] => [1,5,7,4,2,6,3] => ? = 8
[1,2,5,4,7,6,3] => [1,2,6,5,4,7,3] => [6,5,4,7,1,2,3] => [1,6,7,2,5,4,3] => ? = 7
[1,2,5,6,4,3,7] => [1,2,4,6,5,3,7] => [6,4,5,1,2,3,7] => [1,4,6,7,2,5,3] => ? = 4
[1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => [1,5,7,2,6,3,4] => ? = 6
[1,2,5,6,7,4,3] => [1,2,4,7,5,6,3] => [4,7,5,6,1,2,3] => [4,5,7,1,6,2,3] => ? = 6
[1,2,5,7,4,6,3] => [1,2,6,7,5,4,3] => [6,7,5,4,1,2,3] => [1,6,7,5,4,2,3] => ? = 9
[1,2,5,7,6,3,4] => [1,2,6,3,5,7,4] => [6,1,5,7,2,3,4] => [1,6,7,2,3,5,4] => ? = 5
[1,2,5,7,6,4,3] => [1,2,4,6,5,7,3] => [6,4,5,7,1,2,3] => [4,6,7,1,2,5,3] => ? = 5
[1,2,6,4,5,3,7] => [1,2,5,6,4,3,7] => [5,6,4,1,2,3,7] => [1,5,6,7,4,2,3] => ? = 5
[1,2,6,4,5,7,3] => [1,2,7,6,4,5,3] => [4,7,6,5,1,2,3] => [1,4,7,6,5,2,3] => ? = 9
[1,2,6,4,7,5,3] => [1,2,5,7,6,4,3] => [7,5,6,4,1,2,3] => [1,5,7,6,2,4,3] => ? = 8
[1,2,6,5,4,3,7] => [1,2,4,5,6,3,7] => [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => ? = 3
[1,2,6,5,4,7,3] => [1,2,7,5,6,4,3] => [5,7,6,4,1,2,3] => [1,5,7,6,4,2,3] => ? = 9
[1,2,6,5,7,4,3] => [1,2,4,7,6,5,3] => [7,6,4,5,1,2,3] => [1,4,7,2,6,5,3] => ? = 7
[1,2,6,7,4,5,3] => [1,2,5,7,4,6,3] => [7,5,4,6,1,2,3] => [1,5,7,2,6,4,3] => ? = 7
[1,2,6,7,5,3,4] => [1,2,5,3,7,6,4] => [7,1,5,6,2,3,4] => [1,5,7,2,3,6,4] => ? = 5
[1,2,6,7,5,4,3] => [1,2,4,5,7,6,3] => [7,4,5,6,1,2,3] => [4,5,7,1,2,6,3] => ? = 5
[1,2,7,3,5,6,4] => [1,2,6,7,3,5,4] => [1,6,7,5,2,3,4] => [1,6,7,5,2,3,4] => ? = 7
[1,2,7,3,6,5,4] => [1,2,5,6,7,3,4] => [5,6,1,7,2,3,4] => [1,5,6,2,7,3,4] => ? = 6
[1,2,7,4,5,6,3] => [1,2,6,7,4,5,3] => [6,4,7,5,1,2,3] => [1,6,7,4,2,5,3] => ? = 8
[1,2,7,4,6,3,5] => [1,2,6,7,4,3,5] => [1,6,7,4,2,3,5] => [1,6,7,4,2,3,5] => ? = 7
[1,2,7,4,6,5,3] => [1,2,5,6,7,4,3] => [5,6,7,4,1,2,3] => [5,6,7,4,1,2,3] => ? = 7
[1,2,7,5,4,6,3] => [1,2,6,5,7,4,3] => [6,5,7,4,1,2,3] => [1,6,7,5,2,4,3] => ? = 8
[1,2,7,5,6,4,3] => [1,2,4,6,7,5,3] => [6,7,4,5,1,2,3] => [1,6,7,4,5,2,3] => ? = 6
[1,2,7,6,3,4,5] => [1,2,6,3,4,7,5] => [1,2,6,7,3,4,5] => [1,6,7,2,3,4,5] => ? = 4
[1,2,7,6,3,5,4] => [1,2,5,6,3,7,4] => [5,1,6,7,2,3,4] => [1,5,6,2,3,7,4] => ? = 5
[1,2,7,6,4,5,3] => [1,2,5,6,4,7,3] => [5,6,4,7,1,2,3] => [5,6,7,1,4,2,3] => ? = 6
[1,2,7,6,5,3,4] => [1,2,5,3,6,7,4] => [1,5,6,7,2,3,4] => [5,6,7,1,2,3,4] => ? = 4
[1,2,7,6,5,4,3] => [1,2,4,5,6,7,3] => [4,5,6,7,1,2,3] => [4,5,6,1,2,3,7] => ? = 4
[1,3,2,5,6,7,4] => [1,3,2,7,5,6,4] => [3,7,5,6,1,2,4] => [3,5,7,1,6,2,4] => ? = 6
[1,3,2,5,7,6,4] => [1,3,2,6,5,7,4] => [6,3,5,7,1,2,4] => [3,6,7,1,2,5,4] => ? = 5
[1,3,2,6,5,4,7] => [1,3,2,5,6,4,7] => [3,5,6,1,2,4,7] => [3,5,6,7,1,2,4] => ? = 3
[1,3,2,6,7,5,4] => [1,3,2,5,7,6,4] => [7,3,5,6,1,2,4] => [3,5,7,1,2,6,4] => ? = 5
[1,3,2,7,5,6,4] => [1,3,2,6,7,5,4] => [6,7,3,5,1,2,4] => [1,6,7,3,5,2,4] => ? = 6
[1,3,2,7,6,5,4] => [1,3,2,5,6,7,4] => [3,5,6,7,1,2,4] => [3,5,6,1,2,4,7] => ? = 4
Description
The comajor index of a permutation. This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Matching statistic: St000018
(load all 6 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●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] => [2,1,3] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [3,2,1] => [1,2,3] => [3,2,1] => 3
[3,1,2] => [3,1,2] => [2,1,3] => [2,3,1] => 2
[3,2,1] => [2,3,1] => [1,3,2] => [3,1,2] => 2
[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] => [2,1,3,4] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 2
[1,4,3,2] => [1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 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] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 3
[2,3,4,1] => [4,2,3,1] => [1,3,2,4] => [4,2,3,1] => 5
[2,4,1,3] => [4,2,1,3] => [3,1,2,4] => [2,4,3,1] => 4
[2,4,3,1] => [3,2,4,1] => [1,4,2,3] => [4,1,3,2] => 4
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 2
[3,1,4,2] => [4,3,1,2] => [2,1,3,4] => [3,4,2,1] => 5
[3,2,1,4] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 2
[3,2,4,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 6
[3,4,1,2] => [4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 4
[3,4,2,1] => [2,4,3,1] => [1,3,4,2] => [4,2,1,3] => 4
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [2,3,4,1] => 3
[4,1,3,2] => [3,4,1,2] => [2,1,4,3] => [3,4,1,2] => 4
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 3
[4,2,3,1] => [3,4,2,1] => [1,2,4,3] => [4,3,1,2] => 5
[4,3,1,2] => [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 3
[4,3,2,1] => [2,3,4,1] => [1,4,3,2] => [4,1,2,3] => 3
[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] => [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] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => 3
[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] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => 3
[1,3,4,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => [4,2,3,1,5] => 5
[1,3,5,2,4] => [1,5,3,2,4] => [4,2,3,5,1] => [2,4,3,1,5] => 4
[1,3,5,4,2] => [1,4,3,5,2] => [2,5,3,4,1] => [4,1,3,2,5] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [3,2,4,5,1] => [3,4,2,1,5] => 5
[1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 6
[1,4,5,2,3] => [1,5,2,4,3] => [3,4,2,5,1] => [3,2,4,1,5] => 4
[1,4,5,3,2] => [1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => 4
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 1
[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] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,4,3,2,5,6,7] => ? = 3
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [4,2,3,1,5,6,7] => ? = 5
[1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 4
[1,2,3,5,7,6,4] => [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [4,1,3,2,5,6,7] => ? = 4
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,3,4,2,5,6,7] => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 5
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,4,2,3,5,6,7] => ? = 2
[1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 4
[1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [4,2,1,3,5,6,7] => ? = 4
[1,2,3,7,5,6,4] => [1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [4,3,1,2,5,6,7] => ? = 5
[1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => ? = 3
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 1
[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] => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,3,2,5,4,6,7] => ? = 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 4
[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] => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [3,1,2,5,4,6,7] => ? = 3
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 3
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 4
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [7,3,5,4,6,2,1] => [1,5,3,4,2,6,7] => ? = 5
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [3,6,5,4,7,2,1] => [5,2,3,4,1,6,7] => ? = 7
[1,2,4,5,7,3,6] => [1,2,7,4,5,3,6] => [6,3,5,4,7,2,1] => [2,5,3,4,1,6,7] => ? = 6
[1,2,4,5,7,6,3] => [1,2,6,4,5,7,3] => [3,7,5,4,6,2,1] => [5,1,3,4,2,6,7] => ? = 6
[1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => [1,3,5,4,2,6,7] => ? = 4
[1,2,4,6,3,7,5] => [1,2,7,4,6,3,5] => [5,3,6,4,7,2,1] => [3,5,2,4,1,6,7] => ? = 7
[1,2,4,6,5,3,7] => [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [1,5,2,4,3,6,7] => ? = 4
[1,2,4,6,5,7,3] => [1,2,7,4,6,5,3] => [3,5,6,4,7,2,1] => [5,3,2,4,1,6,7] => ? = 8
[1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => ? = 6
[1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [5,2,1,4,3,6,7] => ? = 6
[1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 5
[1,2,4,7,3,6,5] => [1,2,6,4,7,3,5] => [5,3,7,4,6,2,1] => [3,5,1,4,2,6,7] => ? = 6
[1,2,4,7,5,3,6] => [1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [2,5,1,4,3,6,7] => ? = 5
[1,2,4,7,5,6,3] => [1,2,6,4,7,5,3] => [3,5,7,4,6,2,1] => [5,3,1,4,2,6,7] => ? = 7
[1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => [3,1,5,4,2,6,7] => ? = 5
[1,2,4,7,6,5,3] => [1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [5,1,2,4,3,6,7] => ? = 5
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,4,5,3,6,7] => ? = 2
[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] => ? = 3
[1,2,5,3,6,4,7] => [1,2,6,5,3,4,7] => [7,4,3,5,6,2,1] => [1,4,5,3,2,6,7] => ? = 5
[1,2,5,3,6,7,4] => [1,2,7,5,3,6,4] => [4,6,3,5,7,2,1] => [4,2,5,3,1,6,7] => ? = 7
[1,2,5,3,7,4,6] => [1,2,7,5,3,4,6] => [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => ? = 6
[1,2,5,3,7,6,4] => [1,2,6,5,3,7,4] => [4,7,3,5,6,2,1] => [4,1,5,3,2,6,7] => ? = 6
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,5,3,4,6,7] => ? = 2
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,1,5,3,4,6,7] => ? = 3
[1,2,5,4,6,3,7] => [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => ? = 6
[1,2,5,4,6,7,3] => [1,2,7,5,4,6,3] => [3,6,4,5,7,2,1] => [5,2,4,3,1,6,7] => ? = 8
[1,2,5,4,7,3,6] => [1,2,7,5,4,3,6] => [6,3,4,5,7,2,1] => [2,5,4,3,1,6,7] => ? = 7
[1,2,5,4,7,6,3] => [1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => [5,1,4,3,2,6,7] => ? = 7
[1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => [1,4,3,5,2,6,7] => ? = 4
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000246
(load all 4 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [2,3,1] => 1
[2,3,1] => [3,2,1] => [1,2,3] => [1,2,3] => 3
[3,1,2] => [3,1,2] => [2,1,3] => [2,1,3] => 2
[3,2,1] => [2,3,1] => [1,3,2] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [4,2,1,3] => 2
[1,4,3,2] => [1,3,4,2] => [2,4,3,1] => [4,1,3,2] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[2,3,1,4] => [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 3
[2,3,4,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 5
[2,4,1,3] => [4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 4
[2,4,3,1] => [3,2,4,1] => [1,4,2,3] => [1,3,4,2] => 4
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 2
[3,1,4,2] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 5
[3,2,1,4] => [2,3,1,4] => [4,1,3,2] => [2,4,3,1] => 2
[3,2,4,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 6
[3,4,1,2] => [4,1,3,2] => [2,3,1,4] => [3,1,2,4] => 4
[3,4,2,1] => [2,4,3,1] => [1,3,4,2] => [1,4,2,3] => 4
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 3
[4,1,3,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 4
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 3
[4,2,3,1] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 5
[4,3,1,2] => [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 3
[4,3,2,1] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => 3
[1,3,4,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => [5,1,3,2,4] => 5
[1,3,5,2,4] => [1,5,3,2,4] => [4,2,3,5,1] => [5,2,3,1,4] => 4
[1,3,5,4,2] => [1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [3,2,4,5,1] => [5,2,1,3,4] => 5
[1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [5,1,2,3,4] => 6
[1,4,5,2,3] => [1,5,2,4,3] => [3,4,2,5,1] => [5,3,1,2,4] => 4
[1,4,5,3,2] => [1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => 4
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,6,5,4,2,3,1] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [7,6,5,4,1,3,2] => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,5,3,4,2,1] => ? = 1
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,6,5,2,3,4,1] => ? = 3
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [7,6,5,1,3,2,4] => ? = 5
[1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [7,6,5,2,3,1,4] => ? = 4
[1,2,3,5,7,6,4] => [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [7,6,5,1,3,4,2] => ? = 4
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,6,5,3,2,4,1] => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => ? = 5
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,6,5,2,4,3,1] => ? = 2
[1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => ? = 4
[1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [7,6,5,1,4,2,3] => ? = 4
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => ? = 3
[1,2,3,7,4,6,5] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [7,6,5,2,1,4,3] => ? = 4
[1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [7,6,5,2,4,1,3] => ? = 3
[1,2,3,7,5,6,4] => [1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [7,6,5,1,2,4,3] => ? = 5
[1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [7,6,5,3,1,4,2] => ? = 3
[1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [7,6,5,1,4,3,2] => ? = 3
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,4,5,3,2,1] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [7,6,4,5,3,1,2] => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,6,4,5,2,3,1] => ? = 2
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [7,6,4,5,2,1,3] => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [7,6,4,5,1,3,2] => ? = 3
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,3,4,5,2,1] => ? = 3
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [7,6,3,4,5,1,2] => ? = 4
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [7,3,5,4,6,2,1] => [7,6,2,4,3,5,1] => ? = 5
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [3,6,5,4,7,2,1] => [7,6,1,4,3,2,5] => ? = 7
[1,2,4,5,7,3,6] => [1,2,7,4,5,3,6] => [6,3,5,4,7,2,1] => [7,6,2,4,3,1,5] => ? = 6
[1,2,4,5,7,6,3] => [1,2,6,4,5,7,3] => [3,7,5,4,6,2,1] => [7,6,1,4,3,5,2] => ? = 6
[1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => [7,6,3,4,2,5,1] => ? = 4
[1,2,4,6,3,7,5] => [1,2,7,4,6,3,5] => [5,3,6,4,7,2,1] => [7,6,2,4,1,3,5] => ? = 7
[1,2,4,6,5,3,7] => [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [7,6,2,4,5,3,1] => ? = 4
[1,2,4,6,5,7,3] => [1,2,7,4,6,5,3] => [3,5,6,4,7,2,1] => [7,6,1,4,2,3,5] => ? = 8
[1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 6
[1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [7,6,1,4,5,2,3] => ? = 6
[1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => [6,5,3,4,7,2,1] => [7,6,3,4,2,1,5] => ? = 5
[1,2,4,7,3,6,5] => [1,2,6,4,7,3,5] => [5,3,7,4,6,2,1] => [7,6,2,4,1,5,3] => ? = 6
[1,2,4,7,5,3,6] => [1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [7,6,2,4,5,1,3] => ? = 5
[1,2,4,7,5,6,3] => [1,2,6,4,7,5,3] => [3,5,7,4,6,2,1] => [7,6,1,4,2,5,3] => ? = 7
[1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => [7,6,3,4,1,5,2] => ? = 5
[1,2,4,7,6,5,3] => [1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [7,6,1,4,5,3,2] => ? = 5
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 2
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [7,6,4,3,5,1,2] => ? = 3
[1,2,5,3,6,4,7] => [1,2,6,5,3,4,7] => [7,4,3,5,6,2,1] => [7,6,3,2,4,5,1] => ? = 5
[1,2,5,3,6,7,4] => [1,2,7,5,3,6,4] => [4,6,3,5,7,2,1] => [7,6,3,1,4,2,5] => ? = 7
[1,2,5,3,7,4,6] => [1,2,7,5,3,4,6] => [6,4,3,5,7,2,1] => [7,6,3,2,4,1,5] => ? = 6
[1,2,5,3,7,6,4] => [1,2,6,5,3,7,4] => [4,7,3,5,6,2,1] => [7,6,3,1,4,5,2] => ? = 6
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ? = 2
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [7,6,3,5,4,1,2] => ? = 3
Description
The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000004
(load all 5 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 3
[3,1,2] => [3,1,2] => [1,3,2] => 2
[3,2,1] => [2,3,1] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,3,4,2] => [3,4,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => 5
[2,4,1,3] => [4,2,1,3] => [2,1,4,3] => 4
[2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 4
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [4,3,1,2] => [1,4,3,2] => 5
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 6
[3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 4
[3,4,2,1] => [2,4,3,1] => [4,2,3,1] => 4
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 3
[4,1,3,2] => [3,4,1,2] => [3,1,4,2] => 4
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 3
[4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 5
[4,3,1,2] => [3,1,4,2] => [1,3,4,2] => 3
[4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 3
[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] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [4,5,1,2,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => 3
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => 5
[1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,2,4] => 4
[1,3,5,4,2] => [1,4,3,5,2] => [4,3,5,1,2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 5
[1,4,3,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => 6
[1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 4
[1,4,5,3,2] => [1,3,5,4,2] => [5,3,4,1,2] => 4
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 3
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [6,7,1,2,3,4,5] => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ? = 3
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => ? = 5
[1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 4
[1,2,3,5,7,6,4] => [1,2,3,6,5,7,4] => [6,5,7,1,2,3,4] => ? = 4
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [1,7,6,2,3,4,5] => ? = 5
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 2
[1,2,3,6,5,7,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 6
[1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 4
[1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => [7,5,6,1,2,3,4] => ? = 4
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => ? = 3
[1,2,3,7,4,6,5] => [1,2,3,6,7,4,5] => [6,1,7,2,3,4,5] => ? = 4
[1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => [7,5,1,2,3,4,6] => ? = 3
[1,2,3,7,5,6,4] => [1,2,3,6,7,5,4] => [6,7,5,1,2,3,4] => ? = 5
[1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 3
[1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 3
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,7,1,2,3,5,6] => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [4,6,1,2,3,5,7] => ? = 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [7,4,6,1,2,3,5] => ? = 4
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [7,4,1,2,3,5,6] => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [4,6,7,1,2,3,5] => ? = 3
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => ? = 3
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [5,4,7,1,2,3,6] => ? = 4
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => ? = 5
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => ? = 7
[1,2,4,5,7,3,6] => [1,2,7,4,5,3,6] => [1,7,4,5,2,3,6] => ? = 6
[1,2,4,5,7,6,3] => [1,2,6,4,5,7,3] => [4,6,5,7,1,2,3] => ? = 6
[1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => [6,1,4,2,3,5,7] => ? = 4
[1,2,4,6,3,7,5] => [1,2,7,4,6,3,5] => [7,6,1,4,2,3,5] => ? = 7
[1,2,4,6,5,3,7] => [1,2,5,4,6,3,7] => [5,4,6,1,2,3,7] => ? = 4
[1,2,4,6,5,7,3] => [1,2,7,4,6,5,3] => [7,4,6,5,1,2,3] => ? = 8
[1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => [1,7,4,6,2,3,5] => ? = 6
[1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => [5,7,4,6,1,2,3] => ? = 6
[1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => [7,1,2,4,3,5,6] => ? = 5
[1,2,4,7,3,6,5] => [1,2,6,4,7,3,5] => [6,7,1,4,2,3,5] => ? = 6
[1,2,4,7,5,3,6] => [1,2,5,4,7,3,6] => [5,7,4,1,2,3,6] => ? = 5
[1,2,4,7,5,6,3] => [1,2,6,4,7,5,3] => [4,6,7,5,1,2,3] => ? = 7
[1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => [6,1,4,7,2,3,5] => ? = 5
[1,2,4,7,6,5,3] => [1,2,5,4,6,7,3] => [5,4,6,7,1,2,3] => ? = 5
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [1,5,7,2,3,4,6] => ? = 3
Description
The major index of a permutation. This is the sum of the positions of its descents, $$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$ Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$. A statistic equidistributed with the major index is called '''Mahonian statistic'''.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000305The inverse major index of a permutation. St001311The cyclomatic number of a graph. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St001428The number of B-inversions of a signed permutation. St001622The number of join-irreducible elements of a lattice. St000450The number of edges minus the number of vertices plus 2 of a graph. St001621The number of atoms of a lattice. St001875The number of simple modules with projective dimension at most 1. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001433The flag major index of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2.