Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000018
(load all 6 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => 3
[3,1,2] => [3,1,2] => 2
[3,2,1] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => 3
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,3,4,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => 3
[2,3,4,1] => [4,2,3,1] => 5
[2,4,1,3] => [4,2,1,3] => 4
[2,4,3,1] => [3,2,4,1] => 4
[3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [4,3,1,2] => 5
[3,2,1,4] => [2,3,1,4] => 2
[3,2,4,1] => [4,3,2,1] => 6
[3,4,1,2] => [4,1,3,2] => 4
[3,4,2,1] => [2,4,3,1] => 4
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [3,4,1,2] => 4
[4,2,1,3] => [2,4,1,3] => 3
[4,2,3,1] => [3,4,2,1] => 5
[4,3,1,2] => [3,1,4,2] => 3
[4,3,2,1] => [2,3,4,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => 3
[1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,4,5,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => 3
[1,3,4,5,2] => [1,5,3,4,2] => 5
[1,3,5,2,4] => [1,5,3,2,4] => 4
[1,3,5,4,2] => [1,4,3,5,2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,5,4,2,3] => 5
[1,4,3,2,5] => [1,3,4,2,5] => 2
[1,4,3,5,2] => [1,5,4,3,2] => 6
[1,4,5,2,3] => [1,5,2,4,3] => 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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[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
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'''.
Matching statistic: St000246
(load all 4 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [3,2,1] => [1,2,3] => 3
[3,1,2] => [3,1,2] => [1,3,2] => 2
[3,2,1] => [2,3,1] => [2,1,3] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 3
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 2
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 5
[2,4,1,3] => [4,2,1,3] => [1,3,4,2] => 4
[2,4,3,1] => [3,2,4,1] => [2,3,1,4] => 4
[3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 2
[3,1,4,2] => [4,3,1,2] => [1,2,4,3] => 5
[3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 2
[3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 6
[3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 4
[3,4,2,1] => [2,4,3,1] => [3,1,2,4] => 4
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 3
[4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 4
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 3
[4,2,3,1] => [3,4,2,1] => [2,1,3,4] => 5
[4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 3
[4,3,2,1] => [2,3,4,1] => [3,2,1,4] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 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] => [5,4,1,3,2] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 3
[1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => 5
[1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,4,2] => 4
[1,3,5,4,2] => [1,4,3,5,2] => [5,2,3,1,4] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [5,1,2,4,3] => 5
[1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 6
[1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 4
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: 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] => [1] => [1] => [[1]]
 => 0
[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
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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[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
[] => [] => ([],0)
 => ? = 0
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: 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] => [1] => [1] => [1] => 0
[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
[] => [] => [] => [] => ? = 0
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: St000305
(load all 2 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[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] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [2,3,1] => [2,3,1] => [3,1,2] => 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] => [2,3,4,1] => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 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] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => [4,1,3,2] => 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] => [4,2,1,3] => 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] => [1,4,3,2] => 5
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 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] => [2,4,3,1] => 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] => [1,2,4,3] => 3
[4,1,3,2] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 4
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 3
[4,2,3,1] => [3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 5
[4,3,1,2] => [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 3
[4,3,2,1] => [2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 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] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => [3,4,5,2,1] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [4,5,1,2,3] => [3,4,5,1,2] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [3,4,1,5,2] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [3,4,2,1,5] => 3
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => [4,5,1,3,2] => 5
[1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,2,4] => [2,4,3,5,1] => 4
[1,3,5,4,2] => [1,4,3,5,2] => [4,3,5,1,2] => [4,5,2,1,3] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => [1,4,5,3,2] => 5
[1,4,3,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => [4,5,3,2,1] => 6
[1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => [2,4,5,3,1] => 4
[] => [] => [] => [] => ? = 0
Description
The inverse major index of a permutation. This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
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] => ? = 0
[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
[] => ? = 0
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: 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] => [1] => [1] => => ? = 0
[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
[] => [] => [] => ? => ? = 0
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000833
(load all 2 compositions to match this statistic)
Mapping
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[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] => [2,3,1] => [2,1,3] => 2
[3,2,1] => [2,3,1] => [1,3,2] => [3,1,2] => 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] => [3,4,1,2] => [3,4,1,2] => 2
[1,4,3,2] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 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] => [1,4,2,3] => [1,4,2,3] => 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] => [3,2,4,1] => [3,2,4,1] => 4
[2,4,3,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 4
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,1,4,2] => [4,3,1,2] => [3,4,2,1] => [3,2,1,4] => 5
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 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] => [4,2,3,1] => [4,2,3,1] => 4
[3,4,2,1] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 4
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [2,1,3,4] => 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] => [1,3,4,2] => [3,1,2,4] => 3
[4,2,3,1] => [3,4,2,1] => [1,4,3,2] => [4,3,1,2] => 5
[4,3,1,2] => [3,1,4,2] => [4,2,1,3] => [2,4,3,1] => 3
[4,3,2,1] => [2,3,4,1] => [1,2,4,3] => [4,1,2,3] => 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] => [4,5,1,2,3] => [1,4,5,2,3] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,1,2,4] => [1,3,5,2,4] => 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] => [2,5,1,3,4] => [2,3,5,1,4] => 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] => [4,3,5,1,2] => [4,5,1,3,2] => 4
[1,3,5,4,2] => [1,4,3,5,2] => [3,2,5,1,4] => [3,5,2,4,1] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => [3,4,5,1,2] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [4,5,3,1,2] => [4,5,3,1,2] => 5
[1,4,3,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => [2,4,5,1,3] => 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] => [5,3,4,1,2] => [3,5,1,4,2] => 4
[1,4,5,3,2] => [1,3,5,4,2] => [5,2,4,1,3] => [2,5,1,4,3] => 4
[] => [] => [] => [] => ? = 0
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 $n$.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000081The number of edges of a graph. 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. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001875The number of simple modules with projective dimension at most 1. 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. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001877Number of indecomposable injective modules with projective dimension 2.