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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St000018
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 1
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [1,2] => 0
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 2
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 2
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,2,3] => 0
[-,3,2] => [2,1,3] => 1
[2,1,+] => [1,3,2] => 1
[2,1,-] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,+,1] => [2,1,3] => 1
[3,-,1] => [1,3,2] => 1
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 3
[+,-,+,+] => [1,3,4,2] => 2
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 4
[-,+,-,+] => [2,4,1,3] => 3
[-,+,+,-] => [2,3,1,4] => 2
[+,-,-,+] => [1,4,2,3] => 2
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 3
[-,-,+,-] => [3,1,2,4] => 2
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => 2
[+,-,4,3] => [1,3,2,4] => 1
[-,-,4,3] => [3,1,2,4] => 2
[+,3,2,+] => [1,2,4,3] => 1
[-,3,2,+] => [2,4,1,3] => 3
[+,3,2,-] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => 1
[+,3,4,2] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => 1
[+,4,2,3] => [1,2,3,4] => 0
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: St001579
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St001579: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001579: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 1
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [1,2] => 0
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 2
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 2
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,2,3] => 0
[-,3,2] => [2,1,3] => 1
[2,1,+] => [1,3,2] => 1
[2,1,-] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,+,1] => [2,1,3] => 1
[3,-,1] => [1,3,2] => 1
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 3
[+,-,+,+] => [1,3,4,2] => 2
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 4
[-,+,-,+] => [2,4,1,3] => 3
[-,+,+,-] => [2,3,1,4] => 2
[+,-,-,+] => [1,4,2,3] => 2
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 3
[-,-,+,-] => [3,1,2,4] => 2
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => 2
[+,-,4,3] => [1,3,2,4] => 1
[-,-,4,3] => [3,1,2,4] => 2
[+,3,2,+] => [1,2,4,3] => 1
[-,3,2,+] => [2,4,1,3] => 3
[+,3,2,-] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => 1
[+,3,4,2] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => 1
[+,4,2,3] => [1,2,3,4] => 0
Description
The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation.
This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by
$$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$
Matching statistic: St000004
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [1,3,2] => 2
[+,-,+] => [1,3,2] => [3,1,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [2,3,1] => 2
[-,+,-] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,1,3] => 1
[2,1,+] => [1,3,2] => [3,1,2] => 1
[2,1,-] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,1,3] => 1
[3,-,1] => [1,3,2] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [1,2,4,3] => 3
[+,-,+,+] => [1,3,4,2] => [2,4,1,3] => 2
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => 4
[-,+,-,+] => [2,4,1,3] => [1,3,4,2] => 3
[-,+,+,-] => [2,3,1,4] => [1,3,2,4] => 2
[+,-,-,+] => [1,4,2,3] => [3,4,1,2] => 2
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [2,3,4,1] => 3
[-,-,+,-] => [3,1,2,4] => [2,3,1,4] => 2
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => [1,3,2,4] => 2
[+,-,4,3] => [1,3,2,4] => [3,1,2,4] => 1
[-,-,4,3] => [3,1,2,4] => [2,3,1,4] => 2
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => 1
[-,3,2,+] => [2,4,1,3] => [1,3,4,2] => 3
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => 1
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => 0
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: St000067
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [[1]]
=> 0
[-] => [1] => [[1]]
=> 0
[+,+] => [1,2] => [[1,0],[0,1]]
=> 0
[-,+] => [2,1] => [[0,1],[1,0]]
=> 1
[+,-] => [1,2] => [[1,0],[0,1]]
=> 0
[-,-] => [1,2] => [[1,0],[0,1]]
=> 0
[2,1] => [1,2] => [[1,0],[0,1]]
=> 0
[+,+,+] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[-,+,+] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[+,-,+] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[+,+,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[-,-,+] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[-,+,-] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[+,-,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[-,-,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[+,3,2] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[-,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,+] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[3,1,2] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[3,+,1] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[3,-,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[+,+,+,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,+,+,+] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[+,-,+,+] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[+,+,-,+] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[+,+,+,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,-,+,+] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 4
[-,+,-,+] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[-,+,+,-] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[+,-,-,+] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[+,-,+,-] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[+,+,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,-,-,+] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[-,-,+,-] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[-,+,-,-] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[+,-,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,-,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[+,+,4,3] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,+,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[+,-,4,3] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[-,-,4,3] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[+,3,2,+] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[-,3,2,+] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[+,3,2,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,3,2,-] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[+,3,4,2] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[+,4,2,3] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
Description
The inversion number of the alternating sign matrix.
If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as
$$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$
When restricted to permutation matrices, this gives the usual inversion number of the permutation.
Matching statistic: St000081
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 0
[-] => [1] => ([],1)
=> 0
[+,+] => [1,2] => ([],2)
=> 0
[-,+] => [2,1] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => ([],2)
=> 0
[-,-] => [1,2] => ([],2)
=> 0
[2,1] => [1,2] => ([],2)
=> 0
[+,+,+] => [1,2,3] => ([],3)
=> 0
[-,+,+] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[+,-,+] => [1,3,2] => ([(1,2)],3)
=> 1
[+,+,-] => [1,2,3] => ([],3)
=> 0
[-,-,+] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[-,+,-] => [2,1,3] => ([(1,2)],3)
=> 1
[+,-,-] => [1,2,3] => ([],3)
=> 0
[-,-,-] => [1,2,3] => ([],3)
=> 0
[+,3,2] => [1,2,3] => ([],3)
=> 0
[-,3,2] => [2,1,3] => ([(1,2)],3)
=> 1
[2,1,+] => [1,3,2] => ([(1,2)],3)
=> 1
[2,1,-] => [1,2,3] => ([],3)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> 0
[3,1,2] => [1,2,3] => ([],3)
=> 0
[3,+,1] => [2,1,3] => ([(1,2)],3)
=> 1
[3,-,1] => [1,3,2] => ([(1,2)],3)
=> 1
[+,+,+,+] => [1,2,3,4] => ([],4)
=> 0
[-,+,+,+] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[+,-,+,+] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[+,+,-,+] => [1,2,4,3] => ([(2,3)],4)
=> 1
[+,+,+,-] => [1,2,3,4] => ([],4)
=> 0
[-,-,+,+] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[-,+,+,-] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[+,-,-,+] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[+,-,+,-] => [1,3,2,4] => ([(2,3)],4)
=> 1
[+,+,-,-] => [1,2,3,4] => ([],4)
=> 0
[-,-,-,+] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[-,-,+,-] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[-,+,-,-] => [2,1,3,4] => ([(2,3)],4)
=> 1
[+,-,-,-] => [1,2,3,4] => ([],4)
=> 0
[-,-,-,-] => [1,2,3,4] => ([],4)
=> 0
[+,+,4,3] => [1,2,3,4] => ([],4)
=> 0
[-,+,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[+,-,4,3] => [1,3,2,4] => ([(2,3)],4)
=> 1
[-,-,4,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[+,3,2,+] => [1,2,4,3] => ([(2,3)],4)
=> 1
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[+,3,2,-] => [1,2,3,4] => ([],4)
=> 0
[-,3,2,-] => [2,1,3,4] => ([(2,3)],4)
=> 1
[+,3,4,2] => [1,2,3,4] => ([],4)
=> 0
[-,3,4,2] => [2,1,3,4] => ([(2,3)],4)
=> 1
[+,4,2,3] => [1,2,3,4] => ([],4)
=> 0
Description
The number of edges of a graph.
Matching statistic: St000246
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 0
[-] => [1] => [1] => 0
[+,+] => [1,2] => [2,1] => 0
[-,+] => [2,1] => [1,2] => 1
[+,-] => [1,2] => [2,1] => 0
[-,-] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[+,+,+] => [1,2,3] => [3,2,1] => 0
[-,+,+] => [2,3,1] => [2,1,3] => 2
[+,-,+] => [1,3,2] => [3,1,2] => 1
[+,+,-] => [1,2,3] => [3,2,1] => 0
[-,-,+] => [3,1,2] => [1,3,2] => 2
[-,+,-] => [2,1,3] => [2,3,1] => 1
[+,-,-] => [1,2,3] => [3,2,1] => 0
[-,-,-] => [1,2,3] => [3,2,1] => 0
[+,3,2] => [1,2,3] => [3,2,1] => 0
[-,3,2] => [2,1,3] => [2,3,1] => 1
[2,1,+] => [1,3,2] => [3,1,2] => 1
[2,1,-] => [1,2,3] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => 0
[3,+,1] => [2,1,3] => [2,3,1] => 1
[3,-,1] => [1,3,2] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => 0
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => 3
[+,-,+,+] => [1,3,4,2] => [4,2,1,3] => 2
[+,+,-,+] => [1,2,4,3] => [4,3,1,2] => 1
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => 4
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 3
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => 2
[+,-,-,+] => [1,4,2,3] => [4,1,3,2] => 2
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => 1
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => 3
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => 2
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => 1
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[+,+,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => 2
[+,-,4,3] => [1,3,2,4] => [4,2,3,1] => 1
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => 2
[+,3,2,+] => [1,2,4,3] => [4,3,1,2] => 1
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => 3
[+,3,2,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => 1
[+,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => 1
[+,4,2,3] => [1,2,3,4] => [4,3,2,1] => 0
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: St000332
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000332: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000332: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [[1]]
=> 0
[-] => [1] => [[1]]
=> 0
[+,+] => [1,2] => [[1,0],[0,1]]
=> 0
[-,+] => [2,1] => [[0,1],[1,0]]
=> 1
[+,-] => [1,2] => [[1,0],[0,1]]
=> 0
[-,-] => [1,2] => [[1,0],[0,1]]
=> 0
[2,1] => [1,2] => [[1,0],[0,1]]
=> 0
[+,+,+] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[-,+,+] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[+,-,+] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[+,+,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[-,-,+] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[-,+,-] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[+,-,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[-,-,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[+,3,2] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[-,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,+] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,-] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[3,1,2] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[3,+,1] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[3,-,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[+,+,+,+] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,+,+,+] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[+,-,+,+] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[+,+,-,+] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[+,+,+,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,-,+,+] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 4
[-,+,-,+] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[-,+,+,-] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[+,-,-,+] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[+,-,+,-] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[+,+,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,-,-,+] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[-,-,+,-] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[-,+,-,-] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[+,-,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,-,-,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[+,+,4,3] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,+,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[+,-,4,3] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[-,-,4,3] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[+,3,2,+] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[-,3,2,+] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[+,3,2,-] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,3,2,-] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[+,3,4,2] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[-,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[+,4,2,3] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
Description
The positive inversions of an alternating sign matrix.
This is defined as
$$\sum_{i > k,j < l} A_{ij}A_{kl} - \text{the number of negative ones in the matrix}.$$
After counter-clockwise rotation, this is also the number of osculations in the corresponding fan of Dyck paths.
Matching statistic: St001397
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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)
=> 0
[-] => [1] => ([],1)
=> 0
[+,+] => [1,2] => ([(0,1)],2)
=> 0
[-,+] => [2,1] => ([],2)
=> 1
[+,-] => [1,2] => ([(0,1)],2)
=> 0
[-,-] => [1,2] => ([(0,1)],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> 0
[+,+,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,+,+] => [2,3,1] => ([(1,2)],3)
=> 2
[+,-,+] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[+,+,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,-,+] => [3,1,2] => ([(1,2)],3)
=> 2
[-,+,-] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[+,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,3,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[2,1,+] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[2,1,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[3,+,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[3,-,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[+,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,+,+] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3
[+,-,+,+] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[+,+,-,+] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[+,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,+,+] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 4
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 3
[-,+,+,-] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[+,-,-,+] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[+,-,+,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[+,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,-,+] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 3
[-,-,+,-] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[-,+,-,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[+,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,4,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[+,-,4,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[-,-,4,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[+,3,2,+] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 3
[+,3,2,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,3,2,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[+,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,3,4,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[+,4,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 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)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00175: Permutations —inverse Foata 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%
Mp00175: Permutations —inverse Foata 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] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [2] => 0
[-,+] => [2,1] => [2,1] => [1,1] => 1
[+,-] => [1,2] => [1,2] => [2] => 0
[-,-] => [1,2] => [1,2] => [2] => 0
[2,1] => [1,2] => [1,2] => [2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [3] => 0
[-,+,+] => [2,3,1] => [2,3,1] => [2,1] => 2
[+,-,+] => [1,3,2] => [3,1,2] => [1,2] => 1
[+,+,-] => [1,2,3] => [1,2,3] => [3] => 0
[-,-,+] => [3,1,2] => [1,3,2] => [2,1] => 2
[-,+,-] => [2,1,3] => [2,1,3] => [1,2] => 1
[+,-,-] => [1,2,3] => [1,2,3] => [3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => [3] => 0
[-,3,2] => [2,1,3] => [2,1,3] => [1,2] => 1
[2,1,+] => [1,3,2] => [3,1,2] => [1,2] => 1
[2,1,-] => [1,2,3] => [1,2,3] => [3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [3] => 0
[3,+,1] => [2,1,3] => [2,1,3] => [1,2] => 1
[3,-,1] => [1,3,2] => [3,1,2] => [1,2] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[+,-,+,+] => [1,3,4,2] => [3,4,1,2] => [2,2] => 2
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [1,2,1] => 4
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => [1,1,2] => 3
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [2,2] => 2
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [3,1] => 3
[-,-,+,-] => [3,1,2,4] => [1,3,2,4] => [2,2] => 2
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[+,-,4,3] => [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[-,-,4,3] => [3,1,2,4] => [1,3,2,4] => [2,2] => 2
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[-,3,2,+] => [2,4,1,3] => [4,2,1,3] => [1,1,2] => 3
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [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)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ 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%
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] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 1
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [2,3,1] => [2,3,1] => [3,1,2] => 2
[+,-,+] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,3,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,+] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,+,1] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,-,1] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[+,-,+,+] => [1,3,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 4
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 3
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 2
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 3
[-,-,+,-] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[+,-,4,3] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[-,-,4,3] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[-,3,2,+] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 3
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
Description
The inverse major index of a permutation.
This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000330The (standard) major index of a standard tableau. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000795The mad of a permutation. St000391The sum of the positions of the ones in a binary word. St000833The comajor index of a permutation. St001428The number of B-inversions of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. 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.
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