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Your data matches 85 different statistics following compositions of up to 3 maps.
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Matching statistic: St000018
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
St000018: 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] => 2
[3,1,2] => 2
[3,2,1] => 3
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 4
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 3
[3,2,4,1] => 4
[3,4,1,2] => 4
[3,4,2,1] => 5
[4,1,2,3] => 3
[4,1,3,2] => 4
[4,2,1,3] => 4
[4,2,3,1] => 5
[4,3,1,2] => 5
[4,3,2,1] => 6
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 4
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 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 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
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%
St000004: 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] => [3,1,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => 2
[3,1,2] => [2,3,1] => 2
[3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [4,1,2,3] => 1
[1,3,2,4] => [3,1,2,4] => 1
[1,3,4,2] => [2,4,1,3] => 2
[1,4,2,3] => [3,4,1,2] => 2
[1,4,3,2] => [4,3,1,2] => 3
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,3,2,4] => 2
[2,3,4,1] => [1,2,4,3] => 3
[2,4,1,3] => [1,3,4,2] => 3
[2,4,3,1] => [4,1,3,2] => 4
[3,1,2,4] => [2,3,1,4] => 2
[3,1,4,2] => [4,2,1,3] => 3
[3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [2,1,4,3] => 4
[3,4,1,2] => [3,1,4,2] => 4
[3,4,2,1] => [1,4,3,2] => 5
[4,1,2,3] => [2,3,4,1] => 3
[4,1,3,2] => [4,2,3,1] => 4
[4,2,1,3] => [3,2,4,1] => 4
[4,2,3,1] => [2,4,3,1] => 5
[4,3,1,2] => [3,4,2,1] => 5
[4,3,2,1] => [4,3,2,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [4,1,2,3,5] => 1
[1,2,4,5,3] => [3,5,1,2,4] => 2
[1,2,5,3,4] => [4,5,1,2,3] => 2
[1,2,5,4,3] => [5,4,1,2,3] => 3
[1,3,2,4,5] => [3,1,2,4,5] => 1
[1,3,2,5,4] => [2,5,1,3,4] => 2
[1,3,4,2,5] => [2,4,1,3,5] => 2
[1,3,4,5,2] => [2,3,5,1,4] => 3
[1,3,5,2,4] => [2,4,5,1,3] => 3
[1,3,5,4,2] => [5,2,4,1,3] => 4
[1,4,2,3,5] => [3,4,1,2,5] => 2
[1,4,2,5,3] => [5,3,1,2,4] => 3
[1,4,3,2,5] => [4,3,1,2,5] => 3
[1,4,3,5,2] => [3,2,5,1,4] => 4
[1,4,5,2,3] => [4,2,5,1,3] => 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: St000067
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
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%
St000067: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> 0
[1,2] => [[1,0],[0,1]]
=> 0
[2,1] => [[0,1],[1,0]]
=> 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 3
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 4
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 5
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 4
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 5
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 5
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 6
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 2
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 3
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 4
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 3
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 4
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 4
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
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
Description
The number of edges of a graph.
Matching statistic: St000246
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 1
[1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => 1
[2,1,3] => [2,3,1] => 1
[2,3,1] => [2,1,3] => 2
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,2,3] => 3
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [4,2,1,3] => 2
[1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2] => [4,1,2,3] => 3
[2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,2,4,1] => 2
[2,3,4,1] => [3,2,1,4] => 3
[2,4,1,3] => [3,1,4,2] => 3
[2,4,3,1] => [3,1,2,4] => 4
[3,1,2,4] => [2,4,3,1] => 2
[3,1,4,2] => [2,4,1,3] => 3
[3,2,1,4] => [2,3,4,1] => 3
[3,2,4,1] => [2,3,1,4] => 4
[3,4,1,2] => [2,1,4,3] => 4
[3,4,2,1] => [2,1,3,4] => 5
[4,1,2,3] => [1,4,3,2] => 3
[4,1,3,2] => [1,4,2,3] => 4
[4,2,1,3] => [1,3,4,2] => 4
[4,2,3,1] => [1,3,2,4] => 5
[4,3,1,2] => [1,2,4,3] => 5
[4,3,2,1] => [1,2,3,4] => 6
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [5,4,2,1,3] => 2
[1,2,5,3,4] => [5,4,1,3,2] => 2
[1,2,5,4,3] => [5,4,1,2,3] => 3
[1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [5,3,2,4,1] => 2
[1,3,4,5,2] => [5,3,2,1,4] => 3
[1,3,5,2,4] => [5,3,1,4,2] => 3
[1,3,5,4,2] => [5,3,1,2,4] => 4
[1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => 3
[1,4,3,5,2] => [5,2,3,1,4] => 4
[1,4,5,2,3] => [5,2,1,4,3] => 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: St000332
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
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%
St000332: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> 0
[1,2] => [[1,0],[0,1]]
=> 0
[2,1] => [[0,1],[1,0]]
=> 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 3
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 4
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 5
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 4
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 5
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 5
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 6
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 2
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 3
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 4
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 3
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 4
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 4
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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> 0
[2,1] => ([],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => ([(0,1),(0,2)],3)
=> 1
[2,1,3] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => ([(1,2)],3)
=> 2
[3,1,2] => ([(1,2)],3)
=> 2
[3,2,1] => ([],3)
=> 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 3
[2,4,3,1] => ([(1,2),(1,3)],4)
=> 4
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => ([(1,3),(2,3)],4)
=> 4
[3,4,1,2] => ([(0,3),(1,2)],4)
=> 4
[3,4,2,1] => ([(2,3)],4)
=> 5
[4,1,2,3] => ([(1,2),(2,3)],4)
=> 3
[4,1,3,2] => ([(1,2),(1,3)],4)
=> 4
[4,2,1,3] => ([(1,3),(2,3)],4)
=> 4
[4,2,3,1] => ([(2,3)],4)
=> 5
[4,3,1,2] => ([(2,3)],4)
=> 5
[4,3,2,1] => ([],4)
=> 6
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 4
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 4
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: St001428
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001428: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001428: Signed 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] => [2,3,1] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 3
[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] => [1,3,4,2] => 2
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => 3
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [2,3,4,1] => 3
[2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => 4
[3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [3,2,4,1] => 4
[3,4,1,2] => [3,4,1,2] => 4
[3,4,2,1] => [3,4,2,1] => 5
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,1,3,2] => 4
[4,2,1,3] => [4,2,1,3] => 4
[4,2,3,1] => [4,2,3,1] => 5
[4,3,1,2] => [4,3,1,2] => 5
[4,3,2,1] => [4,3,2,1] => 6
[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] => [1,2,4,5,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,4,3] => 3
[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] => [1,3,4,2,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => 3
[1,3,5,2,4] => [1,3,5,2,4] => 3
[1,3,5,4,2] => [1,3,5,4,2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => 3
[1,4,3,2,5] => [1,4,3,2,5] => 3
[1,4,3,5,2] => [1,4,3,5,2] => 4
[1,4,5,2,3] => [1,4,5,2,3] => 4
Description
The number of B-inversions of a signed permutation.
The number of B-inversions of a signed permutation $\sigma$ of length $n$ is
$$
\operatorname{inv}_B(\sigma) = \big|\{ 1 \leq i < j \leq n \mid \sigma(i) > \sigma(j) \}\big| + \big|\{ 1 \leq i \leq j \leq n \mid \sigma(-i) > \sigma(j) \}\big|,
$$
see [1, Eq. (8.2)]. According to [1, Eq. (8.4)], this is the Coxeter length of $\sigma$.
Matching statistic: St000008
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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
[1,2] => [1,2] => [2] => 0
[2,1] => [2,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [3] => 0
[1,3,2] => [3,1,2] => [1,2] => 1
[2,1,3] => [2,1,3] => [1,2] => 1
[2,3,1] => [1,3,2] => [2,1] => 2
[3,1,2] => [2,3,1] => [2,1] => 2
[3,2,1] => [3,2,1] => [1,1,1] => 3
[1,2,3,4] => [1,2,3,4] => [4] => 0
[1,2,4,3] => [4,1,2,3] => [1,3] => 1
[1,3,2,4] => [3,1,2,4] => [1,3] => 1
[1,3,4,2] => [2,4,1,3] => [2,2] => 2
[1,4,2,3] => [3,4,1,2] => [2,2] => 2
[1,4,3,2] => [4,3,1,2] => [1,1,2] => 3
[2,1,3,4] => [2,1,3,4] => [1,3] => 1
[2,1,4,3] => [1,4,2,3] => [2,2] => 2
[2,3,1,4] => [1,3,2,4] => [2,2] => 2
[2,3,4,1] => [1,2,4,3] => [3,1] => 3
[2,4,1,3] => [1,3,4,2] => [3,1] => 3
[2,4,3,1] => [4,1,3,2] => [1,2,1] => 4
[3,1,2,4] => [2,3,1,4] => [2,2] => 2
[3,1,4,2] => [4,2,1,3] => [1,1,2] => 3
[3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[3,2,4,1] => [2,1,4,3] => [1,2,1] => 4
[3,4,1,2] => [3,1,4,2] => [1,2,1] => 4
[3,4,2,1] => [1,4,3,2] => [2,1,1] => 5
[4,1,2,3] => [2,3,4,1] => [3,1] => 3
[4,1,3,2] => [4,2,3,1] => [1,2,1] => 4
[4,2,1,3] => [3,2,4,1] => [1,2,1] => 4
[4,2,3,1] => [2,4,3,1] => [2,1,1] => 5
[4,3,1,2] => [3,4,2,1] => [2,1,1] => 5
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [5] => 0
[1,2,3,5,4] => [5,1,2,3,4] => [1,4] => 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,4] => 1
[1,2,4,5,3] => [3,5,1,2,4] => [2,3] => 2
[1,2,5,3,4] => [4,5,1,2,3] => [2,3] => 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 3
[1,3,2,4,5] => [3,1,2,4,5] => [1,4] => 1
[1,3,2,5,4] => [2,5,1,3,4] => [2,3] => 2
[1,3,4,2,5] => [2,4,1,3,5] => [2,3] => 2
[1,3,4,5,2] => [2,3,5,1,4] => [3,2] => 3
[1,3,5,2,4] => [2,4,5,1,3] => [3,2] => 3
[1,3,5,4,2] => [5,2,4,1,3] => [1,2,2] => 4
[1,4,2,3,5] => [3,4,1,2,5] => [2,3] => 2
[1,4,2,5,3] => [5,3,1,2,4] => [1,1,3] => 3
[1,4,3,2,5] => [4,3,1,2,5] => [1,1,3] => 3
[1,4,3,5,2] => [3,2,5,1,4] => [1,2,2] => 4
[1,4,5,2,3] => [4,2,5,1,3] => [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: St000228
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 5
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 5
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 5
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 6
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
The following 75 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. St000330The (standard) major index of a standard tableau. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000784The maximum of the length and the largest part of the integer partition. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St001862The number of crossings of a signed permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000450The number of edges minus the number of vertices plus 2 of a graph. St000532The total number of rook placements on a Ferrers board. St001400The total number of Littlewood-Richardson tableaux of given shape. St000005The bounce statistic of a Dyck path. St000009The charge of a standard tableau. St000169The cocharge of a standard tableau. St000304The load of a permutation. St000446The disorder of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St001161The major index north count of a Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000494The number of inversions of distance at most 3 of a permutation. St000795The mad of a permutation. St000391The sum of the positions of the ones in a binary word. St000833The comajor index of a permutation. St000947The major index east count of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001746The coalition number of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000636The hull number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St000681The Grundy value of Chomp on Ferrers diagrams. St001622The number of join-irreducible elements of a lattice. 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. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001822The number of alignments of a signed permutation. St001866The nesting alignments 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. St001621The number of atoms of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000766The number of inversions of an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000072The number of circled entries. St000646The number of big ascents of a permutation. St001330The hat guessing number of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
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