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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000422
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> 0
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([],3)
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([],3)
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([],4)
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> 0
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000896
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 75%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1] => [[1]]
=> ? = 0
{{1,2}}
=> [2,1] => [1,2] => [[1,0],[0,1]]
=> 0
{{1},{2}}
=> [1,2] => [1,2] => [[1,0],[0,1]]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [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}}
=> [2,3,1,4] => [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}}
=> [2,4,3,1] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [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}}
=> [2,1,3,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
{{1},{2,3,4}}
=> [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
{{1},{2,3},{4}}
=> [1,3,2,4] => [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}}
=> [1,4,3,2] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [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}}
=> [1,2,3,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [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
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [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}}
=> [2,3,5,4,1] => [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]]
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [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},{4},{5}}
=> [2,3,1,4,5] => [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
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [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]]
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [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]]
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [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},{3,4},{5}}
=> [2,1,4,3,5] => [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}}
=> [2,1,5,4,3] => [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]]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [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},{4},{5}}
=> [2,1,3,4,5] => [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
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [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]]
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [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]]
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [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]]
=> 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [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]]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [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]]
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [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
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [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},{2,3,4},{5}}
=> [1,3,4,2,5] => [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}}
=> [1,3,5,4,2] => [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]]
=> 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [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},{4},{5}}
=> [1,3,2,4,5] => [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
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [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
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [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]]
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [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]]
=> 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [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},{3,4},{5}}
=> [1,2,4,3,5] => [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}}
=> [1,2,5,4,3] => [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]]
=> 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [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},{4},{5}}
=> [1,2,3,4,5] => [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
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 4
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 4
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 4
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
{{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
{{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
{{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
{{1},{2,5,6},{3,4}}
=> [1,5,4,3,6,2] => [1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 4
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 4
{{1,6},{2},{3},{4,5}}
=> [6,2,3,5,4,1] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
{{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,2,3,5,4,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 4
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,3,6,7,4,5] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 4
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,3,6,7,4,5] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 4
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,3},{4},{5},{6},{7}}
=> [2,3,1,4,5,6,7] => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [1,2,4,5,6,7,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4
{{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => [1,2,4,3,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => [1,2,4,3,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,4},{3,5,7},{6}}
=> [2,4,5,1,7,6,3] => [1,2,4,3,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 4
{{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => [1,2,4,3,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
Description
The number of zeros on the main diagonal of an alternating sign matrix.
Matching statistic: St001892
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001892: Signed permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 75%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001892: Signed permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 4
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 4
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 2
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 2
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 4
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 2
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 2
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 4
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 4
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? = 4
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 4
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 4
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 4
{{1,3},{2},{4,5,6}}
=> [3,2,1,5,6,4] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
{{1,3},{2},{4,5},{6}}
=> [3,2,1,5,4,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
{{1,3},{2},{4,6},{5}}
=> [3,2,1,6,5,4] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 4
{{1,3},{2},{4},{5,6}}
=> [3,2,1,4,6,5] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
{{1,3},{2},{4},{5},{6}}
=> [3,2,1,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ? = 4
{{1,4,5},{2,3},{6}}
=> [4,3,2,5,1,6] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ? = 4
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 4
{{1},{2,3,4,5,6}}
=> [1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
{{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 4
Description
The flag excedance statistic of a signed permutation.
This is the number of negative entries plus twice the number of excedances of the signed permutation. That is,
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
where
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$
It has the same distribution as the flag descent statistic.
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