searching the database
Your data matches 111 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: 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
St000455: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],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)
=> 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001434
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001434: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001434: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [2,3,1,4] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [5,1,3,2,4] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [5,1,2,4,3] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [4,1,2,5,3] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [5,1,4,2,3] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [5,2,1,3,4] => 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,1,5] => [4,2,3,1,5] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,1,5] => [3,2,4,1,5] => 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,1,4,5] => [2,3,1,4,5] => 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,5,1,3,4] => [2,5,1,3,4] => 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,1,5,4] => [2,3,1,5,4] => 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,1,5] => [2,4,3,1,5] => 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,1,5] => [2,3,4,1,5] => 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [5,3,4,2,1] => [5,3,4,2,1] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => ? = 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,5,2,3,6,4] => [1,5,2,3,6,4] => 0
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => ? = 0
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [1,6,3,2,4,5] => [1,6,3,2,4,5] => ? = 0
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,5,6,3,4,2] => [1,5,6,3,4,2] => ? = 0
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [6,1,3,4,2,5] => [6,1,3,4,2,5] => ? = 0
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [6,1,3,2,5,4] => [6,1,3,2,5,4] => ? = 0
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [5,1,3,2,6,4] => [5,1,3,2,6,4] => ? = 0
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [6,1,3,5,2,4] => [6,1,3,5,2,4] => ? = 1
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [6,1,2,4,5,3] => [6,1,2,4,5,3] => ? = 0
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [5,1,2,4,6,3] => [5,1,2,4,6,3] => ? = 0
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [6,1,2,5,4,3] => [6,1,2,5,4,3] => ? = 0
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,1,2,6,5,3] => [4,1,2,6,5,3] => ? = 0
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [6,1,4,3,2,5] => [6,1,4,3,2,5] => ? = 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [5,1,4,2,3,6] => [5,1,4,2,3,6] => ? = 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [6,1,4,2,5,3] => [6,1,4,2,5,3] => ? = 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [5,1,6,2,4,3] => [5,1,6,2,4,3] => ? = 0
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [5,1,4,2,6,3] => [5,1,4,2,6,3] => ? = 0
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [6,1,5,4,2,3] => [6,1,5,4,2,3] => ? = 1
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [6,1,4,5,2,3] => [6,1,4,5,2,3] => ? = 1
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [6,2,1,4,3,5] => [6,2,1,4,3,5] => ? = 0
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [6,2,1,3,5,4] => [6,2,1,3,5,4] => ? = 0
{{1,4},{2,3,5},{6}}
=> [4,3,5,1,2,6] => [5,2,1,3,6,4] => [5,2,1,3,6,4] => ? = 0
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [6,2,1,5,3,4] => [6,2,1,5,3,4] => ? = 1
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [5,2,6,3,4,1] => [5,2,6,3,4,1] => ? = 0
{{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [2,6,1,4,3,5] => [2,6,1,4,3,5] => ? = 0
{{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => [6,5,1,3,2,4] => [6,5,1,3,2,4] => ? = 1
{{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => [2,6,1,3,5,4] => [2,6,1,3,5,4] => ? = 0
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [6,3,1,5,2,4] => [6,3,1,5,2,4] => ? = 1
{{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => [2,6,1,5,3,4] => [2,6,1,5,3,4] => ? = 1
{{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [5,6,2,4,3,1] => [5,6,2,4,3,1] => ? = 0
{{1},{2,4},{3,5,6}}
=> [1,4,5,2,6,3] => [4,6,2,3,5,1] => [4,6,2,3,5,1] => ? = 0
{{1},{2,4},{3,6},{5}}
=> [1,4,6,2,5,3] => [4,6,2,5,3,1] => [4,6,2,5,3,1] => ? = 0
{{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => [2,6,3,1,4,5] => [2,6,3,1,4,5] => ? = 0
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [5,3,6,2,4,1] => [5,3,6,2,4,1] => ? = 0
{{1},{2},{3,5},{4,6}}
=> [1,2,5,6,3,4] => [2,5,6,3,4,1] => [2,5,6,3,4,1] => ? = 0
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 0
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 0
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 0
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 0
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 0
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => ? = 0
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 0
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 0
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => ? = 0
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 0
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => ? = 0
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => ? = 0
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => ? = 0
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 0
Description
The number of negative sum pairs of a signed permutation.
The number of negative sum pairs of a signed permutation $\sigma$ is:
$$\operatorname{nsp}(\sigma)=\big|\{1\leq i < j\leq n \mid \sigma(i)+\sigma(j) < 0\}\big|,$$
see [1, Eq.(8.1)].
Matching statistic: St001198
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3}}
=> [2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [4,5,3,1,2,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3},{2},{4,5,6}}
=> [3,2,1,5,6,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4,5},{6}}
=> [3,2,1,5,4,6] => [3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,5,6},{2,3,4}}
=> [5,3,4,2,6,1] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3}}
=> [2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [4,5,3,1,2,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,3},{2},{4,5,6}}
=> [3,2,1,5,6,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4,5},{6}}
=> [3,2,1,5,4,6] => [3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,5,6},{2,3,4}}
=> [5,3,4,2,6,1] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St001260
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1 = 0 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,2,1,5,3] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,2,4,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [3,2,5,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 1 = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [4,2,5,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [3,5,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [6,1,2,3,5,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [4,1,2,3,6,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 1 = 0 + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [5,1,2,3,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [5,1,2,4,3,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> 1 = 0 + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [6,5,1,2,4,3] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [4,6,1,2,5,3] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [6,1,2,5,3,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [3,1,2,4,6,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 1 = 0 + 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [2,5,6,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [3,1,2,5,4,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 1 = 0 + 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [2,6,1,4,3,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 0 + 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [4,1,2,5,3,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 1 = 0 + 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [2,6,1,4,5,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [3,1,2,6,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [4,1,3,2,5,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 1 = 0 + 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [4,3,1,2,6,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 1 = 0 + 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [5,3,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0 + 1
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [5,4,1,3,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> 1 = 0 + 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [3,6,5,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [4,3,6,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [6,1,3,5,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [3,4,1,2,6,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 1 = 0 + 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [3,5,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [3,5,1,4,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> 1 = 0 + 1
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [4,1,3,6,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [5,1,3,6,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [6,3,5,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [6,5,1,4,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [3,4,6,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [4,6,1,5,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [6,1,5,2,3,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [5,1,3,6,4,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [6,1,4,3,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [6,1,4,3,5,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 + 1
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [1,2,6,5,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [1,6,3,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 0 + 1
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [5,6,1,4,3,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 + 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [2,6,1,5,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1 + 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [1,4,6,3,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1 + 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [1,5,6,3,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [6,1,5,3,4,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 1
{{1,3},{2},{4,5,6}}
=> [3,2,1,5,6,4] => [2,1,4,6,3,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [1,4,6,3,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1 + 1
{{1,3},{2},{4,6},{5}}
=> [3,2,1,6,5,4] => [1,6,2,5,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
{{1,3},{2},{4},{5,6}}
=> [3,2,1,4,6,5] => [2,1,6,3,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0 + 1
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [2,1,5,6,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [2,1,6,4,3,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0 + 1
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [2,1,6,4,5,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1 + 1
{{1,4},{2,3},{5,6}}
=> [4,3,2,1,6,5] => [3,2,1,6,4,5] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [2,5,3,1,6,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0 + 1
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
{{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [2,4,3,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1},{2,3,4,6},{5}}
=> [1,3,4,6,5,2] => [6,2,3,5,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
{{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0 + 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [4,2,3,6,1,5] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [5,2,3,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
{{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => [2,6,3,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
{{1},{2,3,6},{4,5}}
=> [1,3,6,5,4,2] => [6,5,2,4,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
Description
The permanent of an alternating sign matrix.
Matching statistic: St001771
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [2,1] => [2,1] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [2,3,1] => [2,3,1] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,1] => [3,2,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,2] => [3,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1] => [3,2,1] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1] => [2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,2] => [1,3,2] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,1,3] => [2,4,1,3] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,1] => [3,2,4,1] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1] => [3,2,4,1] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,2] => [4,3,1,2] => 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,1] => [4,2,3,1] => 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,2] => [1,3,4,2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 0
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 0
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 0
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 0
{{1},{2,3,4,6},{5}}
=> [1,3,4,6,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,6},{4,5}}
=> [1,3,6,5,4,2] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,6},{4},{5}}
=> [1,3,6,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
{{1},{2,3},{4,6},{5}}
=> [1,3,2,6,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,4,5,6},{3}}
=> [1,4,3,5,6,2] => [1,4,3,5,2] => [1,4,3,5,2] => 0
{{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => [1,4,3,5,2] => [1,4,3,5,2] => 0
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001870
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [2,1] => [2,1] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [2,3,1] => [2,3,1] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,1] => [3,2,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,2] => [3,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1] => [3,2,1] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1] => [2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,2] => [1,3,2] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,1,3] => [2,4,1,3] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,1] => [3,2,4,1] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1] => [3,2,4,1] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,2] => [4,3,1,2] => 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,1] => [4,2,3,1] => 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,2] => [1,3,4,2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 0
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 0
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 0
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 0
{{1},{2,3,4,6},{5}}
=> [1,3,4,6,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,6},{4,5}}
=> [1,3,6,5,4,2] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,6},{4},{5}}
=> [1,3,6,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
{{1},{2,3},{4,6},{5}}
=> [1,3,2,6,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,4,5,6},{3}}
=> [1,4,3,5,6,2] => [1,4,3,5,2] => [1,4,3,5,2] => 0
{{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => [1,4,3,5,2] => [1,4,3,5,2] => 0
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [2,1] => [2,1] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [2,3,1] => [2,3,1] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,1] => [3,2,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,2] => [3,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1] => [3,2,1] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1] => [2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,2] => [1,3,2] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,1,3] => [2,4,1,3] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,1] => [3,2,4,1] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1] => [3,2,4,1] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,2] => [4,3,1,2] => 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,1] => [4,2,3,1] => 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,2] => [1,3,4,2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 0
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 0
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 0
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 0
{{1},{2,3,4,6},{5}}
=> [1,3,4,6,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,6},{4,5}}
=> [1,3,6,5,4,2] => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1},{2,3,6},{4},{5}}
=> [1,3,6,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
{{1},{2,3},{4,6},{5}}
=> [1,3,2,6,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,4,5,6},{3}}
=> [1,4,3,5,6,2] => [1,4,3,5,2] => [1,4,3,5,2] => 0
{{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => [1,4,3,5,2] => [1,4,3,5,2] => 0
Description
The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St001597
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 2 = 0 + 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 2 = 0 + 2
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 2 = 0 + 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> 2 = 0 + 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2 = 0 + 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2 = 0 + 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2 = 0 + 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 2 = 0 + 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> 2 = 0 + 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> 2 = 0 + 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> 2 = 0 + 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> 2 = 0 + 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> 2 = 0 + 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> 2 = 0 + 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> 2 = 0 + 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> 2 = 0 + 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 0 + 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> 2 = 0 + 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> 2 = 0 + 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> 2 = 0 + 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 0 + 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> 2 = 0 + 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> 2 = 0 + 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> 2 = 0 + 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 0 + 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> 2 = 0 + 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 2
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> 2 = 0 + 2
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> 2 = 0 + 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 0 + 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> 2 = 0 + 2
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> 2 = 0 + 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> 2 = 0 + 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> 2 = 0 + 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 2
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> 2 = 0 + 2
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> 2 = 0 + 2
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> 2 = 0 + 2
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5],[3]]
=> 2 = 0 + 2
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> 2 = 0 + 2
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> ? = 0 + 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> 2 = 0 + 2
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> ? = 0 + 2
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> ? = 0 + 2
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> 2 = 0 + 2
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[5,3],[1]]
=> 2 = 0 + 2
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ? = 0 + 2
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> 2 = 0 + 2
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> ? = 0 + 2
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> ? = 0 + 2
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ? = 0 + 2
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> 2 = 0 + 2
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> 2 = 0 + 2
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ? = 0 + 2
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> ? = 0 + 2
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ? = 0 + 2
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 0 + 2
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 0 + 2
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> 2 = 0 + 2
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ? = 0 + 2
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> ? = 0 + 2
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ? = 0 + 2
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> 2 = 0 + 2
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ? = 0 + 2
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> 2 = 0 + 2
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 0 + 2
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? = 0 + 2
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 0 + 2
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? = 0 + 2
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> 2 = 0 + 2
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> 2 = 0 + 2
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> ? = 0 + 2
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2],[3]]
=> 2 = 0 + 2
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[4,3,2],[]]
=> ? = 0 + 2
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 0 + 2
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> ? = 1 + 2
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> 2 = 0 + 2
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2]]
=> 2 = 0 + 2
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> ? = 0 + 2
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> ? = 0 + 2
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[3,3,2,2],[1]]
=> ? = 0 + 2
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ? = 0 + 2
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1]]
=> ? = 0 + 2
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> ? = 1 + 2
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2],[]]
=> ? = 1 + 2
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[4,3,2],[]]
=> ? = 1 + 2
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2],[1]]
=> ? = 0 + 2
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 0 + 2
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> ? = 1 + 2
{{1,3},{2},{4,5,6}}
=> [3,2,1,5,6,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2],[1]]
=> 2 = 0 + 2
{{1,3},{2},{4,5},{6}}
=> [3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> 2 = 0 + 2
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> ? = 1 + 2
{{1,3},{2},{4,6},{5}}
=> [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ? = 0 + 2
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> ? = 0 + 2
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> ? = 0 + 2
{{1,4,5},{2,3},{6}}
=> [4,3,2,5,1,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ? = 0 + 2
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [1,1,1,1,0,0,1,0,0,1,0,0]
=> [[4,3,3],[1]]
=> ? = 0 + 2
{{1,4},{2,3,5},{6}}
=> [4,3,5,1,2,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ? = 0 + 2
Description
The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St000787
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 50%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 50%
Values
{{1,3},{2}}
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 0
{{1},{2,3,5},{4}}
=> [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]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 0
{{1},{2,4,5},{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]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 0
{{1},{2,4},{3,5}}
=> [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]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 0
{{1},{2,4},{3},{5}}
=> [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]]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 0
{{1},{2},{3,5},{4}}
=> [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]]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [(1,12),(2,11),(3,4),(5,6),(7,10),(8,9)]
=> ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)]
=> ? = 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [(1,12),(2,3),(4,7),(5,6),(8,9),(10,11)]
=> ? = 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [(1,12),(2,5),(3,4),(6,9),(7,8),(10,11)]
=> ? = 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
=> ? = 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)]
=> ? = 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? = 0
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? = 0
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]
=> ? = 0
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,12),(2,5),(3,4),(6,11),(7,8),(9,10)]
=> ? = 0
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> ? = 0
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [(1,12),(2,5),(3,4),(6,7),(8,11),(9,10)]
=> ? = 0
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> ? = 0
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,12),(2,5),(3,4),(6,7),(8,11),(9,10)]
=> ? = 0
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> ? = 0
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> ? = 0
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)]
=> ? = 0
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? = 0
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 0
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? = 0
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> ? = 0
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 0
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)]
=> ? = 0
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> ? = 0
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> ? = 0
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> 0
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> 0
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> ? = 0
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> ? = 0
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> ? = 1
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 0
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> ? = 0
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> 0
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 0
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> ? = 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]
=> ? = 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 0
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,3},{2},{4,5,6}}
=> [3,2,1,5,6,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
=> ? = 0
{{1,3},{2},{4,5},{6}}
=> [3,2,1,5,4,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> ? = 0
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? = 1
{{1,3},{2},{4,6},{5}}
=> [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> ? = 0
{{1,3},{2},{4},{5,6}}
=> [3,2,1,4,6,5] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
=> ? = 0
{{1,3},{2},{4},{5},{6}}
=> [3,2,1,4,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> ? = 0
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> 0
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? = 0
{{1,4,5},{2,3},{6}}
=> [4,3,2,5,1,6] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> 0
Description
The number of flips required to make a perfect matching noncrossing.
A crossing in a perfect matching is a pair of arcs $\{a,b\}$ and $\{c,d\}$ such that $a < c < b < d$. Replacing any such pair by either $\{a,c\}$ and $\{b,d\}$ or by $\{a,d\}$, $\{b,c\}$ produces a perfect matching with fewer crossings.
This statistic is the minimal number of such flips required to turn a given matching into a noncrossing matching.
The following 101 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000788The number of nesting-similar perfect matchings of a perfect matching. St000069The number of maximal elements of a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001330The hat guessing number of a graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001381The fertility of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001866The nesting alignments of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001871The number of triconnected components of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001960The number of descents of a permutation minus one if its first entry is not one. St000542The number of left-to-right-minima of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001555The order of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!