searching the database
Your data matches 20 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: St001060
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: 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
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,5,6},{2},{4}}
=> [3,2,5,4,6,1] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
{{1,3,6},{2},{4,5}}
=> [3,2,6,5,4,1] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,6},{2},{4},{5}}
=> [3,2,6,4,5,1] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,4,6},{2,3},{5}}
=> [4,3,2,6,5,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,5,6},{2,3,4}}
=> [5,3,4,2,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,5,6},{2,3},{4}}
=> [5,3,2,4,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
{{1,4,6},{2},{3},{5}}
=> [4,2,3,6,5,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
{{1,5,6},{2,4},{3}}
=> [5,4,3,2,6,1] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,5,6},{2},{3,4}}
=> [5,2,4,3,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000260
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Values
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 2
{{1,4},{2,3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2
{{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3
{{1,3,5},{2,4}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
{{1,3,5},{2},{4}}
=> {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,4,5},{2,3}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3
{{1,5},{2,3,4}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3
{{1,5},{2,3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 3
{{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,4},{2,5},{3}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
{{1,5},{2,4},{3}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,5},{2},{3,4}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 3
{{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 3
{{1,3,4,5,6},{2}}
=> {{1},{2,3,4,5,6}}
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
{{1,3,4,6},{2,5}}
=> {{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,3,4,6},{2},{5}}
=> {{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,3,5,6},{2,4}}
=> {{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,3,6},{2,4,5}}
=> {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,3,6},{2,4},{5}}
=> {{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
{{1,3,5,6},{2},{4}}
=> {{1},{2,4,5,6},{3}}
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,3,5},{2,6},{4}}
=> {{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,3,6},{2,5},{4}}
=> {{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,3,6},{2},{4,5}}
=> {{1},{2,5,6},{3,4}}
=> [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,3,6},{2},{4},{5}}
=> {{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,4,5,6},{2,3}}
=> {{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 2
{{1,4,6},{2,3,5}}
=> {{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,4,6},{2,3},{5}}
=> {{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,5,6},{2,3,4}}
=> {{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? = 2
{{1,6},{2,3,4,5}}
=> {{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 4
{{1,6},{2,3,4},{5}}
=> {{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 4
{{1,5,6},{2,3},{4}}
=> {{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ? = 2
{{1,5},{2,3,6},{4}}
=> {{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,3,5},{4}}
=> {{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,6},{2,3},{4,5}}
=> {{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 4
{{1,6},{2,3},{4},{5}}
=> {{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 4
{{1,4,5,6},{2},{3}}
=> {{1},{2},{3,4,5,6}}
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
{{1,4,5},{2,6},{3}}
=> {{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,4,6},{2,5},{3}}
=> {{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,4},{2,5,6},{3}}
=> {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
{{1,4,6},{2},{3,5}}
=> {{1},{2,4},{3,5,6}}
=> [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2
{{1,4},{2,6},{3,5}}
=> {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,4,6},{2},{3},{5}}
=> {{1},{2},{3,5,6},{4}}
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,4},{2,6},{3},{5}}
=> {{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
{{1,5,6},{2,4},{3}}
=> {{1,3},{2},{4,5,6}}
=> [3,2,1,5,6,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ? = 2
{{1,5},{2,4,6},{3}}
=> {{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
{{1,6},{2,4,5},{3}}
=> {{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,6},{2,4},{3,5}}
=> {{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
{{1,6},{2,4},{3},{5}}
=> {{1,3},{2},{4},{5,6}}
=> [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,5,6},{2},{3,4}}
=> {{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ? = 2
{{1,5},{2,6},{3,4}}
=> {{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,5},{3,4}}
=> {{1,4},{2,3},{5,6}}
=> [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,6},{2},{3,4,5}}
=> {{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 4
{{1,6},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 4
{{1,5,6},{2},{3},{4}}
=> {{1},{2},{3},{4,5,6}}
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? = 2
{{1,5},{2,6},{3},{4}}
=> {{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,5},{2},{3,6},{4}}
=> {{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,6},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5,6}}
=> [4,2,3,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,6},{2},{3,5},{4}}
=> {{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,6},{2},{3},{4,5}}
=> {{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? = 4
{{1,6},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? = 4
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001616
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
{{1,3,4},{2}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2
{{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 2
{{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
{{1,4,5},{2,3}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 3
{{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
{{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
{{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3
{{1,4},{2,5},{3}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
{{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2
{{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3
{{1,3,4,5,6},{2}}
=> {{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 4
{{1,3,4,6},{2,5}}
=> {{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2
{{1,3,4,6},{2},{5}}
=> {{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
{{1,3,5,6},{2,4}}
=> {{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2
{{1,3,6},{2,4,5}}
=> {{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2
{{1,3,6},{2,4},{5}}
=> {{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,3,5,6},{2},{4}}
=> {{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2
{{1,3,5},{2,6},{4}}
=> {{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,3,6},{2,5},{4}}
=> {{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2
{{1,3,6},{2},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,3,6},{2},{4},{5}}
=> {{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,4,5,6},{2,3}}
=> {{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2
{{1,4,6},{2,3,5}}
=> {{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 2
{{1,4,6},{2,3},{5}}
=> {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,5,6},{2,3,4}}
=> {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2
{{1,6},{2,3,4,5}}
=> {{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 4
{{1,6},{2,3,4},{5}}
=> {{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4
{{1,5,6},{2,3},{4}}
=> {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2
{{1,5},{2,3,6},{4}}
=> {{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,6},{2,3,5},{4}}
=> {{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2
{{1,6},{2,3},{4,5}}
=> {{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ? = 4
{{1,6},{2,3},{4},{5}}
=> {{1,6},{2},{3},{4,5}}
=> [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4
{{1,4,5,6},{2},{3}}
=> {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 2
{{1,4,5},{2,6},{3}}
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2
{{1,4,6},{2,5},{3}}
=> {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2
{{1,4},{2,5,6},{3}}
=> {{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
{{1,4,6},{2},{3,5}}
=> {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2
{{1,4},{2,6},{3,5}}
=> {{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,4,6},{2},{3},{5}}
=> {{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,4},{2,6},{3},{5}}
=> {{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,5,6},{2,4},{3}}
=> {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2
{{1,5},{2,4,6},{3}}
=> {{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2
{{1,6},{2,4,5},{3}}
=> {{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2
{{1,6},{2,4},{3,5}}
=> {{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2
{{1,6},{2,4},{3},{5}}
=> {{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2
{{1,5,6},{2},{3,4}}
=> {{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2
{{1,5},{2,6},{3,4}}
=> {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,6},{2,5},{3,4}}
=> {{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2
{{1,6},{2},{3,4,5}}
=> {{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4
{{1,6},{2},{3,4},{5}}
=> {{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ? = 4
{{1,5,6},{2},{3},{4}}
=> {{1,5},{2,6},{3},{4}}
=> [5,6,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2
{{1,5},{2,6},{3},{4}}
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,5},{2},{3,6},{4}}
=> {{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,6},{2,5},{3},{4}}
=> {{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ? = 2
{{1,6},{2},{3,5},{4}}
=> {{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2
{{1,6},{2},{3},{4,5}}
=> {{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4
{{1,6},{2},{3},{4},{5}}
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 4
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001720
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
{{1,3,4},{2}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2
{{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 2
{{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
{{1,4,5},{2,3}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 3
{{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
{{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
{{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3
{{1,4},{2,5},{3}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
{{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2
{{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3
{{1,3,4,5,6},{2}}
=> {{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 4
{{1,3,4,6},{2,5}}
=> {{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2
{{1,3,4,6},{2},{5}}
=> {{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
{{1,3,5,6},{2,4}}
=> {{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2
{{1,3,6},{2,4,5}}
=> {{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2
{{1,3,6},{2,4},{5}}
=> {{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,3,5,6},{2},{4}}
=> {{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2
{{1,3,5},{2,6},{4}}
=> {{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,3,6},{2,5},{4}}
=> {{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2
{{1,3,6},{2},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,3,6},{2},{4},{5}}
=> {{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,4,5,6},{2,3}}
=> {{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2
{{1,4,6},{2,3,5}}
=> {{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 2
{{1,4,6},{2,3},{5}}
=> {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,5,6},{2,3,4}}
=> {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2
{{1,6},{2,3,4,5}}
=> {{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 4
{{1,6},{2,3,4},{5}}
=> {{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4
{{1,5,6},{2,3},{4}}
=> {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2
{{1,5},{2,3,6},{4}}
=> {{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,6},{2,3,5},{4}}
=> {{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2
{{1,6},{2,3},{4,5}}
=> {{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ? = 4
{{1,6},{2,3},{4},{5}}
=> {{1,6},{2},{3},{4,5}}
=> [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4
{{1,4,5,6},{2},{3}}
=> {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 2
{{1,4,5},{2,6},{3}}
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2
{{1,4,6},{2,5},{3}}
=> {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2
{{1,4},{2,5,6},{3}}
=> {{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2
{{1,4,6},{2},{3,5}}
=> {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2
{{1,4},{2,6},{3,5}}
=> {{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,4,6},{2},{3},{5}}
=> {{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,4},{2,6},{3},{5}}
=> {{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,5,6},{2,4},{3}}
=> {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2
{{1,5},{2,4,6},{3}}
=> {{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2
{{1,6},{2,4,5},{3}}
=> {{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2
{{1,6},{2,4},{3,5}}
=> {{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2
{{1,6},{2,4},{3},{5}}
=> {{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2
{{1,5,6},{2},{3,4}}
=> {{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2
{{1,5},{2,6},{3,4}}
=> {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,6},{2,5},{3,4}}
=> {{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2
{{1,6},{2},{3,4,5}}
=> {{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4
{{1,6},{2},{3,4},{5}}
=> {{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ? = 4
{{1,5,6},{2},{3},{4}}
=> {{1,5},{2,6},{3},{4}}
=> [5,6,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2
{{1,5},{2,6},{3},{4}}
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2
{{1,5},{2},{3,6},{4}}
=> {{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2
{{1,6},{2,5},{3},{4}}
=> {{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ? = 2
{{1,6},{2},{3,5},{4}}
=> {{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2
{{1,6},{2},{3},{4,5}}
=> {{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4
{{1,6},{2},{3},{4},{5}}
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 4
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001613
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
{{1,3,4},{2}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 - 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 2 - 1
{{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 2 - 1
{{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
{{1,4,5},{2,3}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 3 - 1
{{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 - 1
{{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
{{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
{{1,4},{2,5},{3}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
{{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 - 1
{{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
{{1,3,4,5,6},{2}}
=> {{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 4 - 1
{{1,3,4,6},{2,5}}
=> {{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2 - 1
{{1,3,4,6},{2},{5}}
=> {{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
{{1,3,5,6},{2,4}}
=> {{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2 - 1
{{1,3,6},{2,4,5}}
=> {{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,3,6},{2,4},{5}}
=> {{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,3,5,6},{2},{4}}
=> {{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
{{1,3,5},{2,6},{4}}
=> {{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,3,6},{2,5},{4}}
=> {{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2 - 1
{{1,3,6},{2},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,3,6},{2},{4},{5}}
=> {{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,4,5,6},{2,3}}
=> {{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2 - 1
{{1,4,6},{2,3,5}}
=> {{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,4,6},{2,3},{5}}
=> {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,5,6},{2,3,4}}
=> {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,6},{2,3,4,5}}
=> {{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 4 - 1
{{1,6},{2,3,4},{5}}
=> {{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4 - 1
{{1,5,6},{2,3},{4}}
=> {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2 - 1
{{1,5},{2,3,6},{4}}
=> {{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,6},{2,3,5},{4}}
=> {{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
{{1,6},{2,3},{4,5}}
=> {{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ? = 4 - 1
{{1,6},{2,3},{4},{5}}
=> {{1,6},{2},{3},{4,5}}
=> [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4 - 1
{{1,4,5,6},{2},{3}}
=> {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 2 - 1
{{1,4,5},{2,6},{3}}
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
{{1,4,6},{2,5},{3}}
=> {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
{{1,4},{2,5,6},{3}}
=> {{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
{{1,4,6},{2},{3,5}}
=> {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2 - 1
{{1,4},{2,6},{3,5}}
=> {{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,4,6},{2},{3},{5}}
=> {{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,4},{2,6},{3},{5}}
=> {{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,5,6},{2,4},{3}}
=> {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
{{1,5},{2,4,6},{3}}
=> {{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2 - 1
{{1,6},{2,4,5},{3}}
=> {{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2 - 1
{{1,6},{2,4},{3,5}}
=> {{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
{{1,6},{2,4},{3},{5}}
=> {{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2 - 1
{{1,5,6},{2},{3,4}}
=> {{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2 - 1
{{1,5},{2,6},{3,4}}
=> {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,6},{2,5},{3,4}}
=> {{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2 - 1
{{1,6},{2},{3,4,5}}
=> {{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4 - 1
{{1,6},{2},{3,4},{5}}
=> {{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ? = 4 - 1
{{1,5,6},{2},{3},{4}}
=> {{1,5},{2,6},{3},{4}}
=> [5,6,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2 - 1
{{1,5},{2,6},{3},{4}}
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,5},{2},{3,6},{4}}
=> {{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,6},{2,5},{3},{4}}
=> {{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ? = 2 - 1
{{1,6},{2},{3,5},{4}}
=> {{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2 - 1
{{1,6},{2},{3},{4,5}}
=> {{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4 - 1
{{1,6},{2},{3},{4},{5}}
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 4 - 1
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001719
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
{{1,3,4},{2}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 - 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 2 - 1
{{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 2 - 1
{{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
{{1,4,5},{2,3}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 3 - 1
{{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 - 1
{{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
{{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
{{1,4},{2,5},{3}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
{{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 - 1
{{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
{{1,3,4,5,6},{2}}
=> {{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 4 - 1
{{1,3,4,6},{2,5}}
=> {{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2 - 1
{{1,3,4,6},{2},{5}}
=> {{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
{{1,3,5,6},{2,4}}
=> {{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2 - 1
{{1,3,6},{2,4,5}}
=> {{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,3,6},{2,4},{5}}
=> {{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,3,5,6},{2},{4}}
=> {{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
{{1,3,5},{2,6},{4}}
=> {{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,3,6},{2,5},{4}}
=> {{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2 - 1
{{1,3,6},{2},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,3,6},{2},{4},{5}}
=> {{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,4,5,6},{2,3}}
=> {{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2 - 1
{{1,4,6},{2,3,5}}
=> {{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,4,6},{2,3},{5}}
=> {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,5,6},{2,3,4}}
=> {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,6},{2,3,4,5}}
=> {{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 4 - 1
{{1,6},{2,3,4},{5}}
=> {{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4 - 1
{{1,5,6},{2,3},{4}}
=> {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2 - 1
{{1,5},{2,3,6},{4}}
=> {{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,6},{2,3,5},{4}}
=> {{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
{{1,6},{2,3},{4,5}}
=> {{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ? = 4 - 1
{{1,6},{2,3},{4},{5}}
=> {{1,6},{2},{3},{4,5}}
=> [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4 - 1
{{1,4,5,6},{2},{3}}
=> {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 2 - 1
{{1,4,5},{2,6},{3}}
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
{{1,4,6},{2,5},{3}}
=> {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
{{1,4},{2,5,6},{3}}
=> {{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
{{1,4,6},{2},{3,5}}
=> {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2 - 1
{{1,4},{2,6},{3,5}}
=> {{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,4,6},{2},{3},{5}}
=> {{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,4},{2,6},{3},{5}}
=> {{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,5,6},{2,4},{3}}
=> {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
{{1,5},{2,4,6},{3}}
=> {{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2 - 1
{{1,6},{2,4,5},{3}}
=> {{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2 - 1
{{1,6},{2,4},{3,5}}
=> {{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
{{1,6},{2,4},{3},{5}}
=> {{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2 - 1
{{1,5,6},{2},{3,4}}
=> {{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2 - 1
{{1,5},{2,6},{3,4}}
=> {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,6},{2,5},{3,4}}
=> {{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2 - 1
{{1,6},{2},{3,4,5}}
=> {{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4 - 1
{{1,6},{2},{3,4},{5}}
=> {{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ? = 4 - 1
{{1,5,6},{2},{3},{4}}
=> {{1,5},{2,6},{3},{4}}
=> [5,6,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2 - 1
{{1,5},{2,6},{3},{4}}
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,5},{2},{3,6},{4}}
=> {{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,6},{2,5},{3},{4}}
=> {{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ? = 2 - 1
{{1,6},{2},{3,5},{4}}
=> {{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2 - 1
{{1,6},{2},{3},{4,5}}
=> {{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4 - 1
{{1,6},{2},{3},{4},{5}}
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 4 - 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001881
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
{{1,3,4},{2}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 - 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 2 - 1
{{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 - 1
{{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 2 - 1
{{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
{{1,4,5},{2,3}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 3 - 1
{{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 - 1
{{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
{{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 3 - 1
{{1,4},{2,5},{3}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
{{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 - 1
{{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 3 - 1
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
{{1,3,4,5,6},{2}}
=> {{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 4 - 1
{{1,3,4,6},{2,5}}
=> {{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2 - 1
{{1,3,4,6},{2},{5}}
=> {{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
{{1,3,5,6},{2,4}}
=> {{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,12),(4,13),(5,13),(6,9),(6,12),(8,9),(9,7),(10,8),(11,8),(12,7),(13,10),(13,11)],14)
=> ? = 2 - 1
{{1,3,6},{2,4,5}}
=> {{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,3,6},{2,4},{5}}
=> {{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,3,5,6},{2},{4}}
=> {{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
{{1,3,5},{2,6},{4}}
=> {{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,3,6},{2,5},{4}}
=> {{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2 - 1
{{1,3,6},{2},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,3,6},{2},{4},{5}}
=> {{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,4,5,6},{2,3}}
=> {{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,10),(5,12),(6,9),(6,11),(8,10),(9,8),(10,7),(11,8),(12,7),(13,9)],14)
=> ? = 2 - 1
{{1,4,6},{2,3,5}}
=> {{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,4,6},{2,3},{5}}
=> {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,5,6},{2,3,4}}
=> {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ? = 2 - 1
{{1,6},{2,3,4,5}}
=> {{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 4 - 1
{{1,6},{2,3,4},{5}}
=> {{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4 - 1
{{1,5,6},{2,3},{4}}
=> {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2 - 1
{{1,5},{2,3,6},{4}}
=> {{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,6},{2,3,5},{4}}
=> {{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
{{1,6},{2,3},{4,5}}
=> {{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ? = 4 - 1
{{1,6},{2,3},{4},{5}}
=> {{1,6},{2},{3},{4,5}}
=> [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4 - 1
{{1,4,5,6},{2},{3}}
=> {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 2 - 1
{{1,4,5},{2,6},{3}}
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> ? = 2 - 1
{{1,4,6},{2,5},{3}}
=> {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
{{1,4},{2,5,6},{3}}
=> {{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 1
{{1,4,6},{2},{3,5}}
=> {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ? = 2 - 1
{{1,4},{2,6},{3,5}}
=> {{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,4,6},{2},{3},{5}}
=> {{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,4},{2,6},{3},{5}}
=> {{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,5,6},{2,4},{3}}
=> {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
{{1,5},{2,4,6},{3}}
=> {{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2 - 1
{{1,6},{2,4,5},{3}}
=> {{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2 - 1
{{1,6},{2,4},{3,5}}
=> {{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 1
{{1,6},{2,4},{3},{5}}
=> {{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2 - 1
{{1,5,6},{2},{3,4}}
=> {{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
=> ? = 2 - 1
{{1,5},{2,6},{3,4}}
=> {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,6},{2,5},{3,4}}
=> {{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2 - 1
{{1,6},{2},{3,4,5}}
=> {{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 4 - 1
{{1,6},{2},{3,4},{5}}
=> {{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ? = 4 - 1
{{1,5,6},{2},{3},{4}}
=> {{1,5},{2,6},{3},{4}}
=> [5,6,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 2 - 1
{{1,5},{2,6},{3},{4}}
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
{{1,5},{2},{3,6},{4}}
=> {{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 2 - 1
{{1,6},{2,5},{3},{4}}
=> {{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ? = 2 - 1
{{1,6},{2},{3,5},{4}}
=> {{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 2 - 1
{{1,6},{2},{3},{4,5}}
=> {{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 4 - 1
{{1,6},{2},{3},{4},{5}}
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ? = 4 - 1
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St001520
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Values
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [4,3,2,1] => 1 = 2 - 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [2,4,3,1] => 1 = 2 - 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,1,2] => 1 = 2 - 1
{{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,3,2,4,1] => 2 = 3 - 1
{{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,1,4,2,3] => 1 = 2 - 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,3,2,1,4] => 1 = 2 - 1
{{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => [4,5,3,1,2] => 2 = 3 - 1
{{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,5,3,1,4] => 2 = 3 - 1
{{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,3,2,1] => 2 = 3 - 1
{{1,4,5},{2},{3}}
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,3,4,2,1] => 2 = 3 - 1
{{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,1,2,3,4] => 1 = 2 - 1
{{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,5,1,4,3] => 1 = 2 - 1
{{1,5},{2},{3,4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,4,5,2,1] => 2 = 3 - 1
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,3,4,1,2] => 2 = 3 - 1
{{1,3,4,5,6},{2}}
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [6,3,2,4,5,1] => ? = 4 - 1
{{1,3,4,6},{2,5}}
=> {{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => [6,1,4,3,5,2] => ? = 2 - 1
{{1,3,4,6},{2},{5}}
=> {{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [6,3,2,4,1,5] => ? = 2 - 1
{{1,3,5,6},{2,4}}
=> {{1,5,6},{2,3,4}}
=> [5,3,4,2,6,1] => [6,5,4,2,3,1] => ? = 2 - 1
{{1,3,6},{2,4,5}}
=> {{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [2,6,4,5,3,1] => ? = 2 - 1
{{1,3,6},{2,4},{5}}
=> {{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [6,5,4,2,1,3] => ? = 2 - 1
{{1,3,5,6},{2},{4}}
=> {{1,3,5,6},{2},{4}}
=> [3,2,5,4,6,1] => [6,3,2,5,4,1] => ? = 2 - 1
{{1,3,5},{2,6},{4}}
=> {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => [2,6,5,1,3,4] => ? = 2 - 1
{{1,3,6},{2,5},{4}}
=> {{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => [6,1,5,2,4,3] => ? = 2 - 1
{{1,3,6},{2},{4,5}}
=> {{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [4,3,2,6,5,1] => ? = 2 - 1
{{1,3,6},{2},{4},{5}}
=> {{1,3,6},{2},{4},{5}}
=> [3,2,6,4,5,1] => [6,3,2,5,1,4] => ? = 2 - 1
{{1,4,5,6},{2,3}}
=> {{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [4,6,3,1,5,2] => ? = 2 - 1
{{1,4,6},{2,3,5}}
=> {{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [6,1,3,4,2,5] => ? = 2 - 1
{{1,4,6},{2,3},{5}}
=> {{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [4,6,3,1,2,5] => ? = 2 - 1
{{1,5,6},{2,3,4}}
=> {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [5,6,3,1,4,2] => ? = 2 - 1
{{1,6},{2,3,4,5}}
=> {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [2,6,3,5,1,4] => ? = 4 - 1
{{1,6},{2,3,4},{5}}
=> {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [5,6,3,1,2,4] => ? = 4 - 1
{{1,5,6},{2,3},{4}}
=> {{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [4,6,3,5,1,2] => ? = 2 - 1
{{1,5},{2,3,6},{4}}
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [6,1,3,2,5,4] => ? = 2 - 1
{{1,6},{2,3,5},{4}}
=> {{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [2,6,3,1,4,5] => ? = 2 - 1
{{1,6},{2,3},{4,5}}
=> {{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [5,4,3,6,1,2] => ? = 4 - 1
{{1,6},{2,3},{4},{5}}
=> {{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [4,6,3,5,2,1] => ? = 4 - 1
{{1,4,5,6},{2},{3}}
=> {{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => [6,3,4,2,5,1] => ? = 2 - 1
{{1,4,5},{2,6},{3}}
=> {{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => [2,6,5,3,4,1] => ? = 2 - 1
{{1,4,6},{2,5},{3}}
=> {{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => [6,1,5,3,2,4] => ? = 2 - 1
{{1,4},{2,5,6},{3}}
=> {{1,5,6},{2,4},{3}}
=> [5,4,3,2,6,1] => [6,5,2,3,4,1] => ? = 2 - 1
{{1,4,6},{2},{3,5}}
=> {{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [6,4,1,5,3,2] => ? = 2 - 1
{{1,4},{2,6},{3,5}}
=> {{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => [2,6,1,3,5,4] => ? = 2 - 1
{{1,4,6},{2},{3},{5}}
=> {{1,4,6},{2},{3},{5}}
=> [4,2,3,6,5,1] => [6,3,4,2,1,5] => ? = 2 - 1
{{1,4},{2,6},{3},{5}}
=> {{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => [6,5,2,3,1,4] => ? = 2 - 1
{{1,5,6},{2,4},{3}}
=> {{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [5,6,1,4,3,2] => ? = 2 - 1
{{1,5},{2,4,6},{3}}
=> {{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [6,1,5,4,3,2] => ? = 2 - 1
{{1,6},{2,4,5},{3}}
=> {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => [2,6,5,4,1,3] => ? = 2 - 1
{{1,6},{2,4},{3,5}}
=> {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => [3,1,6,5,4,2] => ? = 2 - 1
{{1,6},{2,4},{3},{5}}
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [5,6,1,4,2,3] => ? = 2 - 1
{{1,5,6},{2},{3,4}}
=> {{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => [5,4,6,2,1,3] => ? = 2 - 1
{{1,5},{2,6},{3,4}}
=> {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => [3,6,2,4,1,5] => ? = 2 - 1
{{1,6},{2,5},{3,4}}
=> {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => [2,3,6,4,5,1] => ? = 2 - 1
{{1,6},{2},{3,4,5}}
=> {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [3,4,6,2,1,5] => ? = 4 - 1
{{1,6},{2},{3,4},{5}}
=> {{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => [5,4,6,2,3,1] => ? = 4 - 1
{{1,5,6},{2},{3},{4}}
=> {{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => [6,3,4,5,2,1] => ? = 2 - 1
{{1,5},{2,6},{3},{4}}
=> {{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => [6,1,4,2,3,5] => ? = 2 - 1
{{1,5},{2},{3,6},{4}}
=> {{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => [6,5,1,3,4,2] => ? = 2 - 1
{{1,6},{2,5},{3},{4}}
=> {{1,5},{2,6},{3},{4}}
=> [5,6,3,4,1,2] => [2,6,4,1,5,3] => ? = 2 - 1
{{1,6},{2},{3,5},{4}}
=> {{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => [3,5,6,1,2,4] => ? = 2 - 1
{{1,6},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => [4,3,5,6,2,1] => ? = 4 - 1
{{1,6},{2},{3},{4},{5}}
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => [6,3,4,5,1,2] => ? = 4 - 1
Description
The number of strict 3-descents.
A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Matching statistic: St001863
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Values
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => 3 = 2 + 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => 4 = 3 + 1
{{1,5},{2,3,4}}
=> {{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => 4 = 3 + 1
{{1,5},{2,3},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,4,2] => 4 = 3 + 1
{{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,4,5,3] => 4 = 3 + 1
{{1,4},{2,5},{3}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => ? = 2 + 1
{{1,5},{2,4},{3}}
=> {{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => 3 = 2 + 1
{{1,5},{2},{3,4}}
=> {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,5,2] => 4 = 3 + 1
{{1,5},{2},{3},{4}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => 4 = 3 + 1
{{1,3,4,5,6},{2}}
=> {{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 4 + 1
{{1,3,4,6},{2,5}}
=> {{1},{2,5},{3,6},{4}}
=> [1,5,6,4,2,3] => [1,5,6,4,2,3] => ? = 2 + 1
{{1,3,4,6},{2},{5}}
=> {{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 2 + 1
{{1,3,5,6},{2,4}}
=> {{1},{2},{3,5},{4,6}}
=> [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2 + 1
{{1,3,6},{2,4,5}}
=> {{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? = 2 + 1
{{1,3,6},{2,4},{5}}
=> {{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [1,3,5,6,2,4] => ? = 2 + 1
{{1,3,5,6},{2},{4}}
=> {{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 2 + 1
{{1,3,5},{2,6},{4}}
=> {{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => [5,6,4,3,1,2] => ? = 2 + 1
{{1,3,6},{2,5},{4}}
=> {{1},{2,5},{3,4,6}}
=> [1,5,4,6,2,3] => [1,5,4,6,2,3] => ? = 2 + 1
{{1,3,6},{2},{4,5}}
=> {{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 2 + 1
{{1,3,6},{2},{4},{5}}
=> {{1},{2,3,4},{5,6}}
=> [1,3,4,2,6,5] => [1,3,4,2,6,5] => ? = 2 + 1
{{1,4,5,6},{2,3}}
=> {{1},{2},{3},{4,6},{5}}
=> [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2 + 1
{{1,4,6},{2,3,5}}
=> {{1},{2,4},{3,6},{5}}
=> [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2 + 1
{{1,4,6},{2,3},{5}}
=> {{1},{2,3},{4,6},{5}}
=> [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 2 + 1
{{1,5,6},{2,3,4}}
=> {{1},{2},{3,6},{4},{5}}
=> [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2 + 1
{{1,6},{2,3,4,5}}
=> {{1},{2,6},{3},{4},{5}}
=> [1,6,3,4,5,2] => [1,6,3,4,5,2] => ? = 4 + 1
{{1,6},{2,3,4},{5}}
=> {{1},{2,3,6},{4},{5}}
=> [1,3,6,4,5,2] => [1,3,6,4,5,2] => ? = 4 + 1
{{1,5,6},{2,3},{4}}
=> {{1},{2},{3,4,6},{5}}
=> [1,2,4,6,5,3] => [1,2,4,6,5,3] => ? = 2 + 1
{{1,5},{2,3,6},{4}}
=> {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [3,6,4,1,5,2] => ? = 2 + 1
{{1,6},{2,3,5},{4}}
=> {{1},{2,6},{3,4},{5}}
=> [1,6,4,3,5,2] => [1,6,4,3,5,2] => ? = 2 + 1
{{1,6},{2,3},{4,5}}
=> {{1},{2,4,6},{3},{5}}
=> [1,4,3,6,5,2] => [1,4,3,6,5,2] => ? = 4 + 1
{{1,6},{2,3},{4},{5}}
=> {{1},{2,3,4,6},{5}}
=> [1,3,4,6,5,2] => [1,3,4,6,5,2] => ? = 4 + 1
{{1,4,5,6},{2},{3}}
=> {{1},{2},{3},{4,5,6}}
=> [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? = 2 + 1
{{1,4,5},{2,6},{3}}
=> {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => [4,6,3,5,1,2] => ? = 2 + 1
{{1,4,6},{2,5},{3}}
=> {{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [1,4,6,5,2,3] => ? = 2 + 1
{{1,4},{2,5,6},{3}}
=> {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [4,2,6,5,1,3] => ? = 2 + 1
{{1,4,6},{2},{3,5}}
=> {{1},{2,4},{3,5,6}}
=> [1,4,5,2,6,3] => [1,4,5,2,6,3] => ? = 2 + 1
{{1,4},{2,6},{3,5}}
=> {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => [5,4,6,2,1,3] => ? = 2 + 1
{{1,4,6},{2},{3},{5}}
=> {{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [1,3,2,5,6,4] => ? = 2 + 1
{{1,4},{2,6},{3},{5}}
=> {{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [4,3,6,5,1,2] => ? = 2 + 1
{{1,5,6},{2,4},{3}}
=> {{1},{2},{3,6},{4,5}}
=> [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 2 + 1
{{1,5},{2,4,6},{3}}
=> {{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [3,6,1,5,4,2] => ? = 2 + 1
{{1,6},{2,4,5},{3}}
=> {{1},{2,6},{3},{4,5}}
=> [1,6,3,5,4,2] => [1,6,3,5,4,2] => ? = 2 + 1
{{1,6},{2,4},{3,5}}
=> {{1},{2,4,6},{3,5}}
=> [1,4,5,6,3,2] => [1,4,5,6,3,2] => ? = 2 + 1
{{1,6},{2,4},{3},{5}}
=> {{1},{2,3,6},{4,5}}
=> [1,3,6,5,4,2] => [1,3,6,5,4,2] => ? = 2 + 1
{{1,5,6},{2},{3,4}}
=> {{1},{2},{3,5,6},{4}}
=> [1,2,5,4,6,3] => [1,2,5,4,6,3] => ? = 2 + 1
{{1,5},{2,6},{3,4}}
=> {{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [3,6,5,4,1,2] => ? = 2 + 1
{{1,6},{2,5},{3,4}}
=> {{1},{2,6},{3,5},{4}}
=> [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2 + 1
{{1,6},{2},{3,4,5}}
=> {{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [1,5,3,4,6,2] => ? = 4 + 1
{{1,6},{2},{3,4},{5}}
=> {{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [1,3,5,4,6,2] => ? = 4 + 1
{{1,5,6},{2},{3},{4}}
=> {{1},{2},{3,4,5,6}}
=> [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? = 2 + 1
{{1,5},{2,6},{3},{4}}
=> {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [3,6,4,5,1,2] => ? = 2 + 1
{{1,5},{2},{3,6},{4}}
=> {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [3,5,4,1,6,2] => ? = 2 + 1
{{1,6},{2,5},{3},{4}}
=> {{1},{2,6},{3,4,5}}
=> [1,6,4,5,3,2] => [1,6,4,5,3,2] => ? = 2 + 1
{{1,6},{2},{3,5},{4}}
=> {{1},{2,5,6},{3,4}}
=> [1,5,4,3,6,2] => [1,5,4,3,6,2] => ? = 2 + 1
{{1,6},{2},{3},{4,5}}
=> {{1},{2,4,5,6},{3}}
=> [1,4,3,5,6,2] => [1,4,3,5,6,2] => ? = 4 + 1
{{1,6},{2},{3},{4},{5}}
=> {{1},{2,3,4,5,6}}
=> [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? = 4 + 1
Description
The number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
Matching statistic: St001846
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Values
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 0 = 2 - 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 0 = 2 - 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 0 = 2 - 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 2 - 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 3 - 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 3 - 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 3 - 2
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 3 - 2
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 3 - 2
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [2,6,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 4 - 2
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ? = 2 - 2
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [2,5,6,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 2 - 2
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [6,5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 2 - 2
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [5,2,6,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 2
{{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [5,6,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,12),(8,9),(9,10),(9,12),(10,11),(12,11)],13)
=> ? = 2 - 2
{{1,3,5,6},{2},{4}}
=> [3,2,5,4,6,1] => [2,4,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 - 2
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [4,5,1,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 2 - 2
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [4,6,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 0 = 2 - 2
{{1,3,6},{2},{4,5}}
=> [3,2,6,5,4,1] => [2,5,4,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 - 2
{{1,3,6},{2},{4},{5}}
=> [3,2,6,4,5,1] => [2,4,5,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? = 2 - 2
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [3,2,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 2 - 2
{{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ? = 2 - 2
{{1,4,6},{2,3},{5}}
=> [4,3,2,6,5,1] => [3,2,5,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,11),(11,10)],12)
=> ? = 2 - 2
{{1,5,6},{2,3,4}}
=> [5,3,4,2,6,1] => [4,3,2,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 2 - 2
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => [5,4,3,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 4 - 2
{{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [4,3,2,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? = 4 - 2
{{1,5,6},{2,3},{4}}
=> [5,3,2,4,6,1] => [3,2,4,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 2 - 2
{{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
{{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => [4,5,3,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 2 - 2
{{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
=> ? = 4 - 2
{{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => [3,2,4,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 4 - 2
{{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => [2,3,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? = 2 - 2
{{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => [3,5,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
{{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [3,6,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 2 - 2
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [3,6,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0 = 2 - 2
{{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => [2,6,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 - 2
{{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => [5,1,6,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ? = 2 - 2
{{1,4,6},{2},{3},{5}}
=> [4,2,3,6,5,1] => [2,3,5,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 2 - 2
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [3,5,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0 = 2 - 2
{{1,5,6},{2,4},{3}}
=> [5,4,3,2,6,1] => [3,4,2,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,11),(11,10)],12)
=> ? = 2 - 2
{{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
{{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => [3,5,4,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 - 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [5,2,4,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 - 2
{{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => [3,4,2,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 2 - 2
{{1,5,6},{2},{3,4}}
=> [5,2,4,3,6,1] => [2,4,3,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
=> ? = 2 - 2
{{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0 = 2 - 2
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 - 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [2,5,4,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 4 - 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [2,4,3,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 4 - 2
{{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => [2,3,4,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 2 - 2
{{1,5},{2,6},{3},{4}}
=> [5,6,3,4,1,2] => [3,4,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 0 = 2 - 2
{{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => [2,4,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
{{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => [3,4,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? = 2 - 2
{{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => [2,4,5,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 2 - 2
{{1,6},{2},{3},{4,5}}
=> [6,2,3,5,4,1] => [2,3,5,4,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 4 - 2
{{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => [2,3,4,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 4 - 2
Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001198The 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 $eA$. St001206The 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001820The size of the image of the pop stack sorting operator. St001498The normalised height of a Nakayama algebra with magnitude 1. St000259The diameter of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000068The number of minimal elements in a poset.
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!