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Your data matches 49 different statistics following compositions of up to 3 maps.
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Matching statistic: St000097
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000147
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [2]
=> 2
[1,0,1,0]
=> [3,1,2] => [3,2,1] => [3]
=> 3
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [2,1]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => [4]
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [3,1]
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => [3,1]
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => [3,1]
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5]
=> 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => [4,1]
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [4,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => [4,1]
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [3,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [4,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => [4,1]
=> 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => [3,2]
=> 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => [3,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [3,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => [3,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => [3,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => [6]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => [5,1]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => [5,1]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => [5,1]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => [4,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => [5,1]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => [4,1,1]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => [5,1]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => [4,2]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => [4,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => [4,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => [4,1,1]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => [4,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => [3,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => [5,1]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => [4,1,1]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => [4,1,1]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => [4,1,1]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => [3,1,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => [5,1]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => [4,1,1]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => [4,2]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => [3,2,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => [4,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => [4,1,1]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => [4,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => [3,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [8,3,1,2,6,4,5,7] => [8,7,5,6,4,2,3,1] => ?
=> ? = 6
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [1,1]
=> 2
[1,0,1,0]
=> [3,1,2] => [3,2,1] => [1,1,1]
=> 3
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [2,1]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => [1,1,1,1]
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [2,1,1]
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => [2,1,1]
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => [2,1,1]
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [3,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [2,1,1,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [3,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [2,1,1,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => [2,2,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => [3,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [3,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => [2,2,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => [3,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [4,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => [1,1,1,1,1,1]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => [2,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => [2,1,1,1,1]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => [2,1,1,1,1]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => [3,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => [2,1,1,1,1]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => [3,1,1,1]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => [2,1,1,1,1]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => [2,2,1,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => [3,1,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => [3,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => [2,2,1,1]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => [3,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => [4,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => [2,1,1,1,1]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => [3,1,1,1]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => [3,1,1,1]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => [2,2,1,1]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => [4,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => [2,1,1,1,1]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => [3,1,1,1]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => [2,2,1,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => [2,2,2]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => [3,2,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => [3,1,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => [2,2,1,1]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => [3,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => [4,1,1]
=> 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [8,7,4,3,2,1,5,6] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [8,7,4,5,3,2,1,6] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,8,6,3,5,7] => [8,7,5,6,3,2,1,4] => ?
=> ? = 6
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [8,1,4,2,3,7,5,6] => [8,6,7,5,3,4,2,1] => ?
=> ? = 6
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => [8,5,2,1,3,4,6,7] => ?
=> ? = 4
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [8,3,1,2,6,4,5,7] => [8,7,5,6,4,2,3,1] => ?
=> ? = 6
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [8,3,1,5,2,4,6,7] => [8,7,6,4,5,2,3,1] => ?
=> ? = 6
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,7,1,4,5,8,6] => [8,6,5,4,1,2,3,7] => ?
=> ? = 5
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => [8,7,5,4,1,2,3,6] => ?
=> ? = 5
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [2,3,8,5,1,4,6,7] => [8,7,6,4,5,1,2,3] => ?
=> ? = 5
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,3,6,5,1,8,4,7] => [8,7,4,5,1,2,3,6] => ?
=> ? = 4
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [2,7,4,5,1,3,8,6] => [8,6,3,4,5,1,2,7] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [9,1,2,5,3,4,6,7,8] => [9,8,7,6,4,5,3,2,1] => ?
=> ? = 8
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [9,1,4,2,3,5,6,7,8] => [9,8,7,6,5,3,4,2,1] => ?
=> ? = 8
Description
The length of the partition.
Matching statistic: St000098
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => [8,5,6,3,2,1,4,7] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,7,2,6,4,8,5] => [8,5,6,4,2,1,3,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [8,5,6,3,4,2,1,7] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,4,7,6,2,8,5] => [8,5,6,2,1,3,4,7] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,7,5,6,2,8,4] => [8,4,5,6,2,1,3,7] => ([(0,7),(1,2),(1,3),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [8,4,5,6,7,2,1,3] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,7,1,5,3,4,8,6] => [8,6,4,5,3,1,2,7] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,1,7,6,3,8,5] => [8,5,6,3,1,2,4,7] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [7,3,1,2,6,4,8,5] => [8,5,6,4,2,3,1,7] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [6,3,1,5,2,7,8,4] => [8,4,5,2,3,1,6,7] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [7,4,1,6,2,3,8,5] => [8,5,2,4,6,3,1,7] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,1,7,6,2,8,5] => [8,5,6,2,3,1,4,7] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [7,3,1,5,6,2,8,4] => [8,4,5,6,2,3,1,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,3,7,1,6,4,8,5] => [8,5,6,4,1,2,3,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [2,7,4,1,3,5,8,6] => [8,6,5,3,4,1,2,7] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [2,6,4,1,3,7,8,5] => [8,5,3,4,1,2,6,7] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,5,4,1,7,3,8,6] => [8,6,3,4,1,2,5,7] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,5,4,1,6,7,8,3] => [8,3,4,1,2,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,7,5,1,2,3,8,4] => [8,4,1,6,3,5,2,7] => ([(0,7),(1,5),(1,7),(2,4),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [7,3,4,1,6,2,8,5] => [8,5,6,2,3,4,1,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [2,3,7,5,1,4,8,6] => [8,6,4,5,1,2,3,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [2,3,6,5,1,7,8,4] => [8,4,5,1,2,3,6,7] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [2,7,4,5,1,3,8,6] => [8,6,3,4,5,1,2,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,6,4,5,1,7,8,3] => [8,3,4,5,1,2,6,7] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => [7,5,8,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 3
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,3,4,7,6,1,8,5] => [8,5,6,1,2,3,4,7] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [2,3,7,5,6,1,8,4] => [8,4,5,6,1,2,3,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,7,4,5,6,1,8,3] => [8,3,4,5,6,1,2,7] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,3,4,8,6,7,1,5] => [8,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => [8,4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St001029
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 70%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7,6,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [4,2,7,6,3,1,5] => ([(0,4),(0,5),(1,2),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [5,2,3,7,6,4,1] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [8,6,5,4,3,2,1,7] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [8,7,5,4,3,2,1,6] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [8,5,4,3,2,1,6,7] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [8,7,6,4,3,2,1,5] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [8,6,4,3,2,1,5,7] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [8,1,2,3,6,4,5,7] => [8,7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => [7,5,8,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [7,1,2,3,6,4,8,5] => [8,5,6,4,3,2,1,7] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [8,7,4,3,2,1,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [8,1,2,3,6,7,4,5] => [8,5,6,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [8,4,3,2,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [8,7,6,5,3,2,1,4] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [8,6,5,3,2,1,4,7] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [8,7,5,3,2,1,4,6] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => [8,6,7,5,3,2,1,4] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [8,5,3,2,1,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => [8,7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,1,2,5,3,4,8,6] => [8,6,4,5,3,2,1,7] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,1,2,6,3,4,5,7] => [6,4,8,7,5,3,2,1] => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [8,7,4,5,3,2,1,6] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [8,1,2,5,3,7,4,6] => [8,6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [8,1,2,6,3,7,4,5] => [7,4,6,8,5,3,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [6,1,2,5,3,7,8,4] => [8,4,5,3,2,1,6,7] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => [8,7,6,3,2,1,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [8,6,3,2,1,4,5,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,8,6,3,5,7] => [8,7,5,6,3,2,1,4] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,1,2,7,6,3,8,5] => [8,5,6,3,2,1,4,7] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [8,1,2,5,6,3,4,7] => [8,7,4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => [7,4,5,8,6,3,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => [7,4,8,5,6,3,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [8,4,5,6,3,2,1,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => [8,7,3,2,1,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [8,1,2,5,6,7,3,4] => [8,4,5,6,7,3,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [8,3,2,1,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [8,7,6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [8,6,5,4,2,1,3,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,1,8,2,4,7,5,6] => [8,6,7,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => [8,5,4,2,1,3,6,7] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [8,7,6,4,2,1,3,5] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => [8,6,4,2,1,3,5,7] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,8,2,6,4,5,7] => [8,7,5,6,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,7,2,6,4,8,5] => [8,5,6,4,2,1,3,7] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => [8,7,4,2,1,3,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => [8,4,2,1,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,1,4,2,3,5,6,7] => [8,7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
Description
The size of the core of a graph.
The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St000527
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 80%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [2,1] => [2,1] => ([],2)
=> 2
[1,0,1,0]
=> [3,1,2] => [3,2,1] => ([],3)
=> 3
[1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(1,2)],3)
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => ([],4)
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => ([(2,3)],4)
=> 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => ([(2,3)],4)
=> 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([],5)
=> 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => ([(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([],6)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => ([(4,5)],6)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(4,5)],6)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => ([(2,5),(3,5),(5,4)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => ([(1,5),(2,5),(3,4)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(4,5)],6)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => ([(2,5),(3,4)],6)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(4,5)],6)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => ([(1,5),(2,3),(2,4),(4,5)],6)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,3,7,6,4,2,1] => ([(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [6,3,5,7,4,2,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
=> ? = 5
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [6,3,4,7,5,2,1] => ([(2,6),(3,4),(4,5),(4,6)],7)
=> ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => ([(3,6),(4,5),(5,6)],7)
=> ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 4
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [6,4,7,5,3,1,2] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ? = 4
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [7,4,5,3,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 4
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [7,6,3,1,2,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 4
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [7,5,6,3,1,2,4] => ([(1,6),(2,4),(3,5),(5,6)],7)
=> ? = 4
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [4,2,7,6,5,3,1] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 5
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [5,3,1,7,6,4,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [6,4,2,7,5,3,1] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [4,2,7,6,3,1,5] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6)],7)
=> ? = 4
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [4,2,7,5,6,3,1] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [6,3,1,5,7,4,2] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(5,6)],7)
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [7,6,5,2,3,1,4] => ([(3,6),(4,5),(5,6)],7)
=> ? = 5
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [7,5,2,3,1,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 4
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [6,4,7,5,2,3,1] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ? = 4
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => [7,4,5,2,3,1,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 4
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [5,2,4,7,6,3,1] => ([(1,5),(1,6),(2,3),(2,4),(4,5),(4,6)],7)
=> ? = 4
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => [7,5,2,4,6,3,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
=> ? = 5
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [7,6,2,3,1,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [7,5,6,2,3,1,4] => ([(1,6),(2,4),(3,5),(5,6)],7)
=> ? = 4
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [6,2,4,5,7,3,1] => ([(1,6),(2,3),(2,4),(4,5),(5,6)],7)
=> ? = 4
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [7,6,4,1,2,3,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [7,5,3,4,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [7,6,3,4,1,2,5] => ([(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [5,2,3,7,6,4,1] => ([(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 4
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2,7,6,3,4,1] => ([(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
=> ? = 4
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => [7,4,1,6,3,5,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(4,6)],7)
=> ? = 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [7,6,2,3,4,1,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [6,2,3,5,7,4,1] => ([(1,3),(2,6),(3,4),(3,5),(5,6)],7)
=> ? = 4
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [6,2,5,7,3,4,1] => ([(1,6),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 4
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [6,2,3,4,7,5,1] => ([(1,3),(2,6),(3,5),(5,4),(5,6)],7)
=> ? = 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [6,2,3,7,4,5,1] => ([(1,3),(2,6),(3,5),(3,6),(5,4)],7)
=> ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [8,6,5,4,3,2,1,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [8,7,5,4,3,2,1,6] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,6,7,5,4,3,2,1] => ([(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [8,5,4,3,2,1,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [8,7,6,4,3,2,1,5] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [8,6,4,3,2,1,5,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [8,1,2,3,6,4,5,7] => [8,7,5,6,4,3,2,1] => ([(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => [7,5,8,6,4,3,2,1] => ([(4,7),(5,6),(5,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [7,1,2,3,6,4,8,5] => [8,5,6,4,3,2,1,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [8,7,4,3,2,1,5,6] => ([(2,7),(3,7),(4,7),(5,7),(7,6)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [8,1,2,3,6,7,4,5] => [8,5,6,7,4,3,2,1] => ([(5,6),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [8,4,3,2,1,5,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(7,5)],8)
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [8,7,6,5,3,2,1,4] => ([(4,7),(5,7),(6,7)],8)
=> ? = 7
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000632
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 80%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [2,1] => [2,1] => ([],2)
=> 1 = 2 - 1
[1,0,1,0]
=> [3,1,2] => [3,2,1] => ([],3)
=> 2 = 3 - 1
[1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => ([],4)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => ([(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => ([(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => ([(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([],6)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => ([(4,5)],6)
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(4,5)],6)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => ([(2,5),(3,5),(5,4)],6)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => ([(1,5),(2,5),(3,4)],6)
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(4,5)],6)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => ([(2,5),(3,4)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(4,5)],6)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => ([(1,5),(2,3),(2,4),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,3,7,6,4,2,1] => ([(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [6,3,5,7,4,2,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
=> ? = 5 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [6,3,4,7,5,2,1] => ([(2,6),(3,4),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => ([(3,6),(4,5),(5,6)],7)
=> ? = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [6,4,7,5,3,1,2] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [7,4,5,3,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [7,6,3,1,2,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [7,5,6,3,1,2,4] => ([(1,6),(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [7,4,5,6,3,1,2] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [4,2,7,6,5,3,1] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [5,3,1,7,6,4,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [6,4,2,7,5,3,1] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [4,2,7,6,3,1,5] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6)],7)
=> ? = 4 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [4,2,7,5,6,3,1] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
=> ? = 4 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [6,3,1,5,7,4,2] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(5,6)],7)
=> ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [7,6,5,2,3,1,4] => ([(3,6),(4,5),(5,6)],7)
=> ? = 5 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [7,5,2,3,1,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [6,4,7,5,2,3,1] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => [7,4,5,2,3,1,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [5,2,4,7,6,3,1] => ([(1,5),(1,6),(2,3),(2,4),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => [7,5,2,4,6,3,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
=> ? = 5 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [7,6,2,3,1,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 4 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [7,5,6,2,3,1,4] => ([(1,6),(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => [7,4,5,6,2,3,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [6,2,4,5,7,3,1] => ([(1,6),(2,3),(2,4),(4,5),(5,6)],7)
=> ? = 4 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [7,2,3,1,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 3 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [7,6,4,1,2,3,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [7,5,6,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [7,5,3,4,1,2,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [7,6,3,4,1,2,5] => ([(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [5,2,3,7,6,4,1] => ([(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 4 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2,7,6,3,4,1] => ([(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
=> ? = 4 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => [7,4,1,6,3,5,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(4,6)],7)
=> ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [7,6,2,3,4,1,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => [7,5,6,2,3,4,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [6,2,3,5,7,4,1] => ([(1,3),(2,6),(3,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [6,2,5,7,3,4,1] => ([(1,6),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => [7,2,3,4,1,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 3 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [7,4,5,1,2,3,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [7,6,3,4,5,1,2] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [7,3,4,5,1,2,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [6,2,3,4,7,5,1] => ([(1,3),(2,6),(3,5),(5,4),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [6,2,3,7,4,5,1] => ([(1,3),(2,6),(3,5),(3,6),(5,4)],7)
=> ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3 - 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St001330
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 80%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,0,1,0]
=> [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [2] => [2] => ([],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [3] => [3] => ([],3)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => ([],4)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5] => ([],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,2,1,1] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001390
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [2,1] => [2,1] => 2
[1,0,1,0]
=> [3,1,2] => [3,2,1] => 3
[1,1,0,0]
=> [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7,5,4,3,2,1,6] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [7,6,4,3,2,1,5] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,5,6,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [7,4,3,2,1,5,6] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [7,6,5,3,2,1,4] => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [7,5,3,2,1,4,6] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,6,4,5,3,2,1] => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [6,4,7,5,3,2,1] => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [7,4,5,3,2,1,6] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7,6,3,2,1,4,5] => ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [7,5,6,3,2,1,4] => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [7,4,5,6,3,2,1] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [7,3,2,1,4,5,6] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [7,6,5,4,2,1,3] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [7,5,4,2,1,3,6] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [7,6,4,2,1,3,5] => ? = 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [7,5,6,4,2,1,3] => ? = 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [7,4,2,1,3,5,6] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,6,5,3,4,2,1] => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [7,5,3,4,2,1,6] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,3,7,6,4,2,1] => ? = 5
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [6,4,2,1,7,5,3] => ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [7,6,3,4,2,1,5] => ? = 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,5,6,3,4,2,1] => ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [6,3,5,7,4,2,1] => ? = 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [7,3,4,2,1,5,6] => ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [7,6,5,2,1,3,4] => ? = 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [7,5,2,1,3,4,6] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [7,6,4,5,2,1,3] => ? = 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [7,4,5,2,1,3,6] => ? = 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,6,3,4,5,2,1] => ? = 5
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [6,3,4,7,5,2,1] => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [6,3,7,4,5,2,1] => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [7,3,4,5,2,1,6] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [7,6,2,1,3,4,5] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [7,5,6,2,1,3,4] => ? = 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [7,4,5,6,2,1,3] => ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,3,4,5,6,2,1] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [7,2,1,3,4,5,6] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [7,6,5,4,3,1,2] => ? = 6
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [7,5,4,3,1,2,6] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [7,6,4,3,1,2,5] => ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [7,5,6,4,3,1,2] => ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [7,4,3,1,2,5,6] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [7,6,4,5,3,1,2] => ? = 5
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [6,4,7,5,3,1,2] => ? = 4
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [7,4,5,3,1,2,6] => ? = 4
Description
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation.
For a given permutation $\pi$, this is the index of the row containing $\pi^{-1}(1)$ of the recording tableau of $\pi$ (obtained by [[Mp00070]]).
Matching statistic: St000062
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [2,1] => [2,1] => [1,2] => 2
[1,0,1,0]
=> [3,1,2] => [3,2,1] => [1,2,3] => 3
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 3
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => [1,3,4,5,2] => 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [1,2,4,5,3] => 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => [1,3,2,4,5] => 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [1,4,5,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [1,2,3,5,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [1,3,5,4,2] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => [1,2,4,3,5] => 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => [2,4,1,3,5] => 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => [1,4,3,5,2] => 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => [1,3,2,5,4] => 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => [1,4,3,2,5] => 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => [1,3,4,5,6,2] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => [1,2,4,5,6,3] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => [1,3,2,4,5,6] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => [1,4,5,6,3,2] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => [1,2,3,5,6,4] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => [1,3,5,6,4,2] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => [2,4,1,3,5,6] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => [1,4,3,5,6,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => [1,2,5,6,4,3] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => [1,3,2,5,6,4] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => [1,4,3,2,5,6] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => [1,5,6,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => [1,3,4,6,5,2] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => [1,2,4,6,5,3] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => [1,3,2,4,6,5] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => [1,4,6,5,3,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => [1,3,5,4,6,2] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => [3,5,1,2,4,6] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => [2,4,6,1,3,5] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => [3,5,1,4,6,2] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => [1,2,5,4,6,3] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => [2,5,3,1,4,6] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => [1,5,4,6,3,2] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7,5,4,3,2,1,6] => [1,3,4,5,6,7,2] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [7,6,4,3,2,1,5] => [1,2,4,5,6,7,3] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,5,6,4,3,2,1] => [1,3,2,4,5,6,7] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [7,4,3,2,1,5,6] => [1,4,5,6,7,3,2] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [7,6,5,3,2,1,4] => [1,2,3,5,6,7,4] => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [7,5,3,2,1,4,6] => [1,3,5,6,7,4,2] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [7,4,5,3,2,1,6] => [1,4,3,5,6,7,2] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7,6,3,2,1,4,5] => [1,2,5,6,7,4,3] => ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [7,5,6,3,2,1,4] => [1,3,2,5,6,7,4] => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [7,4,5,6,3,2,1] => [1,4,3,2,5,6,7] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [7,3,2,1,4,5,6] => [1,5,6,7,4,3,2] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [7,5,4,2,1,3,6] => [1,3,4,6,7,5,2] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [7,6,4,2,1,3,5] => [1,2,4,6,7,5,3] => ? = 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [7,5,6,4,2,1,3] => [1,3,2,4,6,7,5] => ? = 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [7,4,2,1,3,5,6] => [1,4,6,7,5,3,2] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [7,5,3,4,2,1,6] => [1,3,5,4,6,7,2] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => ? = 5
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [6,4,2,1,7,5,3] => [2,4,6,7,1,3,5] => ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [7,6,3,4,2,1,5] => [1,2,5,4,6,7,3] => ? = 5
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,5,6,3,4,2,1] => [1,3,2,5,4,6,7] => ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [6,3,5,7,4,2,1] => [2,5,3,1,4,6,7] => ? = 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [7,3,4,2,1,5,6] => [1,5,4,6,7,3,2] => ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [7,6,5,2,1,3,4] => [1,2,3,6,7,5,4] => ? = 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [7,5,2,1,3,4,6] => [1,3,6,7,5,4,2] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [7,6,4,5,2,1,3] => [1,2,4,3,6,7,5] => ? = 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [7,4,5,2,1,3,6] => [1,4,3,6,7,5,2] => ? = 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 5
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [6,3,4,7,5,2,1] => [2,5,4,1,3,6,7] => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [6,3,7,4,5,2,1] => [2,5,1,4,3,6,7] => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [7,3,4,5,2,1,6] => [1,5,4,3,6,7,2] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [7,6,2,1,3,4,5] => [1,2,6,7,5,4,3] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [7,5,6,2,1,3,4] => [1,3,2,6,7,5,4] => ? = 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [7,4,5,6,2,1,3] => [1,4,3,2,6,7,5] => ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [7,2,1,3,4,5,6] => [1,6,7,5,4,3,2] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => ? = 6
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [7,5,4,3,1,2,6] => [1,3,4,5,7,6,2] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [7,6,4,3,1,2,5] => [1,2,4,5,7,6,3] => ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [7,5,6,4,3,1,2] => [1,3,2,4,5,7,6] => ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [7,4,3,1,2,5,6] => [1,4,5,7,6,3,2] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => [1,2,3,5,7,6,4] => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => [1,3,5,7,6,4,2] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [7,6,4,5,3,1,2] => [1,2,4,3,5,7,6] => ? = 5
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [6,4,7,5,3,1,2] => [2,4,1,3,5,7,6] => ? = 4
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [7,4,5,3,1,2,6] => [1,4,3,5,7,6,2] => ? = 4
Description
The length of the longest increasing subsequence of the permutation.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001720The minimal length of a chain of small intervals in a lattice. St001863The number of weak excedances of a signed permutation. St000390The number of runs of ones in a binary word. St000291The number of descents of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St000317The cycle descent number of a permutation. St000710The number of big deficiencies of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001896The number of right descents of a signed permutations. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001712The number of natural descents of a standard Young tableau. St000670The reversal length of a permutation. St000942The number of critical left to right maxima of the parking functions. St000983The length of the longest alternating subword. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001624The breadth of a lattice. St000662The staircase size of the code of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation.
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