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Matching statistic: St000264
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Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,2,4,5,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,5,3,4,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,5,4,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,2,5,6,4] => [1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,2,6,4,5] => [1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,2,6,5,4] => [1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,4,2,5,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,4,2,6,5] => [1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4
[1,3,4,5,2,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,4,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4
[1,3,5,2,4,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 6
[1,3,5,2,6,4] => [1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,5,4,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,5,4,6,2] => [1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,3,6,2,4,5] => [1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,2,3,6,5] => [1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4
[1,4,2,5,3,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,2,6,3,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4
[1,4,3,2,5,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,3,2,6,5] => [1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4
[1,4,3,5,2,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4
[1,4,5,2,3,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,5,2,6,3] => [1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,5,3,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,5,3,6,2] => [1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,5,2,3,4,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000379
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 4 - 4
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 4 - 4
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 4 - 4
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 4 - 4
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 4 - 4
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 4 - 4
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 4 - 4
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 4 - 4
[2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 4 - 4
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 4 - 4
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 4 - 4
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 4 - 4
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 4 - 4
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 4 - 4
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 4 - 4
[1,2,3,5,4,6] => [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)
=> ([],1)
=> ? = 4 - 4
[1,2,4,3,5,6] => [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)
=> ([],1)
=> ? = 4 - 4
[1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,2,5,3,4,6] => [6,4,3,5,2,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,2,5,4,3,6] => [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)
=> ([],1)
=> ? = 4 - 4
[1,3,2,4,5,6] => [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)
=> ([],1)
=> ? = 4 - 4
[1,3,2,4,6,5] => [5,6,4,2,3,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,3,2,5,4,6] => [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)
=> ([],1)
=> ? = 4 - 4
[1,3,2,5,6,4] => [4,6,5,2,3,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,3,2,6,4,5] => [5,4,6,2,3,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,3,2,6,5,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,3,4,2,5,6] => [6,5,2,4,3,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,3,4,2,6,5] => [5,6,2,4,3,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,3,4,5,2,6] => [6,2,5,4,3,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,3,4,6,2,5] => [5,2,6,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(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)
=> ([],1)
=> ? = 6 - 4
[1,3,5,2,6,4] => [4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,3,5,4,2,6] => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,3,5,4,6,2] => [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,3,6,2,4,5] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,3,6,2,5,4] => [4,5,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,4,2,3,5,6] => [6,5,3,2,4,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,4,2,3,6,5] => [5,6,3,2,4,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,4,2,5,3,6] => [6,3,5,2,4,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,4,2,6,3,5] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,4,3,2,5,6] => [6,5,2,3,4,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,4,3,2,6,5] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,4,3,5,2,6] => [6,2,5,3,4,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,4,3,6,2,5] => [5,2,6,3,4,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,4,5,2,3,6] => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,4,5,2,6,3] => [3,6,2,5,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,4,5,3,2,6] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,4,5,3,6,2] => [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,5,2,3,4,6] => [6,4,3,2,5,1] => ([(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)
=> ([],1)
=> ? = 4 - 4
[1,5,2,4,3,6] => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,5,3,2,4,6] => [6,4,2,3,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 4 - 4
[1,5,3,4,6,2] => [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,5,4,3,6,2] => [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,6,3,2,4,5] => [5,4,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,6,3,2,5,4] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[2,3,5,4,1,6] => [6,1,4,5,3,2] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[2,3,5,4,6,1] => [1,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> 0 = 4 - 4
[2,4,3,6,1,5] => [5,1,6,3,4,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[2,4,3,6,5,1] => [1,5,6,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> 0 = 4 - 4
[2,6,1,4,3,5] => [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[2,6,1,4,5,3] => [3,5,4,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[2,6,1,5,3,4] => [4,3,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[2,6,1,5,4,3] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[3,1,5,4,6,2] => [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[3,2,5,4,1,6] => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[3,2,5,4,6,1] => [1,6,4,5,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> 0 = 4 - 4
[3,4,1,6,5,2] => [2,5,6,1,4,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[3,4,2,6,1,5] => [5,1,6,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[3,4,2,6,5,1] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> 0 = 4 - 4
[4,1,5,3,6,2] => [2,6,3,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[4,2,3,6,1,5] => [5,1,6,3,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[4,2,3,6,5,1] => [1,5,6,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> 0 = 4 - 4
[4,3,1,6,5,2] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[4,3,2,6,1,5] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[4,3,2,6,5,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> 0 = 4 - 4
[5,1,3,2,4,6] => [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)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[5,1,3,2,6,4] => [4,6,2,3,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[5,1,3,4,2,6] => [6,2,4,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[5,1,3,4,6,2] => [2,6,4,3,1,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 4 - 4
[5,1,4,2,3,6] => [6,3,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[5,1,4,2,6,3] => [3,6,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[5,1,4,3,2,6] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[5,1,4,3,6,2] => [2,6,3,4,1,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 4 - 4
[5,2,1,6,3,4] => [4,3,6,1,2,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[5,2,1,6,4,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[6,1,3,2,4,5] => [5,4,2,3,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> 0 = 4 - 4
[6,1,3,2,5,4] => [4,5,2,3,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> 0 = 4 - 4
[6,2,1,4,3,5] => [5,3,4,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> 0 = 4 - 4
[6,2,1,4,5,3] => [3,5,4,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> 0 = 4 - 4
[6,2,1,5,3,4] => [4,3,5,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> 0 = 4 - 4
[6,2,1,5,4,3] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> 0 = 4 - 4
[1,3,4,6,5,7,2] => [2,7,5,6,4,3,1] => ([(0,6),(1,3),(1,4),(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)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,3,5,4,6,7,2] => [2,7,6,4,5,3,1] => ([(0,6),(1,3),(1,4),(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)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,3,5,4,7,6,2] => [2,6,7,4,5,3,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,3,5,6,4,7,2] => [2,7,4,6,5,3,1] => ([(0,6),(1,4),(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)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,3,6,4,5,7,2] => [2,7,5,4,6,3,1] => ([(0,6),(1,4),(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)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,3,6,5,4,7,2] => [2,7,4,5,6,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,4,3,6,5,7,2] => [2,7,5,6,3,4,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
[1,4,5,3,6,7,2] => [2,7,6,3,5,4,1] => ([(0,6),(1,4),(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)
=> ([(0,2),(1,2)],3)
=> 0 = 4 - 4
Description
The number of Hamiltonian cycles in a graph.
A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$.
Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000031
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 4 - 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 4 - 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 1 = 4 - 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1 = 4 - 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1 = 4 - 3
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1 = 4 - 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1 = 4 - 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1 = 4 - 3
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1 = 4 - 3
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1 = 4 - 3
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1 = 4 - 3
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1 = 4 - 3
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1 = 4 - 3
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1 = 4 - 3
[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1 = 4 - 3
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 1 = 4 - 3
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 4 - 3
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 3
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 3
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 3
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 4 - 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4 - 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 4 - 3
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 3
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 3
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 3
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 3
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 3
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 3
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 6 - 3
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 3
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 3
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 3
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 3
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 3
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 3
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 3
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 3
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 3
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,4,5,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,4,5,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,3,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,4,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,4,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,6,3,2,5,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 4 - 3
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 4 - 3
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 4 - 3
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 4 - 3
[2,1,4,5,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 4 - 3
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[3,4,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[3,4,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,1,2,3,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,1,3,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,1,5,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[4,1,5,2,6,3] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[4,1,5,3,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,2,1,3,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,2,1,6,5,3] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,2,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,2,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,3,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,3,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,3,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,3,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[5,1,3,2,4,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[5,1,3,2,6,4] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[5,1,3,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[5,1,3,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000022
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 4 - 4
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 4 - 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0 = 4 - 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0 = 4 - 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0 = 4 - 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0 = 4 - 4
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 0 = 4 - 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 4 - 4
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 4 - 4
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 4 - 4
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4 - 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 4 - 4
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 6 - 4
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,5,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,5,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,4,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,4,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,6,3,2,5,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 4 - 4
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 4 - 4
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 4 - 4
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 4 - 4
[2,1,4,5,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 4 - 4
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,4,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,4,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,1,2,3,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,1,3,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,1,5,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[4,1,5,2,6,3] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[4,1,5,3,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,2,1,3,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,2,1,6,5,3] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,2,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,2,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,3,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,3,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,3,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,3,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,3,2,4,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[5,1,3,2,6,4] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,3,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[5,1,3,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
Description
The number of fixed points of a permutation.
Matching statistic: St000214
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 4 - 4
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 4 - 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0 = 4 - 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0 = 4 - 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0 = 4 - 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0 = 4 - 4
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 0 = 4 - 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 4 - 4
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 4 - 4
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 4 - 4
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4 - 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 4 - 4
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 6 - 4
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,5,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,5,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,4,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,4,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,6,3,2,5,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 4 - 4
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 4 - 4
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 4 - 4
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 4 - 4
[2,1,4,5,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 4 - 4
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,4,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,4,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,1,2,3,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,1,3,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,1,5,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[4,1,5,2,6,3] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[4,1,5,3,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,2,1,3,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,2,1,6,5,3] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,2,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,2,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,3,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,3,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,3,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,3,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,3,2,4,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[5,1,3,2,6,4] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,3,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[5,1,3,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000366
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 4 - 4
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 4 - 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0 = 4 - 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0 = 4 - 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0 = 4 - 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0 = 4 - 4
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 0 = 4 - 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 4 - 4
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 4 - 4
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 4 - 4
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4 - 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 4 - 4
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 6 - 4
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,5,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,5,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,4,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,4,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,6,3,2,5,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 4 - 4
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 4 - 4
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 4 - 4
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 4 - 4
[2,1,4,5,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 4 - 4
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,4,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,4,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,1,2,3,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,1,3,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,1,5,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[4,1,5,2,6,3] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[4,1,5,3,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,2,1,3,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,2,1,6,5,3] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,2,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,2,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,3,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,3,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,3,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,3,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,3,2,4,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[5,1,3,2,6,4] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,3,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[5,1,3,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St001465
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 4 - 4
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 4 - 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0 = 4 - 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0 = 4 - 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0 = 4 - 4
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 4 - 4
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0 = 4 - 4
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 4 - 4
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 4 - 4
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 0 = 4 - 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 4 - 4
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 4 - 4
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 4
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 4 - 4
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4 - 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 4 - 4
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 4
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 6 - 4
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 4
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 4
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 4
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 4
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,5,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,4,5,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,3,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,4,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 4
[1,5,4,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[1,6,3,2,5,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 4
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 4 - 4
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 4 - 4
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 4 - 4
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 4 - 4
[2,1,4,5,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 4 - 4
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[3,4,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[3,4,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,1,2,3,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,1,3,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,1,5,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[4,1,5,2,6,3] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[4,1,5,3,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,2,1,3,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,2,1,6,5,3] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,2,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,2,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,3,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,3,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[4,3,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 0 = 4 - 4
[4,3,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,3,2,4,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[5,1,3,2,6,4] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,3,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
[5,1,3,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 4 - 4
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 0 = 4 - 4
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 4 - 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 4 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2 = 4 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2 = 4 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2 = 4 - 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 4 - 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 4 - 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 4 - 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2 = 4 - 2
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 4 - 2
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 4 - 2
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 4 - 2
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 4 - 2
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 4 - 2
[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 4 - 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 2 = 4 - 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 4 - 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 4 - 2
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 4 - 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4 - 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 4 - 2
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 2
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 2
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 2
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 2
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 6 - 2
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 2
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 2
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 2
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 2
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 2
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 2
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,4,5,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,4,5,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,5,3,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,5,4,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 2
[1,5,4,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[1,6,3,2,5,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 2
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 4 - 2
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 4 - 2
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 4 - 2
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 4 - 2
[2,1,4,5,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 4 - 2
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 4 - 2
[2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 4 - 2
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 4 - 2
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[3,4,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[3,4,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[4,1,2,3,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[4,1,3,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[4,1,5,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 4 - 2
[4,1,5,2,6,3] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 4 - 2
[4,1,5,3,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[4,2,1,3,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[4,2,1,6,5,3] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[4,2,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[4,2,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[4,3,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[4,3,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[4,3,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 4 - 2
[4,3,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[5,1,3,2,4,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 4 - 2
[5,1,3,2,6,4] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[5,1,3,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 4 - 2
[5,1,3,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 4 - 2
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 4 - 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000036
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 4 - 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 4 - 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 1 = 4 - 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1 = 4 - 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1 = 4 - 3
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1 = 4 - 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1 = 4 - 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1 = 4 - 3
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1 = 4 - 3
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1 = 4 - 3
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1 = 4 - 3
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1 = 4 - 3
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1 = 4 - 3
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1 = 4 - 3
[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1 = 4 - 3
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 1 = 4 - 3
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 4 - 3
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 3
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 3
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 - 3
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 4 - 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4 - 3
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 4 - 3
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 3
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 3
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4 - 3
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 3
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 3
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 3
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 6 - 3
[1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 3
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,3,5,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 3
[1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 4 - 3
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 3
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 3
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 3
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 3
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 - 3
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 3
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,4,5,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,4,5,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,3,4,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,4,2,6,3] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4 - 3
[1,5,4,3,6,2] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[1,6,3,2,5,4] => [1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4 - 3
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 4 - 3
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 4 - 3
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 4 - 3
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 4 - 3
[2,1,4,5,6,3] => [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 4 - 3
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[2,3,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[2,3,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[3,4,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[3,4,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[3,4,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,1,2,3,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,1,3,2,6,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,1,5,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[4,1,5,2,6,3] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[4,1,5,3,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,2,1,3,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,2,1,6,5,3] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,2,3,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,2,3,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,3,1,2,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,3,1,6,5,2] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[4,3,2,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 1 = 4 - 3
[4,3,2,6,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[5,1,3,2,4,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[5,1,3,2,6,4] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[5,1,3,4,2,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
[5,1,3,4,6,2] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 4 - 3
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1 = 4 - 3
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.
Matching statistic: St001330
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 4 - 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 4 - 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 4 - 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 4 - 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 4 - 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 4 - 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 4 - 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 4 - 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 4 - 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 4 - 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 4 - 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 4 - 2
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 4 - 2
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 4 - 2
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,4,6,2,5] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 - 2
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 2
[1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 2
[1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,4,5,2,6,3] => [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,4,5,3,2,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,4,5,3,6,2] => [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,5,3,2,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,5,3,2,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,5,3,4,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 4 - 2
[1,5,4,3,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 4 - 2
[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,1,5,6,4,3] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 4 - 2
[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 4 - 2
[2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,1,6,4,5] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,1,6,5,4] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,5,1,4,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,3,6,1,5,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ([(0,5),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 6 - 2
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ([(0,5),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 2
[3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 4 - 2
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.
The following 299 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000065The number of entries equal to -1 in an alternating sign matrix. St000842The breadth of a permutation. St000352The Elizalde-Pak rank of a permutation. St000546The number of global descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000456The monochromatic index of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001578The minimal number of edges to add or remove to make a graph a line graph. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000069The number of maximal elements of a poset. St000153The number of adjacent cycles of a permutation. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000054The first entry of the permutation. St000218The number of occurrences of the pattern 213 in a permutation. St001947The number of ties in a parking function. St000455The second largest eigenvalue of a graph if it is integral. St000068The number of minimal elements in a poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001434The number of negative sum pairs of a signed permutation. St001271The competition number of a graph. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000759The smallest missing part in an integer partition. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001468The smallest fixpoint of a permutation. St000210Minimum over maximum difference of elements in cycles. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St000640The rank of the largest boolean interval in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001301The first Betti number of the order complex associated with the poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St000234The number of global ascents of a permutation. St001260The permanent of an alternating sign matrix. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001645The pebbling number of a connected graph. St000124The cardinality of the preimage of the Simion-Schmidt map. St000862The number of parts of the shifted shape of a permutation. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001964The interval resolution global dimension of a poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001568The smallest positive integer that does not appear twice in the partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000017The number of inversions of a standard tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001721The degree of a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000016The number of attacking pairs of a standard tableau. St000322The skewness of a graph. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000754The Grundy value for the game of removing nestings in a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000402Half the size of the symmetry class of a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000648The number of 2-excedences of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000078The number of alternating sign matrices whose left key is the permutation. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000717The number of ordinal summands of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000850The number of 1/2-balanced pairs in a poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001472The permanent of the Coxeter matrix of the poset. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000454The largest eigenvalue of a graph if it is integral. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001621The number of atoms of a lattice. St001625The Möbius invariant of a lattice. St001381The fertility of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000003The number of standard Young tableaux of the partition. St000010The length of the partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000346The number of coarsenings of a partition. St000517The Kreweras number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St000783The side length of the largest staircase partition fitting into a partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000897The number of different multiplicities of parts of an integer partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001484The number of singletons of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000257The number of distinct parts of a partition that occur at least twice. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001091The number of parts in an integer partition whose next smaller part has the same size. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001214The aft of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001116The game chromatic number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000666The number of right tethers of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001902The number of potential covers of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001618The cardinality of the Frattini sublattice of a lattice. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001741The largest integer such that all patterns of this size are contained in the permutation. St000401The size of the symmetry class of a permutation. St001850The number of Hecke atoms of a permutation. St001889The size of the connectivity set of a signed permutation. St000097The order of the largest clique of the graph. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001674The number of vertices of the largest induced star graph in the graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001111The weak 2-dynamic chromatic number of a graph. St001654The monophonic hull number of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001884The number of borders of a binary word. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000295The length of the border of a binary word. St000351The determinant of the adjacency matrix of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001826The maximal number of leaves on a vertex of a graph.
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