Your data matches 165 different statistics following compositions of up to 3 maps.
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Matching statistic: St000395
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1,0]
=> 1
[2,1] => [1,1,0,0]
=> [2,1] => [1,1,0,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 4
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,1,5,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,1,5,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,2,5,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,2,5,3,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,3,5,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,3,5,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,2,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,2,3,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,1,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,2,1,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,2,4,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,4,1,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,4,6,1,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,4,6,2,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path.
Mp00240: Permutations weak exceedance partitionSet partitions
Mp00080: Set partitions to permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
St001645: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1] => ([],1)
=> 1
[2,1] => {{1,2}}
=> [2,1] => ([(0,1)],2)
=> 2
[2,3,1] => {{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 4
[3,1,2] => {{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,2,1] => {{1,3},{2,4}}
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[4,3,1,2] => {{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,5,1,2,3] => {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,5,1,3,2] => {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,5,2,1,3] => {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,5,2,3,1] => {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,5,3,1,2] => {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[4,5,3,2,1] => {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[5,3,4,1,2] => {{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,3,4,2,1] => {{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,4,1,2,3] => {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,4,1,3,2] => {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,4,2,1,3] => {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,4,2,3,1] => {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,4,3,1,2] => {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,4,3,2,1] => {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,5,6,1,2,3] => {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[4,5,6,1,3,2] => {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[4,5,6,2,1,3] => {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[4,5,6,2,3,1] => {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[4,5,6,3,1,2] => {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[4,5,6,3,2,1] => {{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[4,6,1,5,2,3] => {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[4,6,1,5,3,2] => {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[4,6,2,5,1,3] => {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[4,6,2,5,3,1] => {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[4,6,3,5,1,2] => {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[4,6,3,5,2,1] => {{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[4,6,5,1,2,3] => {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,6,5,1,3,2] => {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,6,5,2,1,3] => {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,6,5,2,3,1] => {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,6,5,3,1,2] => {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,6,5,3,2,1] => {{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,3,6,1,2,4] => {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[5,3,6,1,4,2] => {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[5,3,6,2,1,4] => {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[5,3,6,2,4,1] => {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[5,3,6,4,1,2] => {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[5,3,6,4,2,1] => {{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[5,4,6,1,2,3] => {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,4,6,1,3,2] => {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,4,6,2,1,3] => {{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
Description
The pebbling number of a connected graph.
Matching statistic: St000008
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00143: Dyck paths inverse promotionDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 3 = 4 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3 = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 4 - 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,1,5,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,1,5,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,2,5,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,2,5,3,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,3,5,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,3,5,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,5,2,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,5,2,3,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,3,6,1,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,3,6,2,1,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,3,6,2,4,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,3,6,4,1,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,4,6,1,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
[5,4,6,2,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => 5 = 6 - 1
Description
The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000294
Mp00066: Permutations inversePermutations
Mp00257: Permutations Alexandersson KebedePermutations
Mp00114: Permutations connectivity setBinary words
St000294: Binary words ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => => ? = 1
[2,1] => [2,1] => [2,1] => 0 => 2
[2,3,1] => [3,1,2] => [1,3,2] => 10 => 4
[3,1,2] => [2,3,1] => [3,2,1] => 00 => 3
[3,2,1] => [3,2,1] => [2,3,1] => 00 => 3
[3,4,1,2] => [3,4,1,2] => [4,3,1,2] => 000 => 4
[3,4,2,1] => [4,3,1,2] => [3,4,1,2] => 000 => 4
[4,3,1,2] => [3,4,2,1] => [4,3,2,1] => 000 => 4
[4,3,2,1] => [4,3,2,1] => [3,4,2,1] => 000 => 4
[4,5,1,2,3] => [3,4,5,1,2] => [4,3,5,1,2] => 0000 => 5
[4,5,1,3,2] => [3,5,4,1,2] => [5,3,4,1,2] => 0000 => 5
[4,5,2,1,3] => [4,3,5,1,2] => [3,4,5,1,2] => 0000 => 5
[4,5,2,3,1] => [5,3,4,1,2] => [3,5,4,1,2] => 0000 => 5
[4,5,3,1,2] => [4,5,3,1,2] => [5,4,3,1,2] => 0000 => 5
[4,5,3,2,1] => [5,4,3,1,2] => [4,5,3,1,2] => 0000 => 5
[5,3,4,1,2] => [4,5,2,3,1] => [5,4,2,3,1] => 0000 => 5
[5,3,4,2,1] => [5,4,2,3,1] => [4,5,2,3,1] => 0000 => 5
[5,4,1,2,3] => [3,4,5,2,1] => [4,3,5,2,1] => 0000 => 5
[5,4,1,3,2] => [3,5,4,2,1] => [5,3,4,2,1] => 0000 => 5
[5,4,2,1,3] => [4,3,5,2,1] => [3,4,5,2,1] => 0000 => 5
[5,4,2,3,1] => [5,3,4,2,1] => [3,5,4,2,1] => 0000 => 5
[5,4,3,1,2] => [4,5,3,2,1] => [5,4,3,2,1] => 0000 => 5
[5,4,3,2,1] => [5,4,3,2,1] => [4,5,3,2,1] => 0000 => 5
[4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,4,6,1,2,3] => 00000 => 6
[4,5,6,1,3,2] => [4,6,5,1,2,3] => [6,4,5,1,2,3] => 00000 => 6
[4,5,6,2,1,3] => [5,4,6,1,2,3] => [4,5,6,1,2,3] => 00000 => 6
[4,5,6,2,3,1] => [6,4,5,1,2,3] => [4,6,5,1,2,3] => 00000 => 6
[4,5,6,3,1,2] => [5,6,4,1,2,3] => [6,5,4,1,2,3] => 00000 => 6
[4,5,6,3,2,1] => [6,5,4,1,2,3] => [5,6,4,1,2,3] => 00000 => 6
[4,6,1,5,2,3] => [3,5,6,1,4,2] => [5,3,6,1,4,2] => 00000 => 6
[4,6,1,5,3,2] => [3,6,5,1,4,2] => [6,3,5,1,4,2] => 00000 => 6
[4,6,2,5,1,3] => [5,3,6,1,4,2] => [3,5,6,1,4,2] => 00000 => 6
[4,6,2,5,3,1] => [6,3,5,1,4,2] => [3,6,5,1,4,2] => 00000 => 6
[4,6,3,5,1,2] => [5,6,3,1,4,2] => [6,5,3,1,4,2] => 00000 => 6
[4,6,3,5,2,1] => [6,5,3,1,4,2] => [5,6,3,1,4,2] => 00000 => 6
[4,6,5,1,2,3] => [4,5,6,1,3,2] => [5,4,6,1,3,2] => 00000 => 6
[4,6,5,1,3,2] => [4,6,5,1,3,2] => [6,4,5,1,3,2] => 00000 => 6
[4,6,5,2,1,3] => [5,4,6,1,3,2] => [4,5,6,1,3,2] => 00000 => 6
[4,6,5,2,3,1] => [6,4,5,1,3,2] => [4,6,5,1,3,2] => 00000 => 6
[4,6,5,3,1,2] => [5,6,4,1,3,2] => [6,5,4,1,3,2] => 00000 => 6
[4,6,5,3,2,1] => [6,5,4,1,3,2] => [5,6,4,1,3,2] => 00000 => 6
[5,3,6,1,2,4] => [4,5,2,6,1,3] => [5,4,2,6,1,3] => 00000 => 6
[5,3,6,1,4,2] => [4,6,2,5,1,3] => [6,4,2,5,1,3] => 00000 => 6
[5,3,6,2,1,4] => [5,4,2,6,1,3] => [4,5,2,6,1,3] => 00000 => 6
[5,3,6,2,4,1] => [6,4,2,5,1,3] => [4,6,2,5,1,3] => 00000 => 6
[5,3,6,4,1,2] => [5,6,2,4,1,3] => [6,5,2,4,1,3] => 00000 => 6
[5,3,6,4,2,1] => [6,5,2,4,1,3] => [5,6,2,4,1,3] => 00000 => 6
[5,4,6,1,2,3] => [4,5,6,2,1,3] => [5,4,6,2,1,3] => 00000 => 6
[5,4,6,1,3,2] => [4,6,5,2,1,3] => [6,4,5,2,1,3] => 00000 => 6
[5,4,6,2,1,3] => [5,4,6,2,1,3] => [4,5,6,2,1,3] => 00000 => 6
[5,4,6,2,3,1] => [6,4,5,2,1,3] => [4,6,5,2,1,3] => 00000 => 6
Description
The number of distinct factors of a binary word. This is also known as the subword complexity of a binary word, see [1].
Matching statistic: St000438
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000438: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1,0]
=> [1] => [1,0]
=> ? = 1
[2,1] => [1,1,0,0]
=> [2,1] => [1,1,0,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 4
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,1,5,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,1,5,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,2,5,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,2,5,3,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,3,5,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,3,5,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,2,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,2,3,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,1,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,2,1,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,2,4,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,4,1,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,4,6,1,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,4,6,2,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,4,6,2,3,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
Description
The position of the last up step in a Dyck path.
Matching statistic: St000518
Mp00066: Permutations inversePermutations
Mp00257: Permutations Alexandersson KebedePermutations
Mp00114: Permutations connectivity setBinary words
St000518: Binary words ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => => ? = 1
[2,1] => [2,1] => [2,1] => 0 => 2
[2,3,1] => [3,1,2] => [1,3,2] => 10 => 4
[3,1,2] => [2,3,1] => [3,2,1] => 00 => 3
[3,2,1] => [3,2,1] => [2,3,1] => 00 => 3
[3,4,1,2] => [3,4,1,2] => [4,3,1,2] => 000 => 4
[3,4,2,1] => [4,3,1,2] => [3,4,1,2] => 000 => 4
[4,3,1,2] => [3,4,2,1] => [4,3,2,1] => 000 => 4
[4,3,2,1] => [4,3,2,1] => [3,4,2,1] => 000 => 4
[4,5,1,2,3] => [3,4,5,1,2] => [4,3,5,1,2] => 0000 => 5
[4,5,1,3,2] => [3,5,4,1,2] => [5,3,4,1,2] => 0000 => 5
[4,5,2,1,3] => [4,3,5,1,2] => [3,4,5,1,2] => 0000 => 5
[4,5,2,3,1] => [5,3,4,1,2] => [3,5,4,1,2] => 0000 => 5
[4,5,3,1,2] => [4,5,3,1,2] => [5,4,3,1,2] => 0000 => 5
[4,5,3,2,1] => [5,4,3,1,2] => [4,5,3,1,2] => 0000 => 5
[5,3,4,1,2] => [4,5,2,3,1] => [5,4,2,3,1] => 0000 => 5
[5,3,4,2,1] => [5,4,2,3,1] => [4,5,2,3,1] => 0000 => 5
[5,4,1,2,3] => [3,4,5,2,1] => [4,3,5,2,1] => 0000 => 5
[5,4,1,3,2] => [3,5,4,2,1] => [5,3,4,2,1] => 0000 => 5
[5,4,2,1,3] => [4,3,5,2,1] => [3,4,5,2,1] => 0000 => 5
[5,4,2,3,1] => [5,3,4,2,1] => [3,5,4,2,1] => 0000 => 5
[5,4,3,1,2] => [4,5,3,2,1] => [5,4,3,2,1] => 0000 => 5
[5,4,3,2,1] => [5,4,3,2,1] => [4,5,3,2,1] => 0000 => 5
[4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,4,6,1,2,3] => 00000 => 6
[4,5,6,1,3,2] => [4,6,5,1,2,3] => [6,4,5,1,2,3] => 00000 => 6
[4,5,6,2,1,3] => [5,4,6,1,2,3] => [4,5,6,1,2,3] => 00000 => 6
[4,5,6,2,3,1] => [6,4,5,1,2,3] => [4,6,5,1,2,3] => 00000 => 6
[4,5,6,3,1,2] => [5,6,4,1,2,3] => [6,5,4,1,2,3] => 00000 => 6
[4,5,6,3,2,1] => [6,5,4,1,2,3] => [5,6,4,1,2,3] => 00000 => 6
[4,6,1,5,2,3] => [3,5,6,1,4,2] => [5,3,6,1,4,2] => 00000 => 6
[4,6,1,5,3,2] => [3,6,5,1,4,2] => [6,3,5,1,4,2] => 00000 => 6
[4,6,2,5,1,3] => [5,3,6,1,4,2] => [3,5,6,1,4,2] => 00000 => 6
[4,6,2,5,3,1] => [6,3,5,1,4,2] => [3,6,5,1,4,2] => 00000 => 6
[4,6,3,5,1,2] => [5,6,3,1,4,2] => [6,5,3,1,4,2] => 00000 => 6
[4,6,3,5,2,1] => [6,5,3,1,4,2] => [5,6,3,1,4,2] => 00000 => 6
[4,6,5,1,2,3] => [4,5,6,1,3,2] => [5,4,6,1,3,2] => 00000 => 6
[4,6,5,1,3,2] => [4,6,5,1,3,2] => [6,4,5,1,3,2] => 00000 => 6
[4,6,5,2,1,3] => [5,4,6,1,3,2] => [4,5,6,1,3,2] => 00000 => 6
[4,6,5,2,3,1] => [6,4,5,1,3,2] => [4,6,5,1,3,2] => 00000 => 6
[4,6,5,3,1,2] => [5,6,4,1,3,2] => [6,5,4,1,3,2] => 00000 => 6
[4,6,5,3,2,1] => [6,5,4,1,3,2] => [5,6,4,1,3,2] => 00000 => 6
[5,3,6,1,2,4] => [4,5,2,6,1,3] => [5,4,2,6,1,3] => 00000 => 6
[5,3,6,1,4,2] => [4,6,2,5,1,3] => [6,4,2,5,1,3] => 00000 => 6
[5,3,6,2,1,4] => [5,4,2,6,1,3] => [4,5,2,6,1,3] => 00000 => 6
[5,3,6,2,4,1] => [6,4,2,5,1,3] => [4,6,2,5,1,3] => 00000 => 6
[5,3,6,4,1,2] => [5,6,2,4,1,3] => [6,5,2,4,1,3] => 00000 => 6
[5,3,6,4,2,1] => [6,5,2,4,1,3] => [5,6,2,4,1,3] => 00000 => 6
[5,4,6,1,2,3] => [4,5,6,2,1,3] => [5,4,6,2,1,3] => 00000 => 6
[5,4,6,1,3,2] => [4,6,5,2,1,3] => [6,4,5,2,1,3] => 00000 => 6
[5,4,6,2,1,3] => [5,4,6,2,1,3] => [4,5,6,2,1,3] => 00000 => 6
[5,4,6,2,3,1] => [6,4,5,2,1,3] => [4,6,5,2,1,3] => 00000 => 6
Description
The number of distinct subsequences in a binary word. In contrast to the subword complexity [[St000294]] this is the cardinality of the set of all subsequences of not necessarily consecutive letters.
Matching statistic: St001721
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00114: Permutations connectivity setBinary words
Mp00105: Binary words complementBinary words
St001721: Binary words ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1] => => => ? = 1 - 1
[2,1] => [1,2] => 1 => 0 => 1 = 2 - 1
[2,3,1] => [1,2,3] => 11 => 00 => 3 = 4 - 1
[3,1,2] => [1,3,2] => 10 => 01 => 2 = 3 - 1
[3,2,1] => [1,3,2] => 10 => 01 => 2 = 3 - 1
[3,4,1,2] => [1,3,2,4] => 101 => 010 => 3 = 4 - 1
[3,4,2,1] => [1,3,2,4] => 101 => 010 => 3 = 4 - 1
[4,3,1,2] => [1,4,2,3] => 100 => 011 => 3 = 4 - 1
[4,3,2,1] => [1,4,2,3] => 100 => 011 => 3 = 4 - 1
[4,5,1,2,3] => [1,4,2,5,3] => 1000 => 0111 => 4 = 5 - 1
[4,5,1,3,2] => [1,4,3,2,5] => 1001 => 0110 => 4 = 5 - 1
[4,5,2,1,3] => [1,4,2,5,3] => 1000 => 0111 => 4 = 5 - 1
[4,5,2,3,1] => [1,4,3,2,5] => 1001 => 0110 => 4 = 5 - 1
[4,5,3,1,2] => [1,4,2,5,3] => 1000 => 0111 => 4 = 5 - 1
[4,5,3,2,1] => [1,4,2,5,3] => 1000 => 0111 => 4 = 5 - 1
[5,3,4,1,2] => [1,5,2,3,4] => 1000 => 0111 => 4 = 5 - 1
[5,3,4,2,1] => [1,5,2,3,4] => 1000 => 0111 => 4 = 5 - 1
[5,4,1,2,3] => [1,5,3,2,4] => 1000 => 0111 => 4 = 5 - 1
[5,4,1,3,2] => [1,5,2,4,3] => 1000 => 0111 => 4 = 5 - 1
[5,4,2,1,3] => [1,5,3,2,4] => 1000 => 0111 => 4 = 5 - 1
[5,4,2,3,1] => [1,5,2,4,3] => 1000 => 0111 => 4 = 5 - 1
[5,4,3,1,2] => [1,5,2,4,3] => 1000 => 0111 => 4 = 5 - 1
[5,4,3,2,1] => [1,5,2,4,3] => 1000 => 0111 => 4 = 5 - 1
[4,5,6,1,2,3] => [1,4,2,5,3,6] => 10001 => 01110 => 5 = 6 - 1
[4,5,6,1,3,2] => [1,4,2,5,3,6] => 10001 => 01110 => 5 = 6 - 1
[4,5,6,2,1,3] => [1,4,2,5,3,6] => 10001 => 01110 => 5 = 6 - 1
[4,5,6,2,3,1] => [1,4,2,5,3,6] => 10001 => 01110 => 5 = 6 - 1
[4,5,6,3,1,2] => [1,4,3,6,2,5] => 10000 => 01111 => 5 = 6 - 1
[4,5,6,3,2,1] => [1,4,3,6,2,5] => 10000 => 01111 => 5 = 6 - 1
[4,6,1,5,2,3] => [1,4,5,2,6,3] => 10000 => 01111 => 5 = 6 - 1
[4,6,1,5,3,2] => [1,4,5,3,2,6] => 10001 => 01110 => 5 = 6 - 1
[4,6,2,5,1,3] => [1,4,5,2,6,3] => 10000 => 01111 => 5 = 6 - 1
[4,6,2,5,3,1] => [1,4,5,3,2,6] => 10001 => 01110 => 5 = 6 - 1
[4,6,3,5,1,2] => [1,4,5,2,6,3] => 10000 => 01111 => 5 = 6 - 1
[4,6,3,5,2,1] => [1,4,5,2,6,3] => 10000 => 01111 => 5 = 6 - 1
[4,6,5,1,2,3] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[4,6,5,1,3,2] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[4,6,5,2,1,3] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[4,6,5,2,3,1] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[4,6,5,3,1,2] => [1,4,3,5,2,6] => 10001 => 01110 => 5 = 6 - 1
[4,6,5,3,2,1] => [1,4,3,5,2,6] => 10001 => 01110 => 5 = 6 - 1
[5,3,6,1,2,4] => [1,5,2,3,6,4] => 10000 => 01111 => 5 = 6 - 1
[5,3,6,1,4,2] => [1,5,4,2,3,6] => 10001 => 01110 => 5 = 6 - 1
[5,3,6,2,1,4] => [1,5,2,3,6,4] => 10000 => 01111 => 5 = 6 - 1
[5,3,6,2,4,1] => [1,5,4,2,3,6] => 10001 => 01110 => 5 = 6 - 1
[5,3,6,4,1,2] => [1,5,2,3,6,4] => 10000 => 01111 => 5 = 6 - 1
[5,3,6,4,2,1] => [1,5,2,3,6,4] => 10000 => 01111 => 5 = 6 - 1
[5,4,6,1,2,3] => [1,5,2,4,3,6] => 10001 => 01110 => 5 = 6 - 1
[5,4,6,1,3,2] => [1,5,3,6,2,4] => 10000 => 01111 => 5 = 6 - 1
[5,4,6,2,1,3] => [1,5,2,4,3,6] => 10001 => 01110 => 5 = 6 - 1
[5,4,6,2,3,1] => [1,5,3,6,2,4] => 10000 => 01111 => 5 = 6 - 1
Description
The degree of a binary word. A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$. Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros.
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00117: Graphs Ore closureGraphs
St000454: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0 = 1 - 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 4 - 1
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,5,1,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,5,1,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,5,2,1,3] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,5,2,3,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,5,3,1,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,3,4,1,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,3,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,1,2,3] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,1,3,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,2,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,2,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,3,1,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,5,6,1,2,3] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,5,6,1,3,2] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,5,6,2,1,3] => [4,6,5,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,5,6,2,3,1] => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,5,6,3,1,2] => [4,6,5,3,1,2] => ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,5,6,3,2,1] => [4,6,5,3,2,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,1,5,2,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,1,5,3,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,2,5,1,3] => [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,2,5,3,1] => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,3,5,1,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,3,5,2,1] => [4,6,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,5,1,2,3] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,5,1,3,2] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,5,2,1,3] => [4,6,5,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,5,2,3,1] => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,5,3,1,2] => [4,6,5,3,1,2] => ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[4,6,5,3,2,1] => [4,6,5,3,2,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,3,6,1,2,4] => [5,3,6,1,4,2] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,3,6,1,4,2] => [5,3,6,1,4,2] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,3,6,2,1,4] => [5,3,6,2,1,4] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,3,6,2,4,1] => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,3,6,4,1,2] => [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,3,6,4,2,1] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,4,6,1,2,3] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,4,6,1,3,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,4,6,2,1,3] => [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,4,6,2,3,1] => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[5,4,6,3,1,2] => [5,4,6,3,1,2] => ([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Mp00069: Permutations complementPermutations
Mp00252: Permutations restrictionPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
St000829: Permutations ⟶ ℤResult quality: 71% values known / values provided: 100%distinct values known / distinct values provided: 71%
Values
[1] => [1] => [] => [] => ? = 1 - 3
[2,1] => [1,2] => [1] => [1] => ? = 2 - 3
[2,3,1] => [2,1,3] => [2,1] => [2,1] => 1 = 4 - 3
[3,1,2] => [1,3,2] => [1,2] => [1,2] => 0 = 3 - 3
[3,2,1] => [1,2,3] => [1,2] => [1,2] => 0 = 3 - 3
[3,4,1,2] => [2,1,4,3] => [2,1,3] => [2,1,3] => 1 = 4 - 3
[3,4,2,1] => [2,1,3,4] => [2,1,3] => [2,1,3] => 1 = 4 - 3
[4,3,1,2] => [1,2,4,3] => [1,2,3] => [1,3,2] => 1 = 4 - 3
[4,3,2,1] => [1,2,3,4] => [1,2,3] => [1,3,2] => 1 = 4 - 3
[4,5,1,2,3] => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 5 - 3
[4,5,1,3,2] => [2,1,5,3,4] => [2,1,3,4] => [2,1,4,3] => 2 = 5 - 3
[4,5,2,1,3] => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2 = 5 - 3
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2 = 5 - 3
[4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => [2,1,4,3] => 2 = 5 - 3
[4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4] => [2,1,4,3] => 2 = 5 - 3
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,4] => [1,4,3,2] => 2 = 5 - 3
[5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4] => [1,4,3,2] => 2 = 5 - 3
[5,4,1,2,3] => [1,2,5,4,3] => [1,2,4,3] => [1,4,3,2] => 2 = 5 - 3
[5,4,1,3,2] => [1,2,5,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 5 - 3
[5,4,2,1,3] => [1,2,4,5,3] => [1,2,4,3] => [1,4,3,2] => 2 = 5 - 3
[5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3] => [1,4,3,2] => 2 = 5 - 3
[5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,4] => [1,4,3,2] => 2 = 5 - 3
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => [1,4,3,2] => 2 = 5 - 3
[4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 3 = 6 - 3
[4,5,6,1,3,2] => [3,2,1,6,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => 3 = 6 - 3
[4,5,6,2,1,3] => [3,2,1,5,6,4] => [3,2,1,5,4] => [3,2,1,5,4] => 3 = 6 - 3
[4,5,6,2,3,1] => [3,2,1,5,4,6] => [3,2,1,5,4] => [3,2,1,5,4] => 3 = 6 - 3
[4,5,6,3,1,2] => [3,2,1,4,6,5] => [3,2,1,4,5] => [3,2,1,5,4] => 3 = 6 - 3
[4,5,6,3,2,1] => [3,2,1,4,5,6] => [3,2,1,4,5] => [3,2,1,5,4] => 3 = 6 - 3
[4,6,1,5,2,3] => [3,1,6,2,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,1,5,3,2] => [3,1,6,2,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,2,5,1,3] => [3,1,5,2,6,4] => [3,1,5,2,4] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,2,5,3,1] => [3,1,5,2,4,6] => [3,1,5,2,4] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,3,5,1,2] => [3,1,4,2,6,5] => [3,1,4,2,5] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,3,5,2,1] => [3,1,4,2,5,6] => [3,1,4,2,5] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,5,1,2,3] => [3,1,2,6,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,5,1,3,2] => [3,1,2,6,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,5,2,1,3] => [3,1,2,5,6,4] => [3,1,2,5,4] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,5,2,3,1] => [3,1,2,5,4,6] => [3,1,2,5,4] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,5,3,1,2] => [3,1,2,4,6,5] => [3,1,2,4,5] => [3,1,5,4,2] => 3 = 6 - 3
[4,6,5,3,2,1] => [3,1,2,4,5,6] => [3,1,2,4,5] => [3,1,5,4,2] => 3 = 6 - 3
[5,3,6,1,2,4] => [2,4,1,6,5,3] => [2,4,1,5,3] => [2,5,1,4,3] => 3 = 6 - 3
[5,3,6,1,4,2] => [2,4,1,6,3,5] => [2,4,1,3,5] => [2,5,1,4,3] => 3 = 6 - 3
[5,3,6,2,1,4] => [2,4,1,5,6,3] => [2,4,1,5,3] => [2,5,1,4,3] => 3 = 6 - 3
[5,3,6,2,4,1] => [2,4,1,5,3,6] => [2,4,1,5,3] => [2,5,1,4,3] => 3 = 6 - 3
[5,3,6,4,1,2] => [2,4,1,3,6,5] => [2,4,1,3,5] => [2,5,1,4,3] => 3 = 6 - 3
[5,3,6,4,2,1] => [2,4,1,3,5,6] => [2,4,1,3,5] => [2,5,1,4,3] => 3 = 6 - 3
[5,4,6,1,2,3] => [2,3,1,6,5,4] => [2,3,1,5,4] => [2,5,1,4,3] => 3 = 6 - 3
[5,4,6,1,3,2] => [2,3,1,6,4,5] => [2,3,1,4,5] => [2,5,1,4,3] => 3 = 6 - 3
[5,4,6,2,1,3] => [2,3,1,5,6,4] => [2,3,1,5,4] => [2,5,1,4,3] => 3 = 6 - 3
[5,4,6,2,3,1] => [2,3,1,5,4,6] => [2,3,1,5,4] => [2,5,1,4,3] => 3 = 6 - 3
[5,4,6,3,1,2] => [2,3,1,4,6,5] => [2,3,1,4,5] => [2,5,1,4,3] => 3 = 6 - 3
Description
The Ulam distance of a permutation to the identity permutation. This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$. In other words, this statistic plus [[St000062]] equals $n$.
Matching statistic: St000806
Mp00252: Permutations restrictionPermutations
Mp00114: Permutations connectivity setBinary words
Mp00097: Binary words delta morphismInteger compositions
St000806: Integer compositions ⟶ ℤResult quality: 57% values known / values provided: 99%distinct values known / distinct values provided: 57%
Values
[1] => [] => => [] => ? = 1 - 1
[2,1] => [1] => => [] => ? = 2 - 1
[2,3,1] => [2,1] => 0 => [1] => ? = 4 - 1
[3,1,2] => [1,2] => 1 => [1] => ? = 3 - 1
[3,2,1] => [2,1] => 0 => [1] => ? = 3 - 1
[3,4,1,2] => [3,1,2] => 00 => [2] => 3 = 4 - 1
[3,4,2,1] => [3,2,1] => 00 => [2] => 3 = 4 - 1
[4,3,1,2] => [3,1,2] => 00 => [2] => 3 = 4 - 1
[4,3,2,1] => [3,2,1] => 00 => [2] => 3 = 4 - 1
[4,5,1,2,3] => [4,1,2,3] => 000 => [3] => 4 = 5 - 1
[4,5,1,3,2] => [4,1,3,2] => 000 => [3] => 4 = 5 - 1
[4,5,2,1,3] => [4,2,1,3] => 000 => [3] => 4 = 5 - 1
[4,5,2,3,1] => [4,2,3,1] => 000 => [3] => 4 = 5 - 1
[4,5,3,1,2] => [4,3,1,2] => 000 => [3] => 4 = 5 - 1
[4,5,3,2,1] => [4,3,2,1] => 000 => [3] => 4 = 5 - 1
[5,3,4,1,2] => [3,4,1,2] => 000 => [3] => 4 = 5 - 1
[5,3,4,2,1] => [3,4,2,1] => 000 => [3] => 4 = 5 - 1
[5,4,1,2,3] => [4,1,2,3] => 000 => [3] => 4 = 5 - 1
[5,4,1,3,2] => [4,1,3,2] => 000 => [3] => 4 = 5 - 1
[5,4,2,1,3] => [4,2,1,3] => 000 => [3] => 4 = 5 - 1
[5,4,2,3,1] => [4,2,3,1] => 000 => [3] => 4 = 5 - 1
[5,4,3,1,2] => [4,3,1,2] => 000 => [3] => 4 = 5 - 1
[5,4,3,2,1] => [4,3,2,1] => 000 => [3] => 4 = 5 - 1
[4,5,6,1,2,3] => [4,5,1,2,3] => 0000 => [4] => 5 = 6 - 1
[4,5,6,1,3,2] => [4,5,1,3,2] => 0000 => [4] => 5 = 6 - 1
[4,5,6,2,1,3] => [4,5,2,1,3] => 0000 => [4] => 5 = 6 - 1
[4,5,6,2,3,1] => [4,5,2,3,1] => 0000 => [4] => 5 = 6 - 1
[4,5,6,3,1,2] => [4,5,3,1,2] => 0000 => [4] => 5 = 6 - 1
[4,5,6,3,2,1] => [4,5,3,2,1] => 0000 => [4] => 5 = 6 - 1
[4,6,1,5,2,3] => [4,1,5,2,3] => 0000 => [4] => 5 = 6 - 1
[4,6,1,5,3,2] => [4,1,5,3,2] => 0000 => [4] => 5 = 6 - 1
[4,6,2,5,1,3] => [4,2,5,1,3] => 0000 => [4] => 5 = 6 - 1
[4,6,2,5,3,1] => [4,2,5,3,1] => 0000 => [4] => 5 = 6 - 1
[4,6,3,5,1,2] => [4,3,5,1,2] => 0000 => [4] => 5 = 6 - 1
[4,6,3,5,2,1] => [4,3,5,2,1] => 0000 => [4] => 5 = 6 - 1
[4,6,5,1,2,3] => [4,5,1,2,3] => 0000 => [4] => 5 = 6 - 1
[4,6,5,1,3,2] => [4,5,1,3,2] => 0000 => [4] => 5 = 6 - 1
[4,6,5,2,1,3] => [4,5,2,1,3] => 0000 => [4] => 5 = 6 - 1
[4,6,5,2,3,1] => [4,5,2,3,1] => 0000 => [4] => 5 = 6 - 1
[4,6,5,3,1,2] => [4,5,3,1,2] => 0000 => [4] => 5 = 6 - 1
[4,6,5,3,2,1] => [4,5,3,2,1] => 0000 => [4] => 5 = 6 - 1
[5,3,6,1,2,4] => [5,3,1,2,4] => 0000 => [4] => 5 = 6 - 1
[5,3,6,1,4,2] => [5,3,1,4,2] => 0000 => [4] => 5 = 6 - 1
[5,3,6,2,1,4] => [5,3,2,1,4] => 0000 => [4] => 5 = 6 - 1
[5,3,6,2,4,1] => [5,3,2,4,1] => 0000 => [4] => 5 = 6 - 1
[5,3,6,4,1,2] => [5,3,4,1,2] => 0000 => [4] => 5 = 6 - 1
[5,3,6,4,2,1] => [5,3,4,2,1] => 0000 => [4] => 5 = 6 - 1
[5,4,6,1,2,3] => [5,4,1,2,3] => 0000 => [4] => 5 = 6 - 1
[5,4,6,1,3,2] => [5,4,1,3,2] => 0000 => [4] => 5 = 6 - 1
[5,4,6,2,1,3] => [5,4,2,1,3] => 0000 => [4] => 5 = 6 - 1
[5,4,6,2,3,1] => [5,4,2,3,1] => 0000 => [4] => 5 = 6 - 1
[5,4,6,3,1,2] => [5,4,3,1,2] => 0000 => [4] => 5 = 6 - 1
[5,4,6,3,2,1] => [5,4,3,2,1] => 0000 => [4] => 5 = 6 - 1
[5,6,1,2,3,4] => [5,1,2,3,4] => 0000 => [4] => 5 = 6 - 1
[5,6,1,2,4,3] => [5,1,2,4,3] => 0000 => [4] => 5 = 6 - 1
Description
The semiperimeter of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
The following 155 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001570The minimal number of edges to add to make a graph Hamiltonian. St000844The size of the largest block in the direct sum decomposition of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St000956The maximal displacement of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000133The "bounce" of a permutation. St000238The number of indices that are not small weak excedances. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001959The product of the heights of the peaks of a Dyck path. St000354The number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000934The 2-degree of an integer partition. 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. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice. St000699The toughness times the least common multiple of 1,. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001622The number of join-irreducible elements of a lattice. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer 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. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. 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. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. 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. St001914The size of the orbit 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St000944The 3-degree of an integer partition. 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. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. St000797The stat`` of a permutation. St000242The number of indices that are not cyclical small weak excedances. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.