Your data matches 358 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
(load all 20 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1] => ([],1)
 => 0
([],2)
 => [2] => [1,1] => ([(0,1)],2)
 => 1
([],3)
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([],4)
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([],5)
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([],6)
 => [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(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)
 => [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [1,2,1,1,1] => ([(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)
 => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(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)
 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => [1,2,1,1,1] => ([(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)
 => 2
([(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)
 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000793
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => {{1}}
 => 0
([],2)
 => [2] => [1,1,0,0]
 => {{1,2}}
 => 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 2
([],6)
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 2
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4,5},{6}}
 => 2
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5,6}}
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5,6}}
 => 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 2
([(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)
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6}}
 => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6}}
 => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 2
([(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)
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6}}
 => 2
([(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)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 2
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition. To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$. Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row. Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$. This statistic is the length of the longest partition on the diagonal of the diagram.
Matching statistic: St001486
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => ([],1)
 => [1] => 1 = 0 + 1
([],2)
 => [2] => ([],2)
 => [2] => 2 = 1 + 1
([],3)
 => [3] => ([],3)
 => [3] => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => [2,1] => 3 = 2 + 1
([],4)
 => [4] => ([],4)
 => [4] => 2 = 1 + 1
([(0,3),(1,2)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
([],5)
 => [5] => ([],5)
 => [5] => 2 = 1 + 1
([(1,4),(2,3)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => [4,1] => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => [4,1] => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1] => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => 3 = 2 + 1
([],6)
 => [6] => ([],6)
 => [6] => 2 = 1 + 1
([(2,5),(3,4)],6)
 => [2,4] => ([(3,5),(4,5)],6)
 => [5,1] => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => ([(3,5),(4,5)],6)
 => [5,1] => 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(0,5),(1,4),(2,3)],6)
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => [5,1] => 3 = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => 3 = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => [5,1] => 3 = 2 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => 3 = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => 3 = 2 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => 3 = 2 + 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => 3 = 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => 3 = 2 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => 3 = 2 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => 3 = 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)
 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => 3 = 2 + 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => 3 = 2 + 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)
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => 3 = 2 + 1
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000444
(load all 3 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1] => [1,0]
 => [1,0]
 => ? = 0
([],2)
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
([],6)
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2
([(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)
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2
([(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)
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
([(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)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 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)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 2
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000253
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1]
 => [[1]]
 => {{1}}
 => ? = 0 - 1
([],2)
 => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 0 = 1 - 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 1 = 2 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => {{1,2},{3,4},{5},{6}}
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => {{1,2,3},{4},{5},{6}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => {{1,2,3},{4,5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
Description
The crossing number of a set partition. This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000254
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1]
 => [[1]]
 => {{1}}
 => ? = 0 - 1
([],2)
 => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 0 = 1 - 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 1 = 2 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => {{1,2},{3,4},{5},{6}}
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => {{1,2,3},{4},{5},{6}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => {{1,2,3},{4,5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
Description
The nesting number of a set partition. This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St000442
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1] => [1,0]
 => [1,0]
 => ? = 0 - 1
([],2)
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
([],6)
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 1 = 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)
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 1 = 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)
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 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)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 1 = 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)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000659
(load all 8 compositions to match this statistic)
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 0 - 1
([],2)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
([],3)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
([],4)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 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)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 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)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 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)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000730
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1]
 => [[1]]
 => {{1}}
 => ? = 0 - 1
([],2)
 => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 0 = 1 - 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 1 = 2 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => {{1,2},{3,4},{5},{6}}
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => {{1,2,3},{4},{5},{6}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => {{1,2,3},{4,5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 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)
 => [6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1 = 2 - 1
Description
The maximal arc length of a set partition. The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
Matching statistic: St000919
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1] => [1,0]
 => [.,.]
 => ? = 0 - 1
([],2)
 => [2] => [1,1,0,0]
 => [.,[.,.]]
 => 0 = 1 - 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1 = 2 - 1
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1 = 2 - 1
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 2 - 1
([],6)
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[[.,[.,.]],[.,[.,.]]],.]
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[[.,[.,.]],.],[.,.]],.]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[[.,[.,.]],.],[.,.]],.]
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => 1 = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[[.,[.,.]],.],[.,.]],.]
 => 1 = 2 - 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[[.,[.,.]],.],[.,.]],.]
 => 1 = 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)
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[[.,[.,[.,.]]],[.,.]],.]
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[[[.,[.,.]],[.,.]],.],.]
 => 1 = 2 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[[[.,[.,.]],[.,.]],.],.]
 => 1 = 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)
 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => 1 = 2 - 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)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[[[.,[.,.]],[.,.]],.],.]
 => 1 = 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)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [[[[.,[.,[.,.]]],.],.],.]
 => 1 = 2 - 1
Description
The number of maximal left branches of a binary tree. A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
The following 348 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001093The detour number of a graph. St000264The girth of a graph, which is not a tree. St001568The smallest positive integer that does not appear twice in the partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. 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. St001657The number of twos in an integer partition. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the 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. St000517The Kreweras number of an integer partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000913The number of ways to refine the partition into singletons. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000142The number of even parts of a partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000995The largest even part of an integer partition. St001092The number of distinct even parts of a partition. St001214The aft of an integer partition. St001252Half the sum of the even parts of a 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. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001730The number of times the path corresponding to a binary word crosses the base line. 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 $eA$. St000182The number of permutations whose cycle type is the given integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001484The number of singletons of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St000455The second largest eigenvalue of a graph if it is integral. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000379The number of Hamiltonian cycles in a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000846The maximal number of elements covering an element of a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000296The length of the symmetric border of a binary word. St000699The toughness times the least common multiple of 1,. St000655The length of the minimal rise of a Dyck path. St000297The number of leading ones in a binary word. St000627The exponent of a binary word. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000552The number of cut vertices of a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001689The number of celebrities in a graph. St001826The maximal number of leaves on a vertex of a graph. St000449The number of pairs of vertices of a graph with distance 4. St001890The maximum magnitude of the Möbius function of a poset. St001637The number of (upper) dissectors of a poset. St001518The number of graphs with the same ordinary spectrum as the given graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000311The number of vertices of odd degree in a graph. St000322The skewness of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001248Sum of the even parts of a 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001732The number of peaks visible from the left. St000376The bounce deficit of a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000284The Plancherel distribution on integer partitions. 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. 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. St000717The number of ordinal summands of a poset. 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. St000906The length of the shortest maximal chain in a poset. St000993The multiplicity of the largest part of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001128The exponens consonantiae of a partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000567The sum of the products of all pairs of parts. St000640The rank of the largest boolean interval in a poset. 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. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000418The number of Dyck paths that are weakly below a Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000762The sum of the positions of the weak records of an integer composition. 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. St000933The number of multipartitions of sizes given by an integer partition. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001389The number of partitions of the same length below the given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001531Number of partial orders contained in the poset determined by the Dyck path. 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. St001959The product of the heights of the peaks of a Dyck path. St000369The dinv deficit of a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. 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. St000658The number of rises of length 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000946The sum of the skew hook positions in a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000976The sum of the positions of double up-steps of a Dyck path. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001139The number of occurrences of hills of size 2 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. 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. St001281The normalized isoperimetric number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001584The area statistic between a Dyck path and its bounce path. St001592The maximal number of simple paths between any two different vertices of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000618The number of self-evacuating tableaux of given shape. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000225Difference between largest and smallest parts in a partition. St000327The number of cover relations in a poset. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001668The number of points of the poset minus the width of the poset. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000287The number of connected components of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000763The sum of the positions of the strong records of an integer composition. St000764The number of strong records in an integer composition. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001060The distinguishing index of a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. 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. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. 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. St000447The number of pairs of vertices of a graph with distance 3. St000768The number of peaks in an integer composition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. 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. St000286The number of connected components of the complement of a graph. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000095The number of triangles of a graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001310The number of induced diamond graphs in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St001471The magnitude of a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001352The number of internal nodes in the modular decomposition of a graph. St001463The number of distinct columns in the nullspace of a graph. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000344The number of strongly connected outdegree sequences of a graph. St000274The number of perfect matchings of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001530The depth of a Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001651The Frankl number of a lattice. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.