Your data matches 23 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001036
(load all 4 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1,0]
 => 0
[[1,2]]
 => [2] => [1,1,0,0]
 => 0
[[1],[2]]
 => [2] => [1,1,0,0]
 => 0
[[1,2,3]]
 => [3] => [1,1,1,0,0,0]
 => 0
[[1,3],[2]]
 => [2,1] => [1,1,0,0,1,0]
 => 1
[[1,2],[3]]
 => [3] => [1,1,1,0,0,0]
 => 0
[[1],[2],[3]]
 => [3] => [1,1,1,0,0,0]
 => 0
[[1,2,3,4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1,3,4],[2]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,4],[3]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[[1,2,3],[4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1,3],[2,4]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2],[3,4]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[[1,4],[2],[3]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[[1,3],[2],[4]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2],[3],[4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,3,4,5],[2]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,4,5],[3]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,2,3,4],[5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,5],[3,4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,3,4],[2,5]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,4],[3,5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,4,5],[2],[3]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,5],[3],[4]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,3,4],[2],[5]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3],[4],[5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2],[3,5],[4]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,3],[2,4],[5]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,3],[2],[4],[5]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2],[3],[4],[5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,2,3,4,5],[6]]
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000388
(load all 6 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St000722
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000722: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
Description
The number of different neighbourhoods in a graph.
Matching statistic: St001352
(load all 6 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St001951
(load all 6 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
Description
The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$. In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.
Matching statistic: St000340
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1,0]
 => 0
[[1,2]]
 => [2] => [1,1] => [1,0,1,0]
 => 0
[[1],[2]]
 => [2] => [1,1] => [1,0,1,0]
 => 0
[[1,2,3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1,3],[2]]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[[1,2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1],[2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1,2,3,4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,3,4],[2]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,2,3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,3],[2,4]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2],[3,4]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,4],[2],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,3],[2],[4]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,3,4,5],[2]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4,5],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,2,3,4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3,4]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3,4],[2,5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3,5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,4,5],[2],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,3,4],[2],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2],[3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,3],[2,4],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3],[2],[4],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,2,3,4,5],[6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 4
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000691
Mapping
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => 1 => 0
[[1,2]]
 => [2] => [1,1] => 11 => 0
[[1],[2]]
 => [2] => [1,1] => 11 => 0
[[1,2,3]]
 => [3] => [1,1,1] => 111 => 0
[[1,3],[2]]
 => [3] => [1,1,1] => 111 => 0
[[1,2],[3]]
 => [2,1] => [1,2] => 110 => 1
[[1],[2],[3]]
 => [3] => [1,1,1] => 111 => 0
[[1,2,3,4]]
 => [4] => [1,1,1,1] => 1111 => 0
[[1,3,4],[2]]
 => [4] => [1,1,1,1] => 1111 => 0
[[1,2,4],[3]]
 => [2,2] => [1,2,1] => 1101 => 2
[[1,2,3],[4]]
 => [3,1] => [1,1,2] => 1110 => 1
[[1,3],[2,4]]
 => [3,1] => [1,1,2] => 1110 => 1
[[1,2],[3,4]]
 => [2,2] => [1,2,1] => 1101 => 2
[[1,4],[2],[3]]
 => [4] => [1,1,1,1] => 1111 => 0
[[1,3],[2],[4]]
 => [3,1] => [1,1,2] => 1110 => 1
[[1,2],[3],[4]]
 => [2,2] => [1,2,1] => 1101 => 2
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1] => 1111 => 0
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,3,4,5],[2]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,2,4,5],[3]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,2,3,5],[4]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2,3,4],[5]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,3,5],[2,4]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2,5],[3,4]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,3,4],[2,5]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,2,4],[3,5]]
 => [2,2,1] => [1,2,2] => 11010 => 3
[[1,2,3],[4,5]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,4,5],[2],[3]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,3,5],[2],[4]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2,5],[3],[4]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,3,4],[2],[5]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,2,4],[3],[5]]
 => [2,2,1] => [1,2,2] => 11010 => 3
[[1,2,3],[4],[5]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,4],[2,5],[3]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,3],[2,5],[4]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2],[3,5],[4]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,3],[2,4],[5]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2],[3,4],[5]]
 => [2,2,1] => [1,2,2] => 11010 => 3
[[1,5],[2],[3],[4]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,4],[2],[3],[5]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,3],[2],[4],[5]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2],[3],[4],[5]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1] => 111111 => 0
[[1,3,4,5,6],[2]]
 => [6] => [1,1,1,1,1,1] => 111111 => 0
[[1,2,4,5,6],[3]]
 => [2,4] => [1,2,1,1,1] => 110111 => 2
[[1,2,3,5,6],[4]]
 => [3,3] => [1,1,2,1,1] => 111011 => 2
[[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,2,1] => 111101 => 2
[[1,2,3,4,5],[6]]
 => [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,3,5,6],[2,4]]
 => [3,3] => [1,1,2,1,1] => 111011 => 2
Description
The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001120
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001120: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1
[[1,2],[3]]
 => [3] => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => [4] => ([],4)
 => 0
[[1,3,4],[2]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
[[1,2,3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1,3],[2,4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3,4]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
[[1,4],[2],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
[[1,3],[2],[4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => [5] => ([],5)
 => 0
[[1,3,4,5],[2]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,2,3,4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,4],[2,5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,4],[3,5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3],[4,5]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,4,5],[2],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,3,4],[2],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,3],[2,4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,4],[2],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3],[2],[4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,5,6],[4]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 1
[[1,2,3,4,5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
Description
The length of a longest path in a graph.
Matching statistic: St000071
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [[1],[]]
 => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => [[5,2],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => [[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => 5 = 4 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000453
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
Description
The number of distinct Laplacian eigenvalues of a graph.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001060The distinguishing index of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001862The number of crossings of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset.