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Your data matches 38 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => 2 = 0 + 2
[2] => 2 = 0 + 2
[1,1,1] => 2 = 0 + 2
[1,2] => 3 = 1 + 2
[2,1] => 3 = 1 + 2
[3] => 2 = 0 + 2
[1,1,1,1] => 2 = 0 + 2
[1,1,2] => 3 = 1 + 2
[1,2,1] => 4 = 2 + 2
[1,3] => 3 = 1 + 2
[2,1,1] => 3 = 1 + 2
[2,2] => 4 = 2 + 2
[3,1] => 3 = 1 + 2
[4] => 2 = 0 + 2
[1,1,1,1,1] => 2 = 0 + 2
[1,1,1,2] => 3 = 1 + 2
[1,1,2,1] => 4 = 2 + 2
[1,1,3] => 3 = 1 + 2
[1,2,1,1] => 4 = 2 + 2
[1,2,2] => 5 = 3 + 2
[1,3,1] => 4 = 2 + 2
[1,4] => 3 = 1 + 2
[2,1,1,1] => 3 = 1 + 2
[2,1,2] => 4 = 2 + 2
[2,2,1] => 5 = 3 + 2
[2,3] => 4 = 2 + 2
[3,1,1] => 3 = 1 + 2
[3,2] => 4 = 2 + 2
[4,1] => 3 = 1 + 2
[5] => 2 = 0 + 2
[1,1,1,1,1,1] => 2 = 0 + 2
[1,1,1,1,2] => 3 = 1 + 2
[1,1,1,2,1] => 4 = 2 + 2
[1,1,1,3] => 3 = 1 + 2
[1,1,2,1,1] => 4 = 2 + 2
[1,1,2,2] => 5 = 3 + 2
[1,1,3,1] => 4 = 2 + 2
[1,1,4] => 3 = 1 + 2
[1,2,1,1,1] => 4 = 2 + 2
[1,2,1,2] => 5 = 3 + 2
[1,2,2,1] => 6 = 4 + 2
[1,2,3] => 5 = 3 + 2
[1,3,1,1] => 4 = 2 + 2
[1,3,2] => 5 = 3 + 2
[1,4,1] => 4 = 2 + 2
[1,5] => 3 = 1 + 2
[2,1,1,1,1] => 3 = 1 + 2
[2,1,1,2] => 4 = 2 + 2
[2,1,2,1] => 5 = 3 + 2
[2,1,3] => 4 = 2 + 2
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: St001036
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
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%
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000388
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2] => ([],2)
=> 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => ([],3)
=> 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4] => ([],4)
=> 1 = 0 + 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4] => ([(3,4)],5)
=> 2 = 1 + 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 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5] => ([],5)
=> 1 = 0 + 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[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)
=> 3 = 2 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[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)
=> 3 = 2 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[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
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 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)
=> 4 = 3 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St001352
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001352: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2] => ([],2)
=> 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => ([],3)
=> 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4] => ([],4)
=> 1 = 0 + 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4] => ([(3,4)],5)
=> 2 = 1 + 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 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5] => ([],5)
=> 1 = 0 + 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[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)
=> 3 = 2 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[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)
=> 3 = 2 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[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
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 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)
=> 4 = 3 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St001951
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001951: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2] => ([],2)
=> 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => ([],3)
=> 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4] => ([],4)
=> 1 = 0 + 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4] => ([(3,4)],5)
=> 2 = 1 + 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 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5] => ([],5)
=> 1 = 0 + 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[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)
=> 3 = 2 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[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)
=> 3 = 2 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[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
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 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)
=> 4 = 3 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 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: St001035
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[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
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[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] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2
[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]
=> 3
[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
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St000071
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2] => [[2],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 4 = 3 + 1
[1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 3 = 2 + 1
[1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 3 = 2 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2 = 1 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> 3 = 2 + 1
[1,1,1,3] => [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 3 = 2 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 4 = 3 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 3 = 2 + 1
[1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2 = 1 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> 3 = 2 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 4 = 3 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 5 = 4 + 1
[1,2,3] => [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 4 = 3 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 4 = 3 + 1
[1,4,1] => [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 3 = 2 + 1
[1,5] => [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2 = 1 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 3 = 2 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4 = 3 + 1
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 3 = 2 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St001499
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [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 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000691
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,2] => 0 => 0
[2] => [1,1,0,0]
=> [2,1] => 1 => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 11 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1111 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 00000 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 00001 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 00010 => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 00011 => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 00100 => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 00101 => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 00110 => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 00111 => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 01000 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 01001 => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 01010 => 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 01011 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 01100 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 01101 => 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 01110 => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 01111 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 10000 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 10001 => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 10010 => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 10011 => 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: St000837
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,5,4,1,6] => 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [6,3,2,5,4,1] => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => 2
Description
The number of ascents of distance 2 of a permutation.
This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001304The number of maximally independent sets of vertices of a graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000340The number of non-final maximal constant sub-paths of length greater than one. St000836The number of descents of distance 2 of a permutation. St000871The number of very big ascents of a permutation. St001388The number of non-attacking neighbors of a permutation. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000552The number of cut vertices of a graph. St001692The number of vertices with higher degree than the average degree in a graph. St000259The diameter of a connected graph. St001120The length of a longest path in a graph. St001093The detour number of a graph. St001083The number of boxed occurrences of 132 in a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000670The reversal length of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001902The number of potential covers of a poset. St001488The number of corners of a skew partition. St001537The number of cyclic crossings of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000831The number of indices that are either descents or recoils. St001649The length of a longest trail in a graph. St000455The second largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St001960The number of descents of a permutation minus one if its first entry is not one. St001637The number of (upper) dissectors of a poset.
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