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Your data matches 109 different statistics following compositions of up to 3 maps.
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Matching statistic: St000483
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
St000483: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 1
[1,3,2,4,5] => 2
[1,3,2,5,4] => 3
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 1
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 2
Description
The number of times a permutation switches from increasing to decreasing or decreasing to increasing.
This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]
Matching statistic: St001036
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00071: Permutations —descent 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%
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
[2,1] => [1,1] => [1,0,1,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [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
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000388
Mp00071: Permutations —descent 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%
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
[2,1] => [1,1] => ([(0,1)],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
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [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
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => ([(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
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St001352
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent 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%
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
[2,1] => [1,1] => ([(0,1)],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
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [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
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => ([(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
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St001951
Mp00071: Permutations —descent 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%
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
[2,1] => [1,1] => ([(0,1)],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
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [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
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => ([(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
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 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: St000071
Mp00071: Permutations —descent 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%
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
[2,1] => [1,1] => [[1,1],[]]
=> ([(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
[2,1,3] => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[2,3,1] => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [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
[1,3,4,2] => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(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
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(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
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000340
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
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
Mp00109: Permutations —descent word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => ? = 0
[1,2] => 0 => 0
[2,1] => 1 => 0
[1,2,3] => 00 => 0
[1,3,2] => 01 => 1
[2,1,3] => 10 => 1
[2,3,1] => 01 => 1
[3,1,2] => 10 => 1
[3,2,1] => 11 => 0
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 1
[1,3,2,4] => 010 => 2
[1,3,4,2] => 001 => 1
[1,4,2,3] => 010 => 2
[1,4,3,2] => 011 => 1
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 2
[2,3,1,4] => 010 => 2
[2,3,4,1] => 001 => 1
[2,4,1,3] => 010 => 2
[2,4,3,1] => 011 => 1
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 2
[3,2,1,4] => 110 => 1
[3,2,4,1] => 101 => 2
[3,4,1,2] => 010 => 2
[3,4,2,1] => 011 => 1
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 2
[4,2,1,3] => 110 => 1
[4,2,3,1] => 101 => 2
[4,3,1,2] => 110 => 1
[4,3,2,1] => 111 => 0
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => 0010 => 2
[1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => 0010 => 2
[1,2,5,4,3] => 0011 => 1
[1,3,2,4,5] => 0100 => 2
[1,3,2,5,4] => 0101 => 3
[1,3,4,2,5] => 0010 => 2
[1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => 0010 => 2
[1,3,5,4,2] => 0011 => 1
[1,4,2,3,5] => 0100 => 2
[1,4,2,5,3] => 0101 => 3
[1,4,3,2,5] => 0110 => 2
[1,4,3,5,2] => 0101 => 3
[1,4,5,2,3] => 0010 => 2
[1,4,5,3,2] => 0011 => 1
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: St000053
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [] => ?
=> ? = 0
[1,2] => 0 => [1] => [1,0]
=> 0
[2,1] => 1 => [1] => [1,0]
=> 0
[1,2,3] => 00 => [2] => [1,1,0,0]
=> 0
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 1
[2,1,3] => 10 => [1,1] => [1,0,1,0]
=> 1
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 1
[3,1,2] => 10 => [1,1] => [1,0,1,0]
=> 1
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 0
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000272
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [] => ?
=> ? = 0
[1,2] => 0 => [1] => ([],1)
=> 0
[2,1] => 1 => [1] => ([],1)
=> 0
[1,2,3] => 00 => [2] => ([],2)
=> 0
[1,3,2] => 01 => [1,1] => ([(0,1)],2)
=> 1
[2,1,3] => 10 => [1,1] => ([(0,1)],2)
=> 1
[2,3,1] => 01 => [1,1] => ([(0,1)],2)
=> 1
[3,1,2] => 10 => [1,1] => ([(0,1)],2)
=> 1
[3,2,1] => 11 => [2] => ([],2)
=> 0
[1,2,3,4] => 000 => [3] => ([],3)
=> 0
[1,2,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,2,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,4,2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,4,2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,3,2] => 011 => [1,2] => ([(1,2)],3)
=> 1
[2,1,3,4] => 100 => [1,2] => ([(1,2)],3)
=> 1
[2,1,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,3,1,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,3,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,4,1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,4,3,1] => 011 => [1,2] => ([(1,2)],3)
=> 1
[3,1,2,4] => 100 => [1,2] => ([(1,2)],3)
=> 1
[3,1,4,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1,4] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,2,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,4,1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,4,2,1] => 011 => [1,2] => ([(1,2)],3)
=> 1
[4,1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 1
[4,1,3,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[4,2,3,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[4,3,1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[4,3,2,1] => 111 => [3] => ([],3)
=> 0
[1,2,3,4,5] => 0000 => [4] => ([],4)
=> 0
[1,2,3,5,4] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,4,3,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,4,5,3] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,5,3,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,5,4,3] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,5,2] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,5,2,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,5,4,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2,5] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,5,2,3] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 1
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
The following 99 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001812The biclique partition number of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001202Call 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. St001203We associate to 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 Dyck path as follows:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000937The number of positive values of the symmetric group character corresponding to the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001083The number of boxed occurrences of 132 in a permutation. St000259The diameter of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. 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. St000824The sum of the number of descents and the number of recoils of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. 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. 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 $eA$. St001875The number of simple modules with projective dimension at most 1. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001668The number of points of the poset minus the width of the poset. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001644The dimension of a graph. St001792The arboricity of a graph. St001962The proper pathwidth of a graph. St000785The number of distinct colouring schemes of a graph. St001883The mutual visibility number of a graph. St001323The independence gap of a graph. St001638The book thickness of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001822The number of alignments of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St001734The lettericity of a graph. St000077The number of boxed and circled entries. St000310The minimal degree of a vertex of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001642The Prague dimension of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000640The rank of the largest boolean interval in a poset. St001271The competition number of a graph.
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