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Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St000483
(load all 43 compositions to match this statistic)
(load all 43 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 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => 2 = 3 - 1
[4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => 2 = 3 - 1
[4,3,1,2] => 1 = 2 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => 2 = 3 - 1
[1,2,4,5,3] => 1 = 2 - 1
[1,2,5,3,4] => 2 = 3 - 1
[1,2,5,4,3] => 1 = 2 - 1
[1,3,2,4,5] => 2 = 3 - 1
[1,3,2,5,4] => 3 = 4 - 1
[1,3,4,2,5] => 2 = 3 - 1
[1,3,4,5,2] => 1 = 2 - 1
[1,3,5,2,4] => 2 = 3 - 1
[1,3,5,4,2] => 1 = 2 - 1
[1,4,2,3,5] => 2 = 3 - 1
[1,4,2,5,3] => 3 = 4 - 1
[1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => 3 = 4 - 1
[1,4,5,2,3] => 2 = 3 - 1
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: St000388
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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
[1,2] => [2] => ([],2)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
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)
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
[1,2] => [2] => ([],2)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
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)
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
[1,2] => [2] => ([],2)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
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: St001036
(load all 9 compositions to match this statistic)
(load all 9 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 - 1
[1,2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
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
[1,2] => [2] => [[2],[]]
=> ([(0,1)],2)
=> 1
[2,1] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[2,3,1] => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,3,2,4] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[1,3,4,2] => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,2,3] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[2,3,1,4] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[2,3,4,1] => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,1,3] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[3,2,1,4] => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[3,4,1,2] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[4,2,1,3] => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[4,3,1,2] => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 3
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3
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
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 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
[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
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[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
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[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
[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
[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
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[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
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[1,2,4,3,5] => [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
[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
[1,2,5,3,4] => [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
[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
[1,3,2,4,5] => [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
[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
[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
[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
[1,3,5,2,4] => [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
[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
[1,4,2,3,5] => [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
[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
[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
[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
[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
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
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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] => => ? = 1 - 1
[1,2] => 0 => 0 = 1 - 1
[2,1] => 1 => 0 = 1 - 1
[1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => 01 => 1 = 2 - 1
[2,1,3] => 10 => 1 = 2 - 1
[2,3,1] => 01 => 1 = 2 - 1
[3,1,2] => 10 => 1 = 2 - 1
[3,2,1] => 11 => 0 = 1 - 1
[1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => 001 => 1 = 2 - 1
[1,3,2,4] => 010 => 2 = 3 - 1
[1,3,4,2] => 001 => 1 = 2 - 1
[1,4,2,3] => 010 => 2 = 3 - 1
[1,4,3,2] => 011 => 1 = 2 - 1
[2,1,3,4] => 100 => 1 = 2 - 1
[2,1,4,3] => 101 => 2 = 3 - 1
[2,3,1,4] => 010 => 2 = 3 - 1
[2,3,4,1] => 001 => 1 = 2 - 1
[2,4,1,3] => 010 => 2 = 3 - 1
[2,4,3,1] => 011 => 1 = 2 - 1
[3,1,2,4] => 100 => 1 = 2 - 1
[3,1,4,2] => 101 => 2 = 3 - 1
[3,2,1,4] => 110 => 1 = 2 - 1
[3,2,4,1] => 101 => 2 = 3 - 1
[3,4,1,2] => 010 => 2 = 3 - 1
[3,4,2,1] => 011 => 1 = 2 - 1
[4,1,2,3] => 100 => 1 = 2 - 1
[4,1,3,2] => 101 => 2 = 3 - 1
[4,2,1,3] => 110 => 1 = 2 - 1
[4,2,3,1] => 101 => 2 = 3 - 1
[4,3,1,2] => 110 => 1 = 2 - 1
[4,3,2,1] => 111 => 0 = 1 - 1
[1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => 0001 => 1 = 2 - 1
[1,2,4,3,5] => 0010 => 2 = 3 - 1
[1,2,4,5,3] => 0001 => 1 = 2 - 1
[1,2,5,3,4] => 0010 => 2 = 3 - 1
[1,2,5,4,3] => 0011 => 1 = 2 - 1
[1,3,2,4,5] => 0100 => 2 = 3 - 1
[1,3,2,5,4] => 0101 => 3 = 4 - 1
[1,3,4,2,5] => 0010 => 2 = 3 - 1
[1,3,4,5,2] => 0001 => 1 = 2 - 1
[1,3,5,2,4] => 0010 => 2 = 3 - 1
[1,3,5,4,2] => 0011 => 1 = 2 - 1
[1,4,2,3,5] => 0100 => 2 = 3 - 1
[1,4,2,5,3] => 0101 => 3 = 4 - 1
[1,4,3,2,5] => 0110 => 2 = 3 - 1
[1,4,3,5,2] => 0101 => 3 = 4 - 1
[1,4,5,2,3] => 0010 => 2 = 3 - 1
[1,4,5,3,2] => 0011 => 1 = 2 - 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: St000010
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [] => ?
=> ? = 1
[1,2] => 0 => [1] => [1]
=> 1
[2,1] => 1 => [1] => [1]
=> 1
[1,2,3] => 00 => [2] => [2]
=> 1
[1,3,2] => 01 => [1,1] => [1,1]
=> 2
[2,1,3] => 10 => [1,1] => [1,1]
=> 2
[2,3,1] => 01 => [1,1] => [1,1]
=> 2
[3,1,2] => 10 => [1,1] => [1,1]
=> 2
[3,2,1] => 11 => [2] => [2]
=> 1
[1,2,3,4] => 000 => [3] => [3]
=> 1
[1,2,4,3] => 001 => [2,1] => [2,1]
=> 2
[1,3,2,4] => 010 => [1,1,1] => [1,1,1]
=> 3
[1,3,4,2] => 001 => [2,1] => [2,1]
=> 2
[1,4,2,3] => 010 => [1,1,1] => [1,1,1]
=> 3
[1,4,3,2] => 011 => [1,2] => [2,1]
=> 2
[2,1,3,4] => 100 => [1,2] => [2,1]
=> 2
[2,1,4,3] => 101 => [1,1,1] => [1,1,1]
=> 3
[2,3,1,4] => 010 => [1,1,1] => [1,1,1]
=> 3
[2,3,4,1] => 001 => [2,1] => [2,1]
=> 2
[2,4,1,3] => 010 => [1,1,1] => [1,1,1]
=> 3
[2,4,3,1] => 011 => [1,2] => [2,1]
=> 2
[3,1,2,4] => 100 => [1,2] => [2,1]
=> 2
[3,1,4,2] => 101 => [1,1,1] => [1,1,1]
=> 3
[3,2,1,4] => 110 => [2,1] => [2,1]
=> 2
[3,2,4,1] => 101 => [1,1,1] => [1,1,1]
=> 3
[3,4,1,2] => 010 => [1,1,1] => [1,1,1]
=> 3
[3,4,2,1] => 011 => [1,2] => [2,1]
=> 2
[4,1,2,3] => 100 => [1,2] => [2,1]
=> 2
[4,1,3,2] => 101 => [1,1,1] => [1,1,1]
=> 3
[4,2,1,3] => 110 => [2,1] => [2,1]
=> 2
[4,2,3,1] => 101 => [1,1,1] => [1,1,1]
=> 3
[4,3,1,2] => 110 => [2,1] => [2,1]
=> 2
[4,3,2,1] => 111 => [3] => [3]
=> 1
[1,2,3,4,5] => 0000 => [4] => [4]
=> 1
[1,2,3,5,4] => 0001 => [3,1] => [3,1]
=> 2
[1,2,4,3,5] => 0010 => [2,1,1] => [2,1,1]
=> 3
[1,2,4,5,3] => 0001 => [3,1] => [3,1]
=> 2
[1,2,5,3,4] => 0010 => [2,1,1] => [2,1,1]
=> 3
[1,2,5,4,3] => 0011 => [2,2] => [2,2]
=> 2
[1,3,2,4,5] => 0100 => [1,1,2] => [2,1,1]
=> 3
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4
[1,3,4,2,5] => 0010 => [2,1,1] => [2,1,1]
=> 3
[1,3,4,5,2] => 0001 => [3,1] => [3,1]
=> 2
[1,3,5,2,4] => 0010 => [2,1,1] => [2,1,1]
=> 3
[1,3,5,4,2] => 0011 => [2,2] => [2,2]
=> 2
[1,4,2,3,5] => 0100 => [1,1,2] => [2,1,1]
=> 3
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4
[1,4,3,2,5] => 0110 => [1,2,1] => [2,1,1]
=> 3
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4
[1,4,5,2,3] => 0010 => [2,1,1] => [2,1,1]
=> 3
[1,4,5,3,2] => 0011 => [2,2] => [2,2]
=> 2
Description
The length of the partition.
Matching statistic: St000011
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: 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
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [] => ?
=> ? = 1
[1,2] => 0 => [1] => [1,0]
=> 1
[2,1] => 1 => [1] => [1,0]
=> 1
[1,2,3] => 00 => [2] => [1,1,0,0]
=> 1
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 2
[2,1,3] => 10 => [1,1] => [1,0,1,0]
=> 2
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 2
[3,1,2] => 10 => [1,1] => [1,0,1,0]
=> 2
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 1
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
=> 1
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 1
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. 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. St000806The semiperimeter of the associated bargraph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St001083The number of boxed occurrences of 132 in 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. St001537The number of cyclic crossings of a permutation. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St001734The lettericity of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset.
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