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Your data matches 135 different statistics following compositions of up to 3 maps.
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Matching statistic: St001331
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001331: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001331: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[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)
=> 3
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[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)
=> 4
[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)
=> 3
[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
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[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
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 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
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 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)
=> 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[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)
=> 3
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The size of the minimal feedback vertex set.
A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Matching statistic: St001336
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001336: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[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)
=> 3
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[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)
=> 4
[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)
=> 3
[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
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[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
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 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
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 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)
=> 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[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)
=> 3
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The minimal number of vertices in a graph whose complement is triangle-free.
Matching statistic: St000052
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[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]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[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]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 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]
=> 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 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]
=> 4
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[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]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000118
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [.,.]
=> 0
[1,1] => [1,0,1,0]
=> [.,[.,.]]
=> 0
[2] => [1,1,0,0]
=> [[.,.],.]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 0
[3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 0
[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]
=> [.,[.,[.,[.,[.,.]]]]]
=> 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 2
[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]
=> [.,[[.,.],[[.,.],.]]]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 0
[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]
=> [[[.,.],.],[[.,.],.]]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 0
[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]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 4
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> 2
[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]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> 1
[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]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 3
[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]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 2
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St001036
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[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]
=> [1,1,0,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[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]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[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]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 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
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[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]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 3
[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]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001167
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 2
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St000157
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [[1]]
=> 0
[1,1] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[2] => [2]
=> [[1,2]]
=> [[1,2]]
=> 0
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2],[3]]
=> 1
[1,2] => [2,1]
=> [[1,2],[3]]
=> [[1,2,3]]
=> 0
[2,1] => [2,1]
=> [[1,2],[3]]
=> [[1,2,3]]
=> 0
[3] => [3]
=> [[1,2,3]]
=> [[1,2,3]]
=> 0
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> 2
[1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1
[1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1
[1,3] => [3,1]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> 0
[2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> 0
[3,1] => [3,1]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> 0
[4] => [4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> 3
[1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2
[1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2
[1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2
[1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> 0
[2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2
[2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> 0
[3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> 0
[4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> 0
[5] => [5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2],[3],[4],[5],[6]]
=> 4
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[1,1,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3
[2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000931
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [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] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1
[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]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 0
[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]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 4
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001033
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 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]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 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]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 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]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4
[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]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001253
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 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]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 4
[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]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
The following 125 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001843The Z-index of a set partition. St000733The row containing the largest entry of a standard tableau. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000010The length of the partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000519The largest length of a factor maximising the subword complexity. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St000288The number of ones in a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama 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:
St000806The semiperimeter of the associated bargraph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000327The number of cover relations in a poset. St000890The number of nonzero entries in an alternating sign matrix. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000481The number of upper covers of a partition in dominance order. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000065The number of entries equal to -1 in an alternating sign matrix. St000648The number of 2-excedences of a permutation. St000306The bounce count of a Dyck path. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001432The order dimension of the partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001176The size of a partition minus its first part. St000358The number of occurrences of the pattern 31-2. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000731The number of double exceedences of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000292The number of ascents of a binary word. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000359The number of occurrences of the pattern 23-1. St000312The number of leaves in a graph. St000619The number of cyclic descents of a permutation. St000355The number of occurrences of the pattern 21-3. St001727The number of invisible inversions of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000039The number of crossings of a permutation. St000089The absolute variation of a composition. St000217The number of occurrences of the pattern 312 in a permutation. St000317The cycle descent number of a permutation. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000377The dinv defect of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St000047The number of standard immaculate tableaux of a given shape. St000277The number of ribbon shaped standard tableaux. St000340The number of non-final maximal constant sub-paths of length greater than one. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000778The metric dimension of a graph. St000808The number of up steps of the associated bargraph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001917The order of toric promotion on the set of labellings of a graph. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001549The number of restricted non-inversions between exceedances. St001570The minimal number of edges to add to make a graph Hamiltonian. St000782The indicator function of whether a given perfect matching is an L & P matching. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001578The minimal number of edges to add or remove to make a graph a line graph. St001964The interval resolution global dimension of a poset. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St001845The number of join irreducibles minus the rank of a lattice. St001651The Frankl number of a lattice. St000741The Colin de Verdière graph invariant. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation.
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