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Your data matches 856 different statistics following compositions of up to 3 maps.
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Matching statistic: St000409
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St000409: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0 = -1 + 1
[.,[.,.]]
=> 1 = 0 + 1
[[.,.],.]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> 1 = 0 + 1
[[[.,.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> 1 = 0 + 1
Description
The number of pitchforks in a binary tree.
A pitchfork is a subtree of a complete binary tree with exactly three leaves, see Section 3.2 of [1].
Matching statistic: St000261
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
Description
The edge connectivity of a graph.
This is the minimum number of edges that has to be removed to make the graph disconnected.
Matching statistic: St000262
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000262: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000262: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
Description
The vertex connectivity of a graph.
For non-complete graphs, this is the minimum number of vertices that has to be removed to make the graph disconnected.
Matching statistic: St000272
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 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.
Matching statistic: St000310
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000310: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000310: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
Description
The minimal degree of a vertex of a graph.
Matching statistic: St000535
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
Description
The rank-width of a graph.
Matching statistic: St000536
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
Description
The pathwidth of a graph.
Matching statistic: St000537
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000537: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000537: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
Description
The cutwidth of a graph.
This is the minimum possible width of a linear ordering of its vertices, where the width of an ordering $\sigma$ is the maximum, among all the prefixes of $\sigma$, of the number of edges that have exactly one vertex in a prefix.
Matching statistic: St000778
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000778: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000778: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
Description
The metric dimension of a graph.
This is the length of the shortest vector of vertices, such that every vertex is uniquely determined by the vector of distances from these vertices.
Matching statistic: St000845
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0 = -1 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The maximal number of elements covered by an element in a poset.
The following 846 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000846The maximal number of elements covering an element of a poset. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000244The cardinality of the automorphism group of a graph. St000364The exponent of the automorphism group of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic 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. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001625The Möbius invariant of a lattice. St000256The number of parts from which one can substract 2 and still get an integer partition. St000379The number of Hamiltonian cycles in a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000741The Colin de Verdière graph invariant. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001071The beta invariant of the graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001280The number of parts of an integer partition that are at least two. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001638The book thickness of a graph. St001777The number of weak descents in an integer composition. St001826The maximal number of leaves on a vertex of a graph. St001931The weak major index of an integer composition regarded as a word. St000093The cardinality of a maximal independent set of vertices of a graph. St000258The burning number of a graph. St000273The domination number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000378The diagonal inversion number of an integer partition. St000482The (zero)-forcing number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000909The number of maximal chains of maximal size in a poset. St000917The open packing number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001368The number of vertices of maximal degree in a graph. St001399The distinguishing number of a poset. St001486The number of corners of the ribbon associated with an integer composition. St001642The Prague dimension of a graph. St001716The 1-improper chromatic number of a graph. St001746The coalition number of a graph. St001829The common independence number of a graph. St000271The chromatic index of a graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St000351The determinant of the adjacency matrix of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000088The row sums of the character table of the symmetric group. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000481The number of upper covers of a partition in dominance order. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000552The number of cut vertices of a graph. St000647The number of big descents of a permutation. St000651The maximal size of a rise in a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000665The number of rafts of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000834The number of right outer peaks of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001092The number of distinct even parts of a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001512The minimum rank of a graph. St001587Half of the largest even part of an integer partition. St001665The number of pure excedances of a permutation. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001737The number of descents of type 2 in a permutation. St001812The biclique partition number of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001958The degree of the polynomial interpolating the values of a permutation. St000010The length of the partition. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000054The first entry of the permutation. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000236The number of cyclical small weak excedances. St000288The number of ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000346The number of coarsenings of a partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000401The size of the symmetry class of a permutation. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000542The number of left-to-right-minima of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000808The number of up steps of the associated bargraph. St000862The number of parts of the shifted shape of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000916The packing number of a graph. St000918The 2-limited packing number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001093The detour number of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001432The order dimension of the partition. St001481The minimal height of a peak of a Dyck path. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001884The number of borders of a binary word. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000439The position of the first down step of a Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000661The number of rises of length 3 of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001095The number of non-isomorphic posets with precisely one further covering relation. St001141The number of occurrences of hills of size 3 in a Dyck path. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000456The monochromatic index of a connected graph. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000570The Edelman-Greene number of a permutation. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000988The orbit size of a permutation under Foata's bijection. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000842The breadth of a permutation. St000095The number of triangles of a graph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000358The number of occurrences of the pattern 31-2. St000368The Altshuler-Steinberg determinant of a graph. St000369The dinv deficit of a Dyck path. St000370The genus of a graph. St000376The bounce deficit of a Dyck path. St000403The Szeged index minus the Wiener index of a graph. St000407The number of occurrences of the pattern 2143 in a permutation. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000567The sum of the products of all pairs of parts. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000624The normalized sum of the minimal distances to a greater element. St000629The defect of a binary word. St000637The length of the longest cycle in a graph. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000658The number of rises of length 2 of a Dyck path. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000682The Grundy value of Welter's game on a binary word. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000699The toughness times the least common multiple of 1,. St000711The number of big exceedences of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000779The tier of a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000874The position of the last double rise in a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000946The sum of the skew hook positions in a Dyck path. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001130The number of two successive successions in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001377The major index minus the number of inversions of a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001715The number of non-records in a permutation. St001736The total number of cycles in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St001847The number of occurrences of the pattern 1432 in a permutation. St001871The number of triconnected components of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000096The number of spanning trees of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000284The Plancherel distribution on integer partitions. St000287The number of connected components of a graph. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000487The length of the shortest cycle of a permutation. St000544The cop number of a graph. St000568The hook number of a binary tree. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000717The number of ordinal summands of a poset. St000735The last entry on the main diagonal of a standard tableau. 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. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000847The number of standard Young tableaux whose descent set is the binary word. St000886The number of permutations with the same antidiagonal sums. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000948The chromatic discriminant of a graph. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001204Call 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. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001313The number of Dyck paths above the lattice path given by a binary word. St001363The Euler characteristic of a graph according to Knill. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001518The number of graphs with the same ordinary spectrum as the given graph. St001531Number of partial orders contained in the poset determined by the Dyck path. St001546The number of monomials in the Tutte polynomial of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001828The Euler characteristic of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001959The product of the heights of the peaks of a Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000017The number of inversions of a standard tableau. St000065The number of entries equal to -1 in an alternating sign matrix. St000119The number of occurrences of the pattern 321 in a permutation. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000204The number of internal nodes of a binary tree. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000225Difference between largest and smallest parts in a partition. St000226The convexity of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000354The number of recoils of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000478Another weight of a partition according to Alladi. St000491The number of inversions of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000497The lcb statistic of a set partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000628The balance of a binary word. St000653The last descent of a permutation. St000663The number of right floats of a permutation. St000673The number of non-fixed points of a permutation. St000674The number of hills of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000691The number of changes of a binary word. St000693The modular (standard) major index of a standard tableau. St000710The number of big deficiencies of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000768The number of peaks in an integer composition. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000809The reduced reflection length of the permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000830The total displacement of a permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000989The number of final rises of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001114The number of odd descents of a permutation. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001306The number of induced paths on four vertices in a graph. St001323The independence gap of a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001371The length of the longest Yamanouchi prefix of a binary word. St001388The number of non-attacking neighbors of a permutation. St001394The genus of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001411The number of patterns 321 or 3412 in a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001423The number of distinct cubes in a binary word. St001429The number of negative entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001513The number of nested exceedences of a permutation. St001519The pinnacle sum of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001577The minimal number of edges to add or remove to make a graph a cograph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001699The major index of a standard tableau minus the weighted size of its shape. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001731The factorization defect of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000061The number of nodes on the left branch of a binary tree. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000124The cardinality of the preimage of the Simion-Schmidt map. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000286The number of connected components of the complement of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000390The number of runs of ones in a binary word. St000402Half the size of the symmetry class of a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000485The length of the longest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000543The size of the conjugacy class of a binary word. St000618The number of self-evacuating tableaux of given shape. St000619The number of cyclic descents of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000652The maximal difference between successive positions of a permutation. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000690The size of the conjugacy class of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000762The sum of the positions of the weak records of an integer composition. St000770The major index of an integer partition when read from bottom to top. St000781The number of proper colouring schemes of a Ferrers diagram. St000805The number of peaks of the associated bargraph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000823The number of unsplittable factors of the set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000905The number of different multiplicities of parts of an integer composition. St000933The number of multipartitions of sizes given by an integer partition. St000983The length of the longest alternating subword. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001075The minimal size of a block of a set partition. St001081The number of minimal length factorizations of a permutation into star transpositions. St001220The width of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001260The permanent of an alternating sign matrix. St001272The number of graphs with the same degree sequence. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001468The smallest fixpoint of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001732The number of peaks visible from the left. St001735The number of permutations with the same set of runs. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001780The order of promotion on the set of standard tableaux of given shape. St001808The box weight or horizontal decoration of a Dyck path. St001838The number of nonempty primitive factors of a binary word. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000981The length of the longest zigzag subpath. St000806The semiperimeter of the associated bargraph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000296The length of the symmetric border of a binary word. 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. St000045The number of linear extensions of a binary tree. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000219The number of occurrences of the pattern 231 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001545The second Elser number of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001960The number of descents of a permutation minus one if its first entry is not one. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001890The maximum magnitude of the Möbius function of a poset. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000667The greatest common divisor of the parts of the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001933The largest multiplicity of a part in an integer partition. St000142The number of even parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000877The depth of the binary word interpreted as a path. St000944The 3-degree of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001248Sum of the even parts of a partition. St001252Half the sum of the even parts of a partition. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001541The Gini index of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000026The position of the first return of a Dyck path. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000392The length of the longest run of ones in a binary word. St000655The length of the minimal rise of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000995The largest even part of an integer partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000464The Schultz index of a connected graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000671The maximin edge-connectivity for choosing a subgraph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001469The holeyness of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001673The degree of asymmetry of an integer composition. St000006The dinv of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000308The height of the tree associated to a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001488The number of corners of a skew partition. St001649The length of a longest trail in a graph. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001110The 3-dynamic chromatic number of a graph. St000635The number of strictly order preserving maps of a poset into itself. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001948The number of augmented double ascents of a permutation. St001282The number of graphs with the same chromatic polynomial. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001118The acyclic chromatic index of a graph. St000327The number of cover relations in a poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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 $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000767The number of runs in an integer composition. St000820The number of compositions obtained by rotating the composition. St000903The number of different parts of an integer composition. St000920The logarithmic height of a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001720The minimal length of a chain of small intervals in a lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001927Sparre Andersen's number of positives of a signed permutation. St000679The pruning number of an ordered tree. St000396The register function (or Horton-Strahler number) of a binary tree. St000648The number of 2-excedences of a permutation. St000397The Strahler number of a rooted tree. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000454The largest eigenvalue of a graph if it is integral. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001875The number of simple modules with projective dimension at most 1. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001353The number of prime nodes in the modular decomposition of a graph. 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 $eA$. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000171The degree of the graph. St000686The finitistic dominant dimension of a Dyck path. St001112The 3-weak dynamic number of a graph. 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:
St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001725The harmonious chromatic number of a graph. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000253The crossing number of a set partition. St000306The bounce count of a Dyck path. St000441The number of successions of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000703The number of deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001729The number of visible descents of a permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001801Half the number of preimage-image pairs of different parity in a permutation. St001839The number of excedances of a set partition. St000053The number of valleys of the Dyck path. St000335The difference of lower and upper interactions. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001471The magnitude of a Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000458The number of permutations obtained by switching adjacencies or successions. St000638The number of up-down runs of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
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