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Your data matches 371 different statistics following compositions of up to 3 maps.
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Matching statistic: St000228
(load all 2515 compositions to match this statistic)
(load all 2515 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3]
=> 3 = 2 + 1
[1,3,2] => [2,1]
=> 3 = 2 + 1
[2,1,3] => [2,1]
=> 3 = 2 + 1
[2,3,1] => [2,1]
=> 3 = 2 + 1
[3,2,1] => [1,1,1]
=> 3 = 2 + 1
[1,2,3,4] => [4]
=> 4 = 3 + 1
[1,2,4,3] => [3,1]
=> 4 = 3 + 1
[1,3,2,4] => [3,1]
=> 4 = 3 + 1
[1,3,4,2] => [3,1]
=> 4 = 3 + 1
[1,4,3,2] => [2,1,1]
=> 4 = 3 + 1
[2,1,3,4] => [3,1]
=> 4 = 3 + 1
[2,1,4,3] => [2,2]
=> 4 = 3 + 1
[2,3,1,4] => [3,1]
=> 4 = 3 + 1
[2,3,4,1] => [3,1]
=> 4 = 3 + 1
[2,4,1,3] => [2,2]
=> 4 = 3 + 1
[2,4,3,1] => [2,1,1]
=> 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> 4 = 3 + 1
[3,2,4,1] => [2,1,1]
=> 4 = 3 + 1
[3,4,2,1] => [2,1,1]
=> 4 = 3 + 1
[4,3,2,1] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => [5]
=> 5 = 4 + 1
[1,2,3,5,4] => [4,1]
=> 5 = 4 + 1
[1,2,4,3,5] => [4,1]
=> 5 = 4 + 1
[1,2,4,5,3] => [4,1]
=> 5 = 4 + 1
[1,2,5,4,3] => [3,1,1]
=> 5 = 4 + 1
[1,3,2,4,5] => [4,1]
=> 5 = 4 + 1
[1,3,2,5,4] => [3,2]
=> 5 = 4 + 1
[1,3,4,2,5] => [4,1]
=> 5 = 4 + 1
[1,3,4,5,2] => [4,1]
=> 5 = 4 + 1
[1,3,5,2,4] => [3,2]
=> 5 = 4 + 1
[1,3,5,4,2] => [3,1,1]
=> 5 = 4 + 1
[1,4,3,2,5] => [3,1,1]
=> 5 = 4 + 1
[1,4,3,5,2] => [3,1,1]
=> 5 = 4 + 1
[1,4,5,3,2] => [3,1,1]
=> 5 = 4 + 1
[1,5,4,3,2] => [2,1,1,1]
=> 5 = 4 + 1
[2,1,3,4,5] => [4,1]
=> 5 = 4 + 1
[2,1,3,5,4] => [3,2]
=> 5 = 4 + 1
[2,1,4,3,5] => [3,2]
=> 5 = 4 + 1
[2,1,4,5,3] => [3,2]
=> 5 = 4 + 1
[2,1,5,4,3] => [2,2,1]
=> 5 = 4 + 1
[2,3,1,4,5] => [4,1]
=> 5 = 4 + 1
[2,3,1,5,4] => [3,2]
=> 5 = 4 + 1
[2,3,4,1,5] => [4,1]
=> 5 = 4 + 1
[2,3,4,5,1] => [4,1]
=> 5 = 4 + 1
[2,3,5,1,4] => [3,2]
=> 5 = 4 + 1
[2,3,5,4,1] => [3,1,1]
=> 5 = 4 + 1
[2,4,1,3,5] => [3,2]
=> 5 = 4 + 1
[2,4,1,5,3] => [3,2]
=> 5 = 4 + 1
[2,4,3,1,5] => [3,1,1]
=> 5 = 4 + 1
[2,4,3,5,1] => [3,1,1]
=> 5 = 4 + 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000081
(load all 58 compositions to match this statistic)
(load all 58 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00011: Binary trees —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,4,2,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
Description
The number of edges of a graph.
Matching statistic: St000189
(load all 840 compositions to match this statistic)
(load all 840 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => ([(0,1)],2)
=> 2
[1,3,2] => [1,2] => ([(0,1)],2)
=> 2
[2,1,3] => [2,1] => ([],2)
=> 2
[2,3,1] => [2,1] => ([],2)
=> 2
[3,2,1] => [2,1] => ([],2)
=> 2
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> 3
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 3
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 3
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 3
[2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 3
[2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 3
[2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 3
[2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 3
[2,4,3,1] => [2,3,1] => ([(1,2)],3)
=> 3
[3,2,1,4] => [3,2,1] => ([],3)
=> 3
[3,2,4,1] => [3,2,1] => ([],3)
=> 3
[3,4,2,1] => [3,2,1] => ([],3)
=> 3
[4,3,2,1] => [3,2,1] => ([],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 4
[1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 4
[1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 4
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 4
[1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 4
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 4
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 4
[1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 4
[1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 4
[1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 4
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 4
[2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 4
[2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,1,5,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,3,1,4,5] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 4
[2,3,1,5,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 4
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[2,3,5,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 4
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 4
[2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 4
[2,4,3,1,5] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 4
[2,4,3,5,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 4
Description
The number of elements in the poset.
Matching statistic: St000385
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000385: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000385: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 3
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 4
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 4
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> 4
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> 4
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
Description
The number of vertices with out-degree 1 in a binary tree.
See the references for several connections of this statistic.
In particular, the number $T(n,k)$ of binary trees with $n$ vertices and $k$ out-degree $1$ vertices is given by $T(n,k) = 0$ for $n-k$ odd and
$$T(n,k)=\frac{2^k}{n+1}\binom{n+1}{k}\binom{n+1-k}{(n-k)/2}$$
for $n-k$ is even.
Matching statistic: St000393
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3,2,1] => 00 => 2
[1,3,2] => [2,3,1] => 00 => 2
[2,1,3] => [3,1,2] => 00 => 2
[2,3,1] => [1,3,2] => 10 => 2
[3,2,1] => [1,2,3] => 11 => 2
[1,2,3,4] => [4,3,2,1] => 000 => 3
[1,2,4,3] => [3,4,2,1] => 000 => 3
[1,3,2,4] => [4,2,3,1] => 000 => 3
[1,3,4,2] => [2,4,3,1] => 000 => 3
[1,4,3,2] => [2,3,4,1] => 000 => 3
[2,1,3,4] => [4,3,1,2] => 000 => 3
[2,1,4,3] => [3,4,1,2] => 000 => 3
[2,3,1,4] => [4,1,3,2] => 000 => 3
[2,3,4,1] => [1,4,3,2] => 100 => 3
[2,4,1,3] => [3,1,4,2] => 000 => 3
[2,4,3,1] => [1,3,4,2] => 100 => 3
[3,2,1,4] => [4,1,2,3] => 000 => 3
[3,2,4,1] => [1,4,2,3] => 100 => 3
[3,4,2,1] => [1,2,4,3] => 110 => 3
[4,3,2,1] => [1,2,3,4] => 111 => 3
[1,2,3,4,5] => [5,4,3,2,1] => 0000 => 4
[1,2,3,5,4] => [4,5,3,2,1] => 0000 => 4
[1,2,4,3,5] => [5,3,4,2,1] => 0000 => 4
[1,2,4,5,3] => [3,5,4,2,1] => 0000 => 4
[1,2,5,4,3] => [3,4,5,2,1] => 0000 => 4
[1,3,2,4,5] => [5,4,2,3,1] => 0000 => 4
[1,3,2,5,4] => [4,5,2,3,1] => 0000 => 4
[1,3,4,2,5] => [5,2,4,3,1] => 0000 => 4
[1,3,4,5,2] => [2,5,4,3,1] => 0000 => 4
[1,3,5,2,4] => [4,2,5,3,1] => 0000 => 4
[1,3,5,4,2] => [2,4,5,3,1] => 0000 => 4
[1,4,3,2,5] => [5,2,3,4,1] => 0000 => 4
[1,4,3,5,2] => [2,5,3,4,1] => 0000 => 4
[1,4,5,3,2] => [2,3,5,4,1] => 0000 => 4
[1,5,4,3,2] => [2,3,4,5,1] => 0000 => 4
[2,1,3,4,5] => [5,4,3,1,2] => 0000 => 4
[2,1,3,5,4] => [4,5,3,1,2] => 0000 => 4
[2,1,4,3,5] => [5,3,4,1,2] => 0000 => 4
[2,1,4,5,3] => [3,5,4,1,2] => 0000 => 4
[2,1,5,4,3] => [3,4,5,1,2] => 0000 => 4
[2,3,1,4,5] => [5,4,1,3,2] => 0000 => 4
[2,3,1,5,4] => [4,5,1,3,2] => 0000 => 4
[2,3,4,1,5] => [5,1,4,3,2] => 0000 => 4
[2,3,4,5,1] => [1,5,4,3,2] => 1000 => 4
[2,3,5,1,4] => [4,1,5,3,2] => 0000 => 4
[2,3,5,4,1] => [1,4,5,3,2] => 1000 => 4
[2,4,1,3,5] => [5,3,1,4,2] => 0000 => 4
[2,4,1,5,3] => [3,5,1,4,2] => 0000 => 4
[2,4,3,1,5] => [5,1,3,4,2] => 0000 => 4
[2,4,3,5,1] => [1,5,3,4,2] => 1000 => 4
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000414
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000414: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000414: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 3
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 4
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 4
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> 4
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> 4
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
Description
The binary logarithm of the number of binary trees with the same underlying unordered tree.
Matching statistic: St000553
(load all 58 compositions to match this statistic)
(load all 58 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00011: Binary trees —to graph⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,4,2,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
Description
The number of blocks of a graph.
A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
Matching statistic: St000876
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3,2,1] => 00 => 2
[1,3,2] => [3,1,2] => 00 => 2
[2,1,3] => [2,3,1] => 00 => 2
[2,3,1] => [2,1,3] => 01 => 2
[3,2,1] => [1,2,3] => 11 => 2
[1,2,3,4] => [4,3,2,1] => 000 => 3
[1,2,4,3] => [4,3,1,2] => 000 => 3
[1,3,2,4] => [4,2,3,1] => 000 => 3
[1,3,4,2] => [4,2,1,3] => 000 => 3
[1,4,3,2] => [4,1,2,3] => 000 => 3
[2,1,3,4] => [3,4,2,1] => 000 => 3
[2,1,4,3] => [3,4,1,2] => 000 => 3
[2,3,1,4] => [3,2,4,1] => 000 => 3
[2,3,4,1] => [3,2,1,4] => 001 => 3
[2,4,1,3] => [3,1,4,2] => 000 => 3
[2,4,3,1] => [3,1,2,4] => 001 => 3
[3,2,1,4] => [2,3,4,1] => 000 => 3
[3,2,4,1] => [2,3,1,4] => 001 => 3
[3,4,2,1] => [2,1,3,4] => 011 => 3
[4,3,2,1] => [1,2,3,4] => 111 => 3
[1,2,3,4,5] => [5,4,3,2,1] => 0000 => 4
[1,2,3,5,4] => [5,4,3,1,2] => 0000 => 4
[1,2,4,3,5] => [5,4,2,3,1] => 0000 => 4
[1,2,4,5,3] => [5,4,2,1,3] => 0000 => 4
[1,2,5,4,3] => [5,4,1,2,3] => 0000 => 4
[1,3,2,4,5] => [5,3,4,2,1] => 0000 => 4
[1,3,2,5,4] => [5,3,4,1,2] => 0000 => 4
[1,3,4,2,5] => [5,3,2,4,1] => 0000 => 4
[1,3,4,5,2] => [5,3,2,1,4] => 0000 => 4
[1,3,5,2,4] => [5,3,1,4,2] => 0000 => 4
[1,3,5,4,2] => [5,3,1,2,4] => 0000 => 4
[1,4,3,2,5] => [5,2,3,4,1] => 0000 => 4
[1,4,3,5,2] => [5,2,3,1,4] => 0000 => 4
[1,4,5,3,2] => [5,2,1,3,4] => 0000 => 4
[1,5,4,3,2] => [5,1,2,3,4] => 0000 => 4
[2,1,3,4,5] => [4,5,3,2,1] => 0000 => 4
[2,1,3,5,4] => [4,5,3,1,2] => 0000 => 4
[2,1,4,3,5] => [4,5,2,3,1] => 0000 => 4
[2,1,4,5,3] => [4,5,2,1,3] => 0000 => 4
[2,1,5,4,3] => [4,5,1,2,3] => 0000 => 4
[2,3,1,4,5] => [4,3,5,2,1] => 0000 => 4
[2,3,1,5,4] => [4,3,5,1,2] => 0000 => 4
[2,3,4,1,5] => [4,3,2,5,1] => 0000 => 4
[2,3,4,5,1] => [4,3,2,1,5] => 0001 => 4
[2,3,5,1,4] => [4,3,1,5,2] => 0000 => 4
[2,3,5,4,1] => [4,3,1,2,5] => 0001 => 4
[2,4,1,3,5] => [4,2,5,3,1] => 0000 => 4
[2,4,1,5,3] => [4,2,5,1,3] => 0000 => 4
[2,4,3,1,5] => [4,2,3,5,1] => 0000 => 4
[2,4,3,5,1] => [4,2,3,1,5] => 0001 => 4
Description
The number of factors in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000885
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000885: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St000885: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3,2,1] => 00 => 2
[1,3,2] => [3,1,2] => 00 => 2
[2,1,3] => [2,3,1] => 00 => 2
[2,3,1] => [2,1,3] => 01 => 2
[3,2,1] => [1,2,3] => 11 => 2
[1,2,3,4] => [4,3,2,1] => 000 => 3
[1,2,4,3] => [4,3,1,2] => 000 => 3
[1,3,2,4] => [4,2,3,1] => 000 => 3
[1,3,4,2] => [4,2,1,3] => 000 => 3
[1,4,3,2] => [4,1,2,3] => 000 => 3
[2,1,3,4] => [3,4,2,1] => 000 => 3
[2,1,4,3] => [3,4,1,2] => 000 => 3
[2,3,1,4] => [3,2,4,1] => 000 => 3
[2,3,4,1] => [3,2,1,4] => 001 => 3
[2,4,1,3] => [3,1,4,2] => 000 => 3
[2,4,3,1] => [3,1,2,4] => 001 => 3
[3,2,1,4] => [2,3,4,1] => 000 => 3
[3,2,4,1] => [2,3,1,4] => 001 => 3
[3,4,2,1] => [2,1,3,4] => 011 => 3
[4,3,2,1] => [1,2,3,4] => 111 => 3
[1,2,3,4,5] => [5,4,3,2,1] => 0000 => 4
[1,2,3,5,4] => [5,4,3,1,2] => 0000 => 4
[1,2,4,3,5] => [5,4,2,3,1] => 0000 => 4
[1,2,4,5,3] => [5,4,2,1,3] => 0000 => 4
[1,2,5,4,3] => [5,4,1,2,3] => 0000 => 4
[1,3,2,4,5] => [5,3,4,2,1] => 0000 => 4
[1,3,2,5,4] => [5,3,4,1,2] => 0000 => 4
[1,3,4,2,5] => [5,3,2,4,1] => 0000 => 4
[1,3,4,5,2] => [5,3,2,1,4] => 0000 => 4
[1,3,5,2,4] => [5,3,1,4,2] => 0000 => 4
[1,3,5,4,2] => [5,3,1,2,4] => 0000 => 4
[1,4,3,2,5] => [5,2,3,4,1] => 0000 => 4
[1,4,3,5,2] => [5,2,3,1,4] => 0000 => 4
[1,4,5,3,2] => [5,2,1,3,4] => 0000 => 4
[1,5,4,3,2] => [5,1,2,3,4] => 0000 => 4
[2,1,3,4,5] => [4,5,3,2,1] => 0000 => 4
[2,1,3,5,4] => [4,5,3,1,2] => 0000 => 4
[2,1,4,3,5] => [4,5,2,3,1] => 0000 => 4
[2,1,4,5,3] => [4,5,2,1,3] => 0000 => 4
[2,1,5,4,3] => [4,5,1,2,3] => 0000 => 4
[2,3,1,4,5] => [4,3,5,2,1] => 0000 => 4
[2,3,1,5,4] => [4,3,5,1,2] => 0000 => 4
[2,3,4,1,5] => [4,3,2,5,1] => 0000 => 4
[2,3,4,5,1] => [4,3,2,1,5] => 0001 => 4
[2,3,5,1,4] => [4,3,1,5,2] => 0000 => 4
[2,3,5,4,1] => [4,3,1,2,5] => 0001 => 4
[2,4,1,3,5] => [4,2,5,3,1] => 0000 => 4
[2,4,1,5,3] => [4,2,5,1,3] => 0000 => 4
[2,4,3,1,5] => [4,2,3,5,1] => 0000 => 4
[2,4,3,5,1] => [4,2,3,1,5] => 0001 => 4
Description
The number of critical steps in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of critical steps $\ell + m$ in the Catalan factorisation.
The distribution of this statistic on words of length $n$ is
$$
(n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}.
$$
Matching statistic: St000987
(load all 94 compositions to match this statistic)
(load all 94 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00011: Binary trees —to graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,4,2,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
The following 361 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001267The length of the Lyndon factorization of the binary word. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001437The flex of a binary word. St001479The number of bridges of a graph. St001622The number of join-irreducible elements of a lattice. St000293The number of inversions of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000548The number of different non-empty partial sums of an integer partition. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. 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. St001342The number of vertices in the center of a graph. St001554The number of distinct nonempty subtrees of a binary tree. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000050The depth or height of a binary tree. St000054The first entry of the permutation. St000144The pyramid weight of the Dyck path. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St000288The number of ones in a binary word. St000296The length of the symmetric border of a binary word. St000336The leg major index of a standard tableau. St000363The number of minimal vertex covers of a graph. St000441The number of successions of a permutation. St000501The size of the first part in the decomposition of a permutation. St000503The maximal difference between two elements in a common block. St000627The exponent of a binary word. St000672The number of minimal elements in Bruhat order not less than the permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000730The maximal arc length of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000989The number of final rises of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001405The number of bonds in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001512The minimum rank of a graph. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001746The coalition number of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001884The number of borders of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000022The number of fixed points of a permutation. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000060The greater neighbor of the maximum. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000167The number of leaves of an ordered tree. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000308The height of the tree associated to a permutation. St000313The number of degree 2 vertices of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000362The size of a minimal vertex cover of a graph. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000448The number of pairs of vertices of a graph with distance 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000505The biggest entry in the block containing the 1. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. St000528The height of a poset. St000552The number of cut vertices of a graph. St000625The sum of the minimal distances to a greater element. St000636The hull number of a graph. St000653The last descent of a permutation. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000806The semiperimeter of the associated bargraph. St000837The number of ascents of distance 2 of a permutation. St000839The largest opener of a set partition. St000863The length of the first row of the shifted shape of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000873The aix statistic of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000956The maximal displacement of a permutation. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001074The number of inversions of the cyclic embedding of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001093The detour number of a graph. St001130The number of two successive successions in a permutation. St001176The size of a partition minus its first part. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001308The number of induced paths on three vertices in a graph. St001343The dimension of the reduced incidence algebra of a poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001368The number of vertices of maximal degree in a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001521Half the total irregularity of a graph. St001523The degree of symmetry of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001662The length of the longest factor of consecutive numbers in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001778The largest greatest common divisor of an element and its image in a permutation. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000070The number of antichains in a poset. St000447The number of pairs of vertices of a graph with distance 3. St000520The number of patterns in a permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001306The number of induced paths on four vertices in a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001279The sum of the parts of an integer partition that are at least two. St001759The Rajchgot index of a permutation. St001430The number of positive entries in a signed permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000010The length of the partition. St001268The size of the largest ordinal summand in the poset. St000058The order of a permutation. St000719The number of alignments in a perfect matching. St001645The pebbling number of a connected graph. St001725The harmonious chromatic number of a graph. St000890The number of nonzero entries in an alternating sign matrix. St001925The minimal number of zeros in a row of an alternating sign matrix. St000271The chromatic index of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001958The degree of the polynomial interpolating the values of a permutation. St000924The number of topologically connected components of a perfect matching. St000651The maximal size of a rise in a permutation. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000197The number of entries equal to positive one in the alternating sign matrix. St000829The Ulam distance of a permutation to the identity permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000080The rank of the poset. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000652The maximal difference between successive positions of a permutation. St000957The number of Bruhat lower covers of a permutation. St001077The prefix exchange distance of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001304The number of maximally independent sets of vertices of 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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001391The disjunction number of a graph. St001480The number of simple summands of the module J^2/J^3. St001516The number of cyclic bonds of a permutation. St001649The length of a longest trail in a graph. St001963The tree-depth of a graph. St000028The number of stack-sorts needed to sort a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000064The number of one-box pattern of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000304The load of a permutation. St000309The number of vertices with even degree. St000314The number of left-to-right-maxima of a permutation. St000327The number of cover relations in a poset. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000632The jump number of the poset. St000656The number of cuts of a poset. St000673The number of non-fixed points of a permutation. St000680The Grundy value for Hackendot on posets. St000703The number of deficiencies of a permutation. St000717The number of ordinal summands of a poset. St000733The row containing the largest entry of a standard tableau. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000906The length of the shortest maximal chain in a poset. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001118The acyclic chromatic index of a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001717The largest size of an interval in a poset. St001883The mutual visibility number of a graph. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000094The depth of an ordered tree. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000312The number of leaves in a graph. St000451The length of the longest pattern of the form k 1 2. St000456The monochromatic index of a connected graph. St000527The width of the poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001468The smallest fixpoint of a permutation. St001664The number of non-isomorphic subposets of a poset. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001782The order of rowmotion on the set of order ideals of a poset. 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. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000543The size of the conjugacy class of a binary word. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000454The largest eigenvalue of a graph if it is integral. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000147The largest part of an integer partition. St001845The number of join irreducibles minus the rank of a lattice. St000093The cardinality of a maximal independent set of vertices of a graph. St001875The number of simple modules with projective dimension at most 1. St001401The number of distinct entries in a semistandard tableau. St000135The number of lucky cars of the parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001429The number of negative entries in a signed permutation. St000095The number of triangles of a graph. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000744The length of the path to the largest entry in a standard Young tableau. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001742The difference of the maximal and the minimal degree in a graph. St001948The number of augmented double ascents of a permutation. St000029The depth of a permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000067The inversion number of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000450The number of edges minus the number of vertices plus 2 of a graph. St000809The reduced reflection length of the permutation. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001117The game chromatic index of a graph. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001557The number of inversions of the second entry of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001706The number of closed sets in a graph. St000422The energy of a graph, if it is integral. St001434The number of negative sum pairs of a signed permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001621The number of atoms of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001340The cardinality of a minimal non-edge isolating set of a graph. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number 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. St001829The common independence number of a graph. St000778The metric dimension of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001316The domatic number of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001926Sparre Andersen's position of the maximum of a signed permutation. St001673The degree of asymmetry of an integer composition. St000017The number of inversions of a standard tableau. St000035The number of left outer peaks of a permutation. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000168The number of internal nodes of an ordered tree. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000691The number of changes of a binary word. St001409The maximal entry of a semistandard tableau. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St000820The number of compositions obtained by rotating the composition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001095The number of non-isomorphic posets with precisely one further covering relation. St001115The number of even descents of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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