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Your data matches 730 different statistics following compositions of up to 3 maps.
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Matching statistic: St000846
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> ([],1)
=> 0
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[[.,[[.,.],[.,.]]],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 2
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000995
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000995: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000995: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [2,1,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [2,1,1,1,1]
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [2,1,1,1,1]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [2,1,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => [2,2,1,1]
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => [2,1,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => [2,1,1,1,1]
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => [2,1,1,1,1]
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => [2,2,1,1]
=> 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => [2,1,1,1,1]
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [2,1,1,1,1]
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [2,2,1,1]
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [2,1,1,1,1]
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [2,1,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => [2,2,1,1]
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => [2,2,1,1]
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,4,5] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [2,1,1,1,1]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [2,1,1,1,1]
=> 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => [2,1,1,1,1]
=> 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [2,2,1,1]
=> 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [2,1,1,1,1]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [2,1,1,1,1]
=> 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => [2,2,1,1]
=> 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [2,2,1,1]
=> 2
Description
The largest even part of an integer partition.
Matching statistic: St001108
(load all 78 compositions to match this statistic)
(load all 78 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00011: Binary trees —to graph⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> ([],1)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],[.,.]]],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[[.,[[.,.],[.,.]]],.],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
Description
The 2-dynamic chromatic number of a graph.
A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $2$-dynamic chromatic number of a graph.
Matching statistic: St000511
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000511: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000511: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,2]
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,1]
=> 4 = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,1]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> 4 = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [3,2]
=> 4 = 2 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [3,2]
=> 4 = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> 4 = 2 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> 4 = 2 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> 4 = 2 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [5,1]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [4,2]
=> 4 = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [4,2]
=> 4 = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [5,1]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [4,2]
=> 4 = 2 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [5,1]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => [4,2]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [3,3]
=> 4 = 2 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => [4,2]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [3,3]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => [3,3]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [3,3]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [3,3]
=> 4 = 2 + 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [4,2]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => [4,2]
=> 4 = 2 + 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [3,3]
=> 4 = 2 + 2
Description
The number of invariant subsets when acting with a permutation of given cycle type.
Matching statistic: St001814
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 4 = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 4 = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 4 = 2 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 4 = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 4 = 2 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 4 = 2 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 4 = 2 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => [2,2,1,1]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => [2,2,1,1]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [2,2,1,1]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => [2,2,1,1]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => [2,2,1,1]
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> 4 = 2 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => [2,2,1,1]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,4,5] => [2,2,1,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [2,2,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [2,2,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [2,1,1,1,1]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => [2,2,1,1]
=> 4 = 2 + 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [2,2,1,1]
=> 4 = 2 + 2
Description
The number of partitions interlacing the given partition.
Matching statistic: St000007
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [] => 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [4,1,2,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,1,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,4,1,2] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,4] => 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => [3,1,2,5,4] => 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,3] => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => [5,1,2,3,4] => 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,6,4] => [2,1,5,3,4] => 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [2,1,3,5,4] => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => [2,1,4,5,3] => 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [2,1,5,3,4] => 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,4] => 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,2] => 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => [3,1,5,2,4] => 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => [4,1,5,2,3] => 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => [4,1,5,2,3] => 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => [5,1,2,3,4] => 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => [1,5,2,3,4] => 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => [1,4,2,5,3] => 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [1,5,2,3,4] => 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => [1,2,5,3,4] => 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,4] => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,4,5,3] => 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [1,2,5,3,4] => 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => [1,3,5,2,4] => 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,3,4,5,2] => 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => [1,3,5,2,4] => 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => [1,4,5,2,3] => 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => [1,4,5,2,3] => 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [1,5,2,3,4] => 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,4] => 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => [2,3,1,5,4] => 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000259
(load all 65 compositions to match this statistic)
(load all 65 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,6,4,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000793
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> {{1}}
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition.
To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$.
Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row.
Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$.
This statistic is the length of the longest partition on the diagonal of the diagram.
Matching statistic: St000845
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> ([],1)
=> ([],1)
=> 0
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[[.,[[.,.],[.,.]]],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> 2
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St001120
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001120: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001120: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 2
Description
The length of a longest path in a graph.
The following 720 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001484The number of singletons of an integer partition. St000013The height of a Dyck path. St000548The number of different non-empty partial sums of an integer partition. St000691The number of changes of a binary word. St000759The smallest missing part in an integer partition. St000947The major index east count of a Dyck path. St001093The detour number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000973The length of the boundary of an ordered tree. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000253The crossing number of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. 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. St000668The least common multiple of the parts of the partition. St000842The breadth of a permutation. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000254The nesting number of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000264The girth of a graph, which is not a tree. St000392The length of the longest run of ones in a binary word. St000456The monochromatic index of a connected graph. St000504The cardinality of the first block of a set partition. St000570The Edelman-Greene number of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000729The minimal arc length of a set partition. St000730The maximal arc length of a set partition. St000823The number of unsplittable factors of the set partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001592The maximal number of simple paths between any two different vertices of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St000119The number of occurrences of the pattern 321 in a permutation. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000347The inversion sum of a binary word. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000379The number of Hamiltonian cycles in a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. 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. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) 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. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000629The defect of a binary word. St000658The number of rises of length 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000699The toughness times the least common multiple of 1,. St000709The number of occurrences of 14-2-3 or 14-3-2. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000732The number of double deficiencies of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. 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. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. 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. St000931The number of occurrences of the pattern UUU in a Dyck path. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001281The normalized isoperimetric number of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001381The fertility 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001513The number of nested exceedences of a permutation. 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. 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. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. 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. St001847The number of occurrences of the pattern 1432 in a permutation. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000260The radius of a connected graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000402Half the size of the symmetry class of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000542The number of left-to-right-minima of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000981The length of the longest zigzag subpath. St000983The length of the longest alternating subword. St000991The number of right-to-left minima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001371The length of the longest Yamanouchi prefix of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. 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. St000251The number of nonsingleton blocks of a set partition. St000284The Plancherel distribution on integer partitions. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000383The last part of an integer composition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000529The number of permutations whose descent word is the given binary word. St000535The rank-width of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000543The size of the conjugacy class of a binary word. St000618The number of self-evacuating tableaux of given shape. 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. St000627The exponent of a binary word. St000628The balance of a binary word. St000640The rank of the largest boolean interval in a poset. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000667The greatest common divisor of the parts of the partition. 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. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000756The sum of the positions of the left to right maxima of a permutation. 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St000877The depth of the binary word interpreted as a path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000900The minimal number of repetitions of a part in an integer composition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000905The number of different multiplicities of parts of an integer composition. St000919The number of maximal left branches of a binary tree. St000932The number of occurrences of the pattern UDU in a Dyck path. St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001389The number of partitions of the same length below the given integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001432The order dimension of the partition. St001437The flex of a binary word. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001468The smallest fixpoint of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001665The number of pure excedances of a permutation. 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. St001838The number of nonempty primitive factors of a binary word. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001884The number of borders of a binary word. St001885The number of binary words with the same proper border set. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000065The number of entries equal to -1 in an alternating sign matrix. St000095The number of triangles of a graph. 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. 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. St000221The number of strong fixed points of a permutation. St000225Difference between largest and smallest parts in a partition. St000234The number of global ascents of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000348The non-inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000369The dinv deficit of a Dyck path. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000376The bounce deficit of a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000425The number of occurrences of the pattern 132 or of the pattern 213 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. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 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. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} 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. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. 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. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. 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. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. 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. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. 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. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000768The number of peaks in an integer composition. St000779The tier of a permutation. St000792The Grundy value for the game of ruler on a binary word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000934The 2-degree 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. St000940The number of characters of the symmetric group whose value on the partition is zero. St000944The 3-degree of an integer partition. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. 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. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001301The first Betti number of the order complex associated with the poset. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a 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. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. 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. St001413Half the length of the longest even length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001434The number of negative sum pairs of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001525The number of symmetric hooks on the diagonal of a partition. St001537The number of cyclic crossings of a permutation. St001541The Gini index of an integer partition. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. 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. 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. St001587Half of the largest even part of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001727The number of invisible inversions of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000219The number of occurrences of the pattern 231 in a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000232The number of crossings of a set partition. St000455The second largest eigenvalue of a graph if it is integral. St000974The length of the trunk of an ordered tree. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001696The natural major index of a standard Young tableau. 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 $eA$. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000567The sum of the products of all pairs of parts. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St000891The number of distinct diagonal sums of a permutation matrix. St000047The number of standard immaculate tableaux of a given shape. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000766The number of inversions of an integer composition. St000914The sum of the values of the Möbius function of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001052The length of the exterior of a permutation. St000461The rix statistic of a permutation. St000662The staircase size of the code of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000717The number of ordinal summands of a poset. St000701The protection number of a binary tree. St000005The bounce statistic of a Dyck path. St000873The aix statistic of a permutation. St000670The reversal length of a permutation. St000733The row containing the largest entry of a standard tableau. St000876The number of factors in the Catalan decomposition of a binary word. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000042The number of crossings of a perfect matching. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000439The position of the first down step of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St001060The distinguishing index of a graph. St001260The permanent of an alternating sign matrix. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000889The number of alternating sign matrices with the same antidiagonal sums. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules 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. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000354The number of recoils of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000831The number of indices that are either descents or recoils. St000068The number of minimal elements in a poset. St000145The Dyson rank of a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000115The single entry in the last row. 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. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000769The major index of a composition regarded as a word. St000764The number of strong records in an integer composition. St000742The number of big ascents of a permutation after prepending zero. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St000913The number of ways to refine the partition into singletons. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000141The maximum drop size of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000763The sum of the positions of the strong records of an integer composition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000761The number of ascents in an integer composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000026The position of the first return of a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000028The number of stack-sorts needed to sort a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St001561The value of the elementary symmetric function evaluated at 1. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001545The second Elser number of a connected graph. 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 $K[x]/(x^n)$. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001114The number of odd descents of a permutation. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000091The descent variation of a composition. St001651The Frankl number of a lattice. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001890The maximum magnitude of the Möbius function of a poset. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001481The minimal height of a peak of a Dyck path. St000526The number of posets with combinatorially isomorphic order polytopes. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001061The number of indices that are both descents and recoils of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000740The last entry of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000834The number of right outer peaks of a permutation. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St000120The number of left tunnels of a Dyck path. St000311The number of vertices of odd degree in a graph. St000312The number of leaves in a graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000350The sum of the vertex degrees of a graph. St000422The energy of a graph, if it is integral. St000465The first Zagreb index of a graph. St000467The hyper-Wiener index of a connected graph. St000571The F-index (or forgotten topological index) of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000915The Ore degree of a graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001649The length of a longest trail in a graph. St000638The number of up-down runs of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001081The number of minimal length factorizations of a permutation into star transpositions. St001112The 3-weak dynamic number of a graph. St001119The length of a shortest maximal path in a graph. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001303The number of dominating sets of vertices of a graph. St001315The dissociation number of a graph. St001441The number of non-empty connected induced subgraphs of a graph. St001512The minimum rank of a graph. St001530The depth of a Dyck path. St000624The normalized sum of the minimal distances to a greater element. St001110The 3-dynamic chromatic number of a graph. St001486The number of corners of the ribbon associated with an integer composition. St001706The number of closed sets in a graph. St001762The number of convex subsets of vertices in a graph. St001834The number of non-isomorphic minors of a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000061The number of nodes on the left branch of a binary tree. St000464The Schultz index of a connected graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000045The number of linear extensions of a binary tree. St000060The greater neighbor of the maximum. St000472The sum of the ascent bottoms of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000836The number of descents of distance 2 of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001118The acyclic chromatic index of a graph. St001220The width of a permutation. St001948The number of augmented double ascents of a permutation. St000488The number of cycles of a permutation of length at most 2. St000837The number of ascents of distance 2 of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001429The number of negative entries in a signed permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001731The factorization defect of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St000117The number of centered tunnels of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St001116The game chromatic number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001618The cardinality of the Frattini sublattice of a lattice. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. 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. St000893The number of distinct diagonal sums of an alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001462The number of factors of a standard tableaux under concatenation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000272The treewidth of a graph. St000536The pathwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001792The arboricity of a graph. St001962The proper pathwidth of a graph. St001580The acyclic chromatic number of a graph. St001746The coalition number of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000647The number of big descents of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St000154The sum of the descent bottoms of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000494The number of inversions of distance at most 3 of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001488The number of corners of a skew partition. St000299The number of nonisomorphic vertex-induced subtrees. St000401The size of the symmetry class of a permutation. St000509The diagonal index (content) of a partition. St000824The sum of the number of descents and the number of recoils of a permutation. St000928The sum of the coefficients of the character polynomial of an integer partition. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001867The number of alignments of type EN of a signed permutation. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra.
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