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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000093
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000442
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 1 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[5,4,3,8,7,6,2,1] => [[[[[.,.],.],[[[.,.],.],.]],.],.]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[1,7,8,6,5,4,3,2] => [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[1,6,8,7,5,4,3,2] => [.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[1,7,6,8,5,4,3,2] => [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[1,6,7,8,5,4,3,2] => [.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[1,5,8,7,6,4,3,2] => [.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6 - 1
[1,7,6,5,8,4,3,2] => [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[1,6,7,5,8,4,3,2] => [.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[1,5,7,6,8,4,3,2] => [.,[[[[.,[[.,.],[.,.]]],.],.],.]]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 5 - 1
[1,5,6,7,8,4,3,2] => [.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,4,8,7,6,5,3,2] => [.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 - 1
[1,4,7,8,6,5,3,2] => [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 5 - 1
[1,4,6,7,8,5,3,2] => [.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4 - 1
[1,4,5,8,7,6,3,2] => [.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5 - 1
[1,7,6,5,4,8,3,2] => [.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[1,4,7,6,5,8,3,2] => [.,[[[.,[[[.,.],.],[.,.]]],.],.]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 5 - 1
[1,6,5,4,7,8,3,2] => [.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[1,5,4,6,7,8,3,2] => [.,[[[[.,.],[.,[.,[.,.]]]],.],.]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,4,5,6,7,8,3,2] => [.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 3 - 1
[1,3,8,7,6,5,4,2] => [.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 6 - 1
[1,3,6,8,7,5,4,2] => [.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[1,3,6,7,8,5,4,2] => [.,[[.,[[[.,[.,[.,.]]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,3,5,8,7,6,4,2] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5 - 1
[1,3,5,7,8,6,4,2] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,3,6,7,5,8,4,2] => [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,3,6,5,7,8,4,2] => [.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,3,5,6,7,8,4,2] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[1,3,4,8,7,6,5,2] => [.,[[.,[.,[[[[.,.],.],.],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[1,3,4,7,6,8,5,2] => [.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 4 - 1
[1,3,4,5,7,8,6,2] => [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,4,3,6,5,8,7,2] => [.,[[[.,.],[[.,.],[[.,.],.]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,3,4,5,6,8,7,2] => [.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,7,6,5,4,3,8,2] => [.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[1,6,7,5,4,3,8,2] => [.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[1,4,6,7,5,3,8,2] => [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4 - 1
[1,5,6,4,7,3,8,2] => [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,5,4,6,7,3,8,2] => [.,[[[[.,.],[.,[.,.]]],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,4,5,6,7,3,8,2] => [.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 3 - 1
[1,3,6,5,7,4,8,2] => [.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,3,5,6,7,4,8,2] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[1,3,4,5,7,6,8,2] => [.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,5,4,6,3,7,8,2] => [.,[[[[.,.],[.,.]],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,3,4,6,5,7,8,2] => [.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,5,4,3,6,7,8,2] => [.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,4,5,3,6,7,8,2] => [.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 3 - 1
[1,4,3,5,6,7,8,2] => [.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,3,4,5,6,7,8,2] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,2,8,7,6,5,4,3] => [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000147
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1]
=> 1
[1,2] => [.,[.,.]]
=> [2,1] => [1,1]
=> 1
[2,1] => [[.,.],.]
=> [1,2] => [2]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3]
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,1,1]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,1]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,1,1]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,1]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,1]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,1]
=> 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,1]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1]
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4]
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,1,1,1]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,1,1]
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,1,1,1]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,1,1,1]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,1,1]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2
[7,8,6,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 5
[8,7,3,4,5,6,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[8,5,3,4,6,7,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[8,4,3,5,6,7,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[7,6,3,4,5,8,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[7,5,3,4,6,8,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[6,4,3,5,7,8,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[5,4,3,6,7,8,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[6,7,5,4,8,2,1,3] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 5
[5,6,7,8,4,1,2,3] => [[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [8,7,4,3,2,1,5,6] => ?
=> ? = 3
[8,7,4,5,6,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[8,6,4,5,7,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[7,6,4,5,8,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[7,5,4,6,8,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[6,5,4,7,8,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[5,6,7,8,3,1,2,4] => [[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [8,7,4,3,2,1,5,6] => ?
=> ? = 3
[5,6,7,8,2,1,3,4] => [[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [8,7,4,3,2,1,5,6] => ?
=> ? = 3
[7,8,6,3,4,2,1,5] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 5
[8,7,2,3,4,5,1,6] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[8,7,3,4,5,1,2,6] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[8,7,2,3,4,1,5,6] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[8,5,2,3,4,6,1,7] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[8,3,2,4,5,6,1,7] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[8,5,3,4,6,1,2,7] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[8,4,3,5,6,1,2,7] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[8,6,2,3,4,1,5,7] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[8,5,2,3,4,1,6,7] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[8,4,2,3,5,1,6,7] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[6,7,5,3,4,2,1,8] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 5
[6,7,4,3,5,2,1,8] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 5
[5,6,4,3,7,2,1,8] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 5
[7,6,2,3,4,5,1,8] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[7,4,2,3,5,6,1,8] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[6,4,2,3,5,7,1,8] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[4,3,2,5,6,7,1,8] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => ?
=> ? = 4
[4,5,6,7,3,1,2,8] => [[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [8,7,4,3,2,1,5,6] => ?
=> ? = 3
[4,5,6,7,2,1,3,8] => [[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [8,7,4,3,2,1,5,6] => ?
=> ? = 3
[7,6,2,3,4,1,5,8] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[7,5,2,3,4,1,6,8] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[7,4,2,3,5,1,6,8] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[3,4,5,6,2,1,7,8] => [[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [8,7,4,3,2,1,5,6] => ?
=> ? = 3
[6,5,2,3,4,1,7,8] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[4,3,2,5,6,1,7,8] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => ?
=> ? = 4
[1,6,7,5,8,4,3,2] => [.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> [5,3,2,4,6,7,8,1] => ?
=> ? = 5
[1,5,7,6,8,4,3,2] => [.,[[[[.,[[.,.],[.,.]]],.],.],.]]
=> [5,3,4,2,6,7,8,1] => ?
=> ? = 5
[1,4,7,8,6,5,3,2] => [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
=> [4,3,5,6,2,7,8,1] => ?
=> ? = 5
[1,4,7,6,5,8,3,2] => [.,[[[.,[[[.,.],.],[.,.]]],.],.]]
=> [6,3,4,5,2,7,8,1] => ?
=> ? = 5
[1,5,4,6,7,8,3,2] => [.,[[[[.,.],[.,[.,[.,.]]]],.],.]]
=> [6,5,4,2,3,7,8,1] => ?
=> ? = 4
[1,3,6,7,8,5,4,2] => [.,[[.,[[[.,[.,[.,.]]],.],.]],.]]
=> [5,4,3,6,7,2,8,1] => ?
=> ? = 4
[1,4,5,6,7,3,8,2] => [.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [7,5,4,3,2,6,8,1] => ?
=> ? = 3
Description
The largest part of an integer partition.
Matching statistic: St001039
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[7,8,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 6
[6,7,8,5,4,3,1,2] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 5
[7,8,5,6,4,3,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 5
[6,7,5,8,4,3,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 5
[5,6,7,8,4,3,1,2] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 4
[7,8,6,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 5
[6,7,8,4,5,3,1,2] => [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 4
[7,8,4,5,6,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 4
[6,7,4,5,8,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 4
[5,6,4,7,8,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 4
[4,5,6,7,8,3,1,2] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 3
[6,7,8,5,3,4,1,2] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[6,7,5,8,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 3
[6,7,8,4,3,5,1,2] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[7,8,6,3,4,5,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[7,8,4,5,3,6,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[7,8,5,3,4,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 3
[5,6,7,4,3,8,1,2] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[6,7,4,5,3,8,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[5,6,4,7,3,8,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 4
[4,5,6,7,3,8,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 3
[5,6,4,3,7,8,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[4,5,6,3,7,8,1,2] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 3
[5,6,3,4,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 3
[4,5,3,6,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 3
[3,4,5,6,7,8,1,2] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2
[7,8,6,5,4,2,1,3] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 6
[6,7,8,5,4,2,1,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 5
[7,8,5,6,4,2,1,3] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 5
[6,7,5,8,4,2,1,3] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 5
[5,6,7,8,4,2,1,3] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 4
[6,7,5,4,8,2,1,3] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 5
[5,6,7,4,8,2,1,3] => [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 4
[4,5,6,7,8,2,1,3] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 3
[7,8,6,5,4,1,2,3] => [[[[[.,[.,.]],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 5
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[7,8,5,6,4,1,2,3] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[6,7,5,8,4,1,2,3] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[5,6,7,8,4,1,2,3] => [[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 3
[6,7,8,4,5,1,2,3] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
[7,8,5,4,6,1,2,3] => [[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4
[6,7,5,4,8,1,2,3] => [[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4
[5,6,7,4,8,1,2,3] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
[4,5,6,7,8,1,2,3] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[7,8,6,5,3,2,1,4] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 6
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000786
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,3,8,7,6,2,1] => [[[[[.,.],.],[[[.,.],.],.]],.],.]
=> [4,5,6,1,2,3,7,8] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 5
[1,4,3,6,5,8,7,2] => [.,[[[.,.],[[.,.],[[.,.],.]]],.]]
=> [6,7,4,5,2,3,8,1] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,2,4,6,5,8,7,3] => [.,[.,[[.,[[.,.],[[.,.],.]]],.]]]
=> [6,7,4,5,3,8,2,1] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,3,6,5,8,7,4] => [.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [6,7,4,5,8,3,2,1] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 4
[3,4,2,1,7,8,6,5] => [[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,2,1,3,4] => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,4,3,1,6,8,7,5] => [[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [6,7,5,8,2,3,1,4] => ([(0,1),(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,4,3,1,7,6,8,5] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [7,5,6,8,2,3,1,4] => ([(0,1),(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[3,2,4,1,6,8,7,5] => [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [6,7,5,8,3,1,2,4] => ([(0,1),(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,3,4,1,6,7,8,5] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [7,6,5,8,3,2,1,4] => ([(0,1),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,4,3,8,7,6,5] => [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [5,6,7,8,3,4,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,2,4,3,7,8,6,5] => [.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,3,4,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,4,3,6,7,8,5] => [.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,3,4,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,3,2,4,8,7,6,5] => [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [5,6,7,8,4,2,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,3,2,4,7,6,8,5] => [.,[[.,.],[.,[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,4,2,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,3,2,4,6,7,8,5] => [.,[[.,.],[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,2,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[2,1,3,4,8,7,6,5] => [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> [5,6,7,8,4,3,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[2,1,3,4,7,8,6,5] => [[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,4,3,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,3,4,7,6,8,5] => [[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,4,3,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,3,4,6,7,8,5] => [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,3,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,5,4,3,8,7,6] => [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [6,7,8,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,5,4,3,7,8,6] => [.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
=> [7,6,8,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,4,5,3,8,7,6] => [.,[.,[[.,[.,.]],[[[.,.],.],.]]]]
=> [6,7,8,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,4,5,3,7,8,6] => [.,[.,[[.,[.,.]],[[.,[.,.]],.]]]]
=> [7,6,8,4,3,5,2,1] => ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,3,5,4,7,8,6] => [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [7,6,8,4,5,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,4,3,2,5,8,7,6] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,4,3,2,5,7,8,6] => [.,[[[.,.],.],[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,3,4,2,5,7,8,6] => [.,[[.,[.,.]],[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,4,3,5,7,8,6] => [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [6,7,8,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[3,2,1,4,5,7,8,6] => [[[.,.],.],[.,[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,3,1,4,5,7,8,6] => [[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,4,2,1,3] => ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,3,2,4,5,7,8,6] => [.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,4,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[2,1,3,4,5,8,7,6] => [[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,3,1,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,3,4,5,7,8,6] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,4,3,1,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,4,5,6,3,8,7] => [.,[.,[[.,[.,[.,.]]],[[.,.],.]]]]
=> [7,8,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,2,4,3,6,5,8,7] => [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[2,1,3,4,6,5,8,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,3,4,6,5,8,7] => [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,4,5,3,2,6,8,7] => [.,[[[.,[.,.]],.],[.,[[.,.],.]]]]
=> [7,8,6,3,2,4,5,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,3,5,4,2,6,8,7] => [.,[[.,[[.,.],.]],[.,[[.,.],.]]]]
=> [7,8,6,3,4,2,5,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,5,4,3,6,8,7] => [.,[.,[[[.,.],.],[.,[[.,.],.]]]]]
=> [7,8,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,3,5,4,6,8,7] => [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[4,3,2,1,5,6,8,7] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [7,8,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[3,2,4,1,5,6,8,7] => [[[.,.],[.,.]],[.,[.,[[.,.],.]]]]
=> [7,8,6,5,3,1,2,4] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,3,4,1,5,6,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [7,8,6,5,3,2,1,4] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,3,4,2,5,6,8,7] => [.,[[.,[.,.]],[.,[.,[[.,.],.]]]]]
=> [7,8,6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[2,1,4,3,5,6,8,7] => [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> [7,8,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,4,3,5,6,8,7] => [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[7,8,5,6,4,3,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 5
[6,7,5,8,4,3,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 5
[5,6,7,8,4,3,1,2] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 4
[7,8,6,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 5
[8,6,7,4,5,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[6,7,8,4,5,3,1,2] => [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4
[7,8,4,5,6,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 4
[8,5,6,4,7,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[8,6,4,5,7,3,1,2] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[8,5,4,6,7,3,1,2] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[7,5,6,4,8,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[6,5,7,4,8,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[7,6,4,5,8,3,1,2] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[6,7,4,5,8,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 4
[6,5,4,7,8,3,1,2] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[5,6,4,7,8,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 4
[6,7,8,5,3,4,1,2] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 4
[8,7,5,6,3,4,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[8,6,5,7,3,4,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[8,5,6,7,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[7,6,5,8,3,4,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[7,5,6,8,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[6,5,7,8,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[6,7,8,4,3,5,1,2] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 4
[7,8,6,3,4,5,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 4
[8,7,4,5,3,6,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
[7,8,5,3,4,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 4
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 4
[5,6,7,4,3,8,1,2] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 4
[6,4,5,7,3,8,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[4,5,6,7,3,8,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[5,6,4,3,7,8,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 4
[7,8,5,6,4,2,1,3] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 5
[6,7,5,8,4,2,1,3] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 5
[5,6,7,8,4,2,1,3] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 4
[7,6,8,4,5,2,1,3] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[6,7,5,4,8,2,1,3] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 5
[6,5,7,4,8,2,1,3] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5
[5,6,7,4,8,2,1,3] => [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4
[7,6,4,5,8,2,1,3] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[7,5,4,6,8,2,1,3] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[7,8,6,5,4,1,2,3] => [[[[[.,[.,.]],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 4
[7,8,5,6,4,1,2,3] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 4
[6,7,5,8,4,1,2,3] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 4
[8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[7,8,5,4,6,1,2,3] => [[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 4
[8,5,6,4,7,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[6,7,5,4,8,1,2,3] => [[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 4
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St001203
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[7,8,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6
[8,6,7,5,4,3,1,2] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6
[7,6,8,5,4,3,1,2] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6
[6,7,8,5,4,3,1,2] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[8,7,5,6,4,3,1,2] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 6
[7,8,5,6,4,3,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 5
[8,6,5,7,4,3,1,2] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 6
[7,6,5,8,4,3,1,2] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 6
[6,7,5,8,4,3,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 5
[5,6,7,8,4,3,1,2] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 4
[8,7,6,4,5,3,1,2] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 6
[7,8,6,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 5
[8,6,7,4,5,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 5
[6,7,8,4,5,3,1,2] => [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 4
[8,7,5,4,6,3,1,2] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 6
[7,8,4,5,6,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 4
[8,5,6,4,7,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 5
[8,6,4,5,7,3,1,2] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 5
[8,5,4,6,7,3,1,2] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 5
[7,6,5,4,8,3,1,2] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 6
[7,5,6,4,8,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 5
[6,5,7,4,8,3,1,2] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 5
[7,6,4,5,8,3,1,2] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 5
[6,7,4,5,8,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 4
[6,5,4,7,8,3,1,2] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 5
[5,6,4,7,8,3,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 4
[4,5,6,7,8,3,1,2] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 3
[8,7,6,5,3,4,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 6
[6,7,8,5,3,4,1,2] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 4
[8,7,5,6,3,4,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 5
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4
[8,6,5,7,3,4,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 5
[8,5,6,7,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 4
[7,6,5,8,3,4,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 5
[6,7,5,8,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4
[7,5,6,8,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 4
[6,5,7,8,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 4
[5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 3
[6,7,8,4,3,5,1,2] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 4
[8,7,6,3,4,5,1,2] => [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 5
[7,8,6,3,4,5,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 4
[8,7,4,5,3,6,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 5
[7,8,4,5,3,6,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4
[7,8,5,3,4,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 4
[8,7,4,3,5,6,1,2] => [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 5
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 4
[8,7,3,4,5,6,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 4
[7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> ? = 3
[8,6,5,4,3,7,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 6
Description
We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra.
Example:
[5,6,6,6,6] goes into [2,2,2,2,1].
Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra.
The statistic return the global dimension of the CNakayama algebra divided by 2.
Matching statistic: St000306
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,6,3,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,7,3,6] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,5,3,7] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,4,6,5,7,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,2,4,6,7,3,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,7,5,3] => [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,7,5,3,6] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,4,7,5,6,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,2,4,7,6,3,5] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,4,7,6,5,3] => [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,3,6,7,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,3,7,6,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,5,4,6,3,7] => [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,5,4,6,7,3] => [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,2,5,4,7,3,6] => [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,5,4,7,6,3] => [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,4,7,3] => [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,2,5,6,7,3,4] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,7,4,3] => [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,7,4,6,3] => [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,2,5,7,6,3,4] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,5,7,6,4,3] => [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,2,6,3,4,7,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,5,7,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,7,5,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,6,4,5,3,7] => [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,2,6,4,5,7,3] => [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000662
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1] => 0 = 1 - 1
[1,2] => [.,[.,.]]
=> [2,1] => [1,2] => 0 = 1 - 1
[2,1] => [[.,.],.]
=> [1,2] => [2,1] => 1 = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 1 = 2 - 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,3,1] => 1 = 2 - 1
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,4,1] => 2 = 3 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,4,3,1] => 2 = 3 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,4,3,1] => 2 = 3 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,3,4,5] => 1 = 2 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,3,2,4,5] => 1 = 2 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,3,1,4,5] => 1 = 2 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,3,2,4,5] => 1 = 2 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,1,4,5] => 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,4,3,5] => 1 = 2 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,1,4,3,5] => 1 = 2 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,3,4,2,5] => 1 = 2 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,3,4,1,5] => 1 = 2 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,3,4,2,5] => 1 = 2 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2,4,1,5] => 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,1,4,3,5] => 1 = 2 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,1,5] => 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,3,4,2,5] => 1 = 2 - 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 2 - 1
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 3 - 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 2 - 1
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 3 - 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 3 - 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 2 - 1
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 3 - 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => ? = 2 - 1
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 2 - 1
[1,2,4,5,7,6,3] => [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [3,2,4,5,1,6,7] => ? = 3 - 1
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 2 - 1
[1,2,4,6,5,7,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [6,4,5,3,7,2,1] => [2,4,3,5,1,6,7] => ? = 3 - 1
[1,2,4,6,7,5,3] => [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => ? = 3 - 1
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 2 - 1
[1,2,4,7,5,6,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [6,4,5,3,7,2,1] => [2,4,3,5,1,6,7] => ? = 3 - 1
[1,2,4,7,6,5,3] => [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [4,3,2,5,1,6,7] => ? = 4 - 1
[1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,5,3,6,7,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => ? = 2 - 1
[1,2,5,3,7,6,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,4,6,7,3] => [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 3 - 1
[1,2,5,4,7,6,3] => [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => ? = 3 - 1
[1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 2 - 1
[1,2,5,6,4,7,3] => [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => ? = 3 - 1
[1,2,5,6,7,4,3] => [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [3,4,5,2,1,6,7] => ? = 3 - 1
[1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 2 - 1
[1,2,5,7,4,6,3] => [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => ? = 3 - 1
[1,2,5,7,6,4,3] => [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [4,3,5,2,1,6,7] => ? = 4 - 1
[1,2,6,3,4,7,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,6,3,5,7,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => ? = 2 - 1
[1,2,6,3,7,5,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,6,4,3,7,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,6,4,5,7,3] => [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 3 - 1
[1,2,6,4,7,5,3] => [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => ? = 3 - 1
[1,2,6,5,3,7,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,6,5,4,3,7] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,6,5,4,7,3] => [.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [2,5,4,3,1,6,7] => ? = 4 - 1
[1,2,6,5,7,4,3] => [.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [3,5,4,2,1,6,7] => ? = 4 - 1
[1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 2 - 1
[1,2,6,7,4,5,3] => [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => ? = 3 - 1
[1,2,6,7,5,4,3] => [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => ? = 4 - 1
[1,2,7,3,4,6,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,7,3,5,6,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => ? = 2 - 1
[1,2,7,3,6,5,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,7,4,3,6,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,7,4,5,6,3] => [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 3 - 1
[1,2,7,4,6,5,3] => [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => ? = 3 - 1
[1,2,7,5,3,6,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 3 - 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001337
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,5,6,3,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,5,7,3,6] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,6,7,3,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,3,6,4,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,3,6,7,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,3,7,4,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,6,7,3,4] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,6,3,4,5,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,6,3,4,7,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,6,3,5,4,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,6,3,5,7,4] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,6,3,7,4,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,7,3,4,5,6] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,7,3,4,6,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,7,3,5,4,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001338The upper irredundance number of a graph. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St000094The depth of an ordered tree. St000451The length of the longest pattern of the form k 1 2. St001717The largest size of an interval in a poset. St000245The number of ascents of a permutation. St000141The maximum drop size of a permutation. St000308The height of the tree associated to a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001330The hat guessing number of a graph. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000314The number of left-to-right-maxima of a permutation. St000080The rank of the poset. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001651The Frankl number of a lattice. St001589The nesting number of a perfect matching. St001875The number of simple modules with projective dimension at most 1. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001624The breadth of a lattice.
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