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Your data matches 70 different statistics following compositions of up to 3 maps.
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Matching statistic: St000013
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Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
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: St000010
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1]
=> 1
[1,2] => [.,[.,.]]
=> [2,1] => [1,1]
=> 2
[2,1] => [[.,.],.]
=> [1,2] => [2]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,1]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,1]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1]
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1]
=> 4
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,1,1]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1]
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1]
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000147
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1]
=> 1
[1,2] => [.,[.,.]]
=> [2,1] => [2]
=> 2
[2,1] => [[.,.],.]
=> [1,2] => [1,1]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1]
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1]
=> 4
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,1]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,1]
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,1]
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St001203
(load all 2 compositions to match this statistic)
(load all 2 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
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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
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: St001494
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001494: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001494: 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)
=> 2
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 1
[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)
=> 5
[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)
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[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] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
Description
The Alon-Tarsi number of a graph.
Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St000442
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> ? = 1 - 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001039
(load all 2 compositions to match this statistic)
(load all 2 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
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000730
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => {{1}}
=> ? = 1 - 1
[1,2] => [.,[.,.]]
=> [2,1] => {{1,2}}
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => {{1},{2}}
=> 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> 2 = 3 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => {{1},{2,3}}
=> 1 = 2 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => {{1},{2,3}}
=> 1 = 2 - 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 3 = 4 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1 = 2 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1 = 2 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1 = 2 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1 = 2 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1 = 2 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1 = 2 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1 = 2 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 2 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 2 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 4 = 5 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> 3 = 4 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => {{1,3,4},{2,5}}
=> 3 = 4 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => {{1,3,4},{2,5}}
=> 3 = 4 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> 3 = 4 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3 = 4 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3 = 4 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3 = 4 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2 = 3 - 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 2 = 3 - 1
Description
The maximal arc length of a set partition.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
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: 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
St000786: 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
[2,1,5,6,7,4,3] => [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,5,6,7,4,2] => [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1,7,6] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,6,2,1,7,5] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,7,2,1,6,5] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,6,2,1,7,4] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,7,2,1,6,4] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,6,7,2,1,5,4] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,5,6,7,3,2] => [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,5,6,2,1,7,3] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,5,6,3,1,7,2] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,5,7,2,1,6,3] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,5,7,3,1,6,2] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,6,7,2,1,5,3] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,6,7,3,1,5,2] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,1,4,6,7,3,2] => [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,6,7,2,1,4,3] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,6,7,3,1,4,2] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,6,7,4,1,3,2] => [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[6,1,4,5,7,3,2] => [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[7,1,4,5,6,3,2] => [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
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: St001963
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001963: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001963: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 2
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 1
[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)
=> 5
[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)
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[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] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,2,7,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,3,4,2,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,3,4,5,2,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,3,4,5,6,2,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,3,4,5,6,7,2] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[2,3,1,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[2,3,4,1,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[2,3,4,5,1,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[2,3,4,5,6,1,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[2,3,4,5,6,7,1] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
[7,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => ([(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)
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7}
Description
The tree-depth of a graph.
The tree-depth $\operatorname{td}(G)$ of a graph $G$ whose connected components are $G_1,\ldots,G_p$ is recursively defined as
$$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$$
Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once).
Tree-depth is strictly greater than [[St000536|pathwidth]]. A [[St001120|longest path]] in $G$ has at least $\operatorname{td}(G)$ vertices [3].
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000527The width of the poset. St000260The radius of a connected graph. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St001060The distinguishing index of a graph. St000451The length of the longest pattern of the form k 1 2. St000264The girth of a graph, which is not a tree. St000306The bounce count of a Dyck path. St000662The staircase size of the code of a permutation. St000259The diameter of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000245The number of ascents of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000308The height of the tree associated to a permutation. St000454The largest eigenvalue of a graph if it is integral. St000141The maximum drop size of a permutation. St000470The number of runs in a permutation. St001330The hat guessing number of a graph. 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. St000542The number of left-to-right-minima of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000080The rank of the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000143The largest repeated part of a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001589The nesting number of a perfect matching. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001896The number of right descents of a signed permutations. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001946The number of descents in a parking function. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice.
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