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Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St000013
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(load all 4 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000013: 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
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] => [.,[.,.]]
=> [1,0,1,0]
=> 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] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 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] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 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] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 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] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 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] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
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: St000062
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(load all 5 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 1
[1,2] => [.,[.,.]]
=> [2,1] => 1
[2,1] => [[.,.],.]
=> [1,2] => 2
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 2
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000166
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [[]]
=> 1
[1,2] => [.,[.,.]]
=> [[],[]]
=> 1
[2,1] => [[.,.],.]
=> [[[]]]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [[[]],[]]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [[[],[]]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[[],[]]]]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[],[],[],[[]]]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[],[],[[],[]]]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 2
Description
The depth minus 1 of an ordered tree.
The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
Matching statistic: St000308
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3] => [2,1,3] => [3,1,2] => 2
[2,3,1] => [3,1,2] => [2,1,3] => 2
[3,1,2] => [2,3,1] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 2
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 3
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 2
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 2
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => 2
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => 2
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[3,2,4,1] => [4,2,1,3] => [3,1,2,4] => 3
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [4,3,1,2] => [2,1,3,4] => 3
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => 2
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => 3
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => 3
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 3
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,2,5,3,4] => [1,2,4,5,3] => [3,5,4,2,1] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 2
[1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => 2
[1,3,5,2,4] => [1,4,2,5,3] => [3,5,2,4,1] => 2
[1,3,5,4,2] => [1,5,2,4,3] => [3,4,2,5,1] => 2
[1,4,2,3,5] => [1,3,4,2,5] => [5,2,4,3,1] => 2
[1,4,2,5,3] => [1,3,5,2,4] => [4,2,5,3,1] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 3
[1,4,3,5,2] => [1,5,3,2,4] => [4,2,3,5,1] => 3
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => 2
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000094
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [[]]
=> 2 = 1 + 1
[1,2] => [.,[.,.]]
=> [[],[]]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [[[]]]
=> 3 = 2 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 2 = 1 + 1
[1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 3 = 2 + 1
[2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 3 = 2 + 1
[2,3,1] => [[.,.],[.,.]]
=> [[[]],[]]
=> 3 = 2 + 1
[3,1,2] => [[.,[.,.]],.]
=> [[[],[]]]
=> 3 = 2 + 1
[3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 4 = 3 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 2 = 1 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 4 = 3 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3 = 2 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3 = 2 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3 = 2 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3 = 2 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3 = 2 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3 = 2 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3 = 2 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4 = 3 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4 = 3 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3 = 2 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4 = 3 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 3 = 2 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> 4 = 3 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 4 = 3 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 4 = 3 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[[],[]]]]
=> 4 = 3 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 5 = 4 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 3 = 2 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 3 = 2 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[],[],[[],[]]]
=> 3 = 2 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> 4 = 3 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3 = 2 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3 = 2 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3 = 2 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 4 = 3 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 4 = 3 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3 = 2 + 1
Description
The depth of an ordered tree.
Matching statistic: St000093
Mp00072: Permutations —binary search tree: left to right⟶ 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] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(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] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(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] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(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] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(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] => [.,[.,[[.,.],[.,.]]]]
=> [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
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [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] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [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
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [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] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(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: St000147
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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] => [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [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] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,1,1]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [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] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,1]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,1]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [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] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,1,1,1]
=> 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [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] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,1,1,1]
=> 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000314
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1] => 1
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 1
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 2
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 2
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,5,4,3,1] => 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,5,4,3,1] => 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,5,4,2,1] => 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,5,4,2] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,5,4,2,1] => 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,5,4,2,1] => 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,5,4,2] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,5,4,2] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,5,4,1] => 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,5,4,1] => 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 2
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000442
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000442: 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
Mp00199: Dyck paths —prime 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]
=> [1,1,0,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000451
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ 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
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2,1] => 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
Description
The length of the longest pattern of the form k 1 2...(k-1).
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000528The height of a poset. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000141The maximum drop size of a permutation. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000662The staircase size of the code of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001589The nesting number of a perfect matching. St001651The Frankl number of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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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