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Matching statistic: St000662
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Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => 1
[2,1] => [[.,.],.]
=> [1,2] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 2
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000010
Mp00061: Permutations —to increasing tree⟶ 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 = 0 + 1
[1,2] => [.,[.,.]]
=> [2,1] => [1,1]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,2] => [2]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,1]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3 = 2 + 1
[8,1,2,3,4,5,6,9,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
[7,1,2,3,4,5,8,9,6] => [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,8,7,9,6,5,4,3,2] => ?
=> ? = 6 + 1
[9,1,2,3,4,5,6,7,10,8] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 7 + 1
[7,1,2,3,4,5,6,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
[9,1,2,3,4,5,6,8,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
[10,1,2,3,4,5,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 7 + 1
[9,1,2,3,4,5,7,8,6] => [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,8,7,9,6,5,4,3,2] => ?
=> ? = 6 + 1
[8,1,2,3,4,5,6,7,10,9] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 7 + 1
[2,1,3,4,5,6,8,9,7] => [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,8,7,9,6,5,4,3,2] => ?
=> ? = 6 + 1
[2,1,3,4,5,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
[2,1,3,4,5,6,7,8,10,9] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 7 + 1
[3,1,2,4,5,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
Description
The length of the partition.
Matching statistic: St000147
Mp00061: Permutations —to increasing tree⟶ 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 = 0 + 1
[1,2] => [.,[.,.]]
=> [2,1] => [2]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,1]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,1]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2]
=> 3 = 2 + 1
[8,1,2,3,4,5,6,9,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
[7,1,2,3,4,5,8,9,6] => [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,8,7,9,6,5,4,3,2] => ?
=> ? = 6 + 1
[9,1,2,3,4,5,6,7,10,8] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 7 + 1
[7,1,2,3,4,5,6,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
[9,1,2,3,4,5,6,8,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
[10,1,2,3,4,5,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 7 + 1
[9,1,2,3,4,5,7,8,6] => [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,8,7,9,6,5,4,3,2] => ?
=> ? = 6 + 1
[8,1,2,3,4,5,6,7,10,9] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 7 + 1
[2,1,3,4,5,6,8,9,7] => [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,8,7,9,6,5,4,3,2] => ?
=> ? = 6 + 1
[2,1,3,4,5,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
[2,1,3,4,5,6,7,8,10,9] => [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => ?
=> ? = 7 + 1
[3,1,2,4,5,6,7,9,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => ?
=> ? = 6 + 1
Description
The largest part of an integer partition.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[8,5,4,6,3,7,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,8,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,5,6,4,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,5,6,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[6,5,7,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[6,5,7,8,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,7,3,4,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,8,3,4,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,7,5,3,4,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,7,4,3,5,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,5,6,3,4,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,4,3,5,2,7,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,4,5,3,6,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,3,4,5,2,6,7,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,6,5,3,4,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,5,6,3,4,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[6,5,7,3,4,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,4,3,5,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,5,4,3,6,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,4,5,3,6,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[6,4,5,3,7,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[5,4,6,3,7,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,4,5,6,2,3,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[5,4,6,7,2,3,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[6,3,4,5,2,7,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[4,3,5,6,2,7,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[5,6,3,4,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[4,5,3,6,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[6,7,8,5,2,1,3,4] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[6,7,8,4,3,1,2,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[6,7,8,4,2,1,3,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[6,7,8,3,2,1,4,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[7,8,2,3,4,5,1,6] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[6,7,2,3,4,5,1,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[4,5,2,3,6,7,1,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[3,4,2,5,6,7,1,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[4,5,6,3,2,1,7,8] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[4,3,6,5,2,8,7,1] => [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
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: St001039
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 86%●distinct values known / distinct values provided: 80%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 86%●distinct values known / distinct values provided: 80%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 0 + 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[8,5,4,6,3,7,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,5,3,2,4,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,4,3,2,5,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,5,3,4,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,4,3,5,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,6,5,4,3,2,7,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,6,4,3,5,2,7,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,6,5,3,4,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,6,4,3,5,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,5,4,3,6,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,7,6,5,4,2,1,3] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,7,6,5,3,2,1,4] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,7,6,4,3,2,1,5] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,7,5,4,3,2,1,6] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,5,4,3,2,1,7] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
[2,3,4,5,8,7,6,1] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,3,6,5,2,8,7,1] => [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[2,3,4,5,6,8,7,1] => [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[2,3,4,5,7,6,8,1] => [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 5 + 1
[2,3,4,6,5,7,8,1] => [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,6,8,7,5,4,3,2] => [.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,5,8,7,6,4,3,2] => [.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
[4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[2,5,4,3,1,8,7,6] => [[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
[2,4,3,5,1,7,8,6] => [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[2,1,3,4,5,8,7,6] => [[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 4 + 1
[4,6,5,3,2,1,8,7] => [[[[.,[[.,.],.]],.],.],[[.,.],.]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
[2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000451
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 3 = 2 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 3 = 2 + 1
[1,3,7,6,5,4,2] => [.,[[.,[[[[.,.],.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,1,7,2] => ? = 2 + 1
[1,4,6,7,5,3,2] => [.,[[[.,[[.,[.,.]],.]],.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [4,1,5,2,6,7,3] => ? = 3 + 1
[1,4,7,6,5,3,2] => [.,[[[.,[[[.,.],.],.]],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,1,6,7,2] => ? = 2 + 1
[1,5,6,7,4,3,2] => [.,[[[[.,[.,[.,.]]],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 3 + 1
[1,5,7,6,4,3,2] => [.,[[[[.,[[.,.],.]],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? = 2 + 1
[1,6,7,5,4,3,2] => [.,[[[[[.,[.,.]],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 2 + 1
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => ? = 2 + 1
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => ? = 3 + 1
[4,5,6,7,8,1,2,3] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,1,2,3,4,8,6,7] => ? = 4 + 1
[6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 4 + 1
[3,4,5,6,7,1,2,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,1,2,3,4,8,6,7] => ? = 4 + 1
[5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 4 + 1
[2,3,4,5,6,1,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,1,2,3,4,8,6,7] => ? = 4 + 1
[4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 4 + 1
[3,4,5,1,2,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 4 + 1
[2,3,4,1,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 4 + 1
[2,4,5,6,7,3,8,1] => [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ? = 4 + 1
[2,4,5,3,6,7,8,1] => [[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,7,2,4,5,6,8] => ? = 4 + 1
[1,7,8,6,5,4,3,2] => [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 2 + 1
[1,6,8,7,5,4,3,2] => [.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,8,2] => ? = 2 + 1
[1,6,7,8,5,4,3,2] => [.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 3 + 1
[1,5,8,7,6,4,3,2] => [.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,1,6,7,8,2] => ? = 2 + 1
[1,3,5,6,7,8,4,2] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [6,1,2,3,7,4,8,5] => ? = 5 + 1
[1,3,4,5,7,8,6,2] => [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [6,1,7,2,3,4,8,5] => ? = 5 + 1
[1,2,3,5,7,8,6,4] => [.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [6,1,7,2,8,3,4,5] => ? = 5 + 1
[1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,1,2,3,8,4,5,6] => ? = 6 + 1
[3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,3,4,5] => ? = 3 + 1
[1,2,3,4,5,7,8,6] => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,1,8,2,3,4,5,6] => ? = 6 + 1
[2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,5,7,8,2,4,6] => ? = 3 + 1
[3,5,4,6,2,7,1,8] => [[[.,[[.,.],[.,.]]],[.,.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => ? = 2 + 1
[3,4,2,6,5,7,1,8] => [[[.,[.,.]],[[.,.],[.,.]]],[.,.]]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,1,4,6,3,5,8,7] => ? = 2 + 1
[1,4,5,3,6,7,2,8] => [.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,6,2,4,8,5,7] => ? = 3 + 1
[1,2,5,6,4,7,3,8] => [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,8,3,5,7] => ? = 3 + 1
[2,3,4,1,6,7,5,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,2,6,4,8,5,7] => ? = 2 + 1
[1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,5,6,8,1,4,7] => ? = 3 + 1
[3,4,2,5,1,7,6,8] => [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,3,6,8,5,7] => ? = 2 + 1
[2,4,3,5,1,7,6,8] => [[.,[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,6,8,5,7] => ? = 2 + 1
[1,3,2,4,5,7,6,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,6,8,1,3,4,5,7] => ? = 5 + 1
[1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,5,8,1,2,4,6,7] => ? = 5 + 1
[2,4,6,1,3,5,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 4 + 1
[2,4,6,7,8,1,3,5] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,1,2,3,4,8,6,7] => ? = 4 + 1
[2,1,4,3,7,5,8,6] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,5,7,8,2,4,6] => ? = 3 + 1
[2,4,5,1,7,8,3,6] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,2,6,4,8,5,7] => ? = 2 + 1
[2,4,5,7,8,1,3,6] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,1,2,3,4,8,6,7] => ? = 4 + 1
[2,4,8,1,3,5,6,7] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 4 + 1
[2,4,5,1,6,8,3,7] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,2,6,4,8,5,7] => ? = 2 + 1
[2,4,5,6,8,1,3,7] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,1,2,3,4,8,6,7] => ? = 4 + 1
[2,4,5,1,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 4 + 1
[2,4,5,6,7,1,3,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,1,2,3,4,8,6,7] => ? = 4 + 1
[2,1,5,3,6,4,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,5,7,8,2,4,6] => ? = 3 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000442
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 84%●distinct values known / distinct values provided: 70%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 84%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1,0]
=> ? = 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[4,5,6,7,3,8,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[3,4,5,6,7,8,1,2] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 5
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[4,5,6,7,8,1,2,3] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[5,6,7,8,2,3,1,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[6,7,8,5,2,1,3,4] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[6,7,8,4,3,1,2,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[6,7,8,4,2,1,3,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[6,7,8,3,2,1,4,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[4,5,6,7,2,3,1,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[3,4,5,6,2,7,1,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 5
[3,4,5,6,7,1,2,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[4,5,6,7,1,2,3,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[4,5,6,3,2,1,7,8] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[2,3,4,5,6,1,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[3,4,5,6,1,2,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[2,3,4,5,1,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[3,4,5,1,2,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[2,3,4,1,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[2,3,4,6,7,8,5,1] => [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 5
[2,3,4,5,8,7,6,1] => [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 4
[2,3,4,5,7,8,6,1] => [[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 5
[2,3,4,5,6,8,7,1] => [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 5
[2,4,5,6,7,3,8,1] => [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 4
[2,3,4,5,7,6,8,1] => [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[2,3,4,6,5,7,8,1] => [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[2,4,5,3,6,7,8,1] => [[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 4
[1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,7,8,6,5,4,3,2] => [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,8,7,5,4,3,2] => [.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,7,8,5,4,3,2] => [.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,5,8,7,6,4,3,2] => [.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,3,5,7,8,6,4,2] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[1,3,6,7,5,8,4,2] => [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000097
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 1 = 0 + 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 = 4 + 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)
=> 4 = 3 + 1
[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 = 3 + 1
[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)
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[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 = 2 + 1
[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 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[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)
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,6,7,1,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,5,6,7,1,4] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,4,5,6,7,1,3] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,6,5,4,3,1,7] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,7,5,4,3,1,6] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,7,6,4,3,1,5] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,7,6,5,3,1,4] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,7,6,5,4,1,3] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,4,5,6,7,1,2] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,5,1,2,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,6,1,2,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,7,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,7,6,5,4,1,2] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[4,6,1,2,3,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[4,7,1,2,3,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[5,7,1,2,3,4,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[6,5,7,8,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[8,3,4,5,2,6,7,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[7,4,5,6,2,3,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[5,4,6,7,2,3,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[6,3,4,5,2,7,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[4,3,5,6,2,7,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[4,5,6,7,8,1,2,3] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[6,7,8,5,2,1,3,4] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[6,7,8,4,3,1,2,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[6,7,8,4,2,1,3,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[6,7,8,3,2,1,4,5] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [3,2,1,8,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[3,4,5,6,7,1,2,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [3,2,1,8,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[4,5,6,3,2,1,7,8] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 2 + 1
[2,3,4,5,6,1,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [3,2,1,8,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[3,4,5,1,2,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [3,2,1,8,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[2,3,4,1,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [3,2,1,8,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[4,3,6,5,2,8,7,1] => [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,3,4,2,6,7,5,8] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 1 + 1
[2,3,1,5,7,8,6,4] => [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,6,5,7,4,8,3] => ([(0,1),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 1 = 0 + 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 = 4 + 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)
=> 4 = 3 + 1
[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 = 3 + 1
[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)
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[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 = 2 + 1
[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 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[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)
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,4,6,7,1,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,3,5,6,7,1,4] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,4,5,6,7,1,3] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,6,5,4,3,1,7] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,7,5,4,3,1,6] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,7,6,4,3,1,5] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,7,6,5,3,1,4] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,7,6,5,4,1,3] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,4,5,6,7,1,2] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,5,1,2,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,6,1,2,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,7,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[3,7,6,5,4,1,2] => [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[4,6,1,2,3,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[4,7,1,2,3,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[5,7,1,2,3,4,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 0 + 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[8,6,7,4,5,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[8,5,6,4,7,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1 + 1
[7,5,6,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[6,5,7,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[8,5,4,6,3,7,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => ([(6,7)],8)
=> ? = 1 + 1
[8,6,7,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 1 + 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000730
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 70%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1] => {{1}}
=> ? = 0
[1,2] => [.,[.,.]]
=> [2,1] => {{1,2}}
=> 1
[2,1] => [[.,.],.]
=> [1,2] => {{1},{2}}
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => {{1},{2,3}}
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => {{1},{2,3}}
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => {{1,3,4},{2,5}}
=> 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => {{1,3,4},{2,5}}
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 2
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 2
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [4,3,2,1,5,6,7,8] => {{1,4},{2,3},{5},{6},{7},{8}}
=> ? = 3
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[8,6,7,4,5,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 1
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[8,5,6,4,7,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[7,5,6,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 1
[6,5,7,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 1
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [5,4,3,2,1,6,7,8] => {{1,5},{2,4},{3},{6},{7},{8}}
=> ? = 4
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[8,5,4,6,3,7,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [6,5,4,3,2,1,7,8] => {{1,6},{2,5},{3,4},{7},{8}}
=> ? = 5
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[8,6,7,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 1
[7,6,8,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 1
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[7,6,8,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[8,5,6,4,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[7,5,6,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[6,5,7,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[8,7,6,5,3,2,4,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[8,6,7,5,3,2,4,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 1
[7,6,8,5,3,2,4,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 1
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 2
[6,5,7,8,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 2
[8,7,6,4,3,2,5,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[7,6,8,4,3,2,5,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 1
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 1
[8,6,7,3,4,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[7,6,8,3,4,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[8,7,5,3,4,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 1
[8,7,4,3,5,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 1
[8,6,5,4,3,2,7,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[8,5,6,3,4,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[8,6,4,3,5,2,7,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 1
[8,4,5,3,6,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 1
[8,3,4,5,2,6,7,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 2
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[6,5,7,4,3,2,8,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 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 $0$.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000272The treewidth of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core 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:
St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001963The tree-depth of a graph. St000141The maximum drop size of a permutation. St000527The width of the poset. St000306The bounce count of a Dyck path. St000094The depth of an ordered tree. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000308The height of the tree associated to a permutation. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000166The depth minus 1 of an ordered tree. St000021The number of descents of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000062The length of the longest increasing subsequence of the 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. St001330The hat guessing number of a graph. St001717The largest size of an interval in a poset. St001651The Frankl number of a lattice. St000080The rank of the poset. St001589The nesting number of a perfect matching. St001875The number of simple modules with projective dimension at most 1. St001624The breadth of a lattice.
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