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Your data matches 120 different statistics following compositions of up to 3 maps.
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Matching statistic: St000308
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 0 + 1
[1,2] => 2 = 1 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 3 = 2 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 2 = 1 + 1
[3,1,2] => 2 = 1 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => 3 = 2 + 1
[1,3,4,2] => 3 = 2 + 1
[1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => 3 = 2 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 2 = 1 + 1
[2,3,4,1] => 3 = 2 + 1
[2,4,1,3] => 2 = 1 + 1
[2,4,3,1] => 2 = 1 + 1
[3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => 2 = 1 + 1
[3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => 2 = 1 + 1
[3,4,2,1] => 2 = 1 + 1
[4,1,2,3] => 3 = 2 + 1
[4,1,3,2] => 2 = 1 + 1
[4,2,1,3] => 2 = 1 + 1
[4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => 2 = 1 + 1
[4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => 5 = 4 + 1
[1,2,3,5,4] => 4 = 3 + 1
[1,2,4,3,5] => 4 = 3 + 1
[1,2,4,5,3] => 4 = 3 + 1
[1,2,5,3,4] => 4 = 3 + 1
[1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => 4 = 3 + 1
[1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => 3 = 2 + 1
[1,3,4,5,2] => 4 = 3 + 1
[1,3,5,2,4] => 3 = 2 + 1
[1,3,5,4,2] => 3 = 2 + 1
[1,4,2,3,5] => 4 = 3 + 1
[1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => 3 = 2 + 1
[1,4,3,5,2] => 3 = 2 + 1
[1,4,5,2,3] => 3 = 2 + 1
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000141
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000141: 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 maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000662
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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: St000013
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [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]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 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: St000062
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [2,1] => 1 = 0 + 1
[2,1] => [[.,.],.]
=> [1,2] => 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 3 = 2 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 1 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 1 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3 = 2 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 1 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 1 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2 = 1 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 2 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 2 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2 = 1 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 2 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 1 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3 = 2 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3 = 2 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3 = 2 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 3 = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 4 = 3 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 2 = 1 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2 = 1 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2 = 1 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2 = 1 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2 = 1 + 1
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000166
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [[]]
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [[],[]]
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> [[[]]]
=> 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
=> [[[]],[]]
=> 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
=> [[[],[]]]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 3 = 2 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 2 = 1 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 3 = 2 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2 = 1 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 2 = 1 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 2 = 1 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2 = 1 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2 = 1 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 2 = 1 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 3 = 2 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 3 = 2 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 3 = 2 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 2 = 1 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> 3 = 2 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 3 = 2 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 3 = 2 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[[],[]]]]
=> 3 = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 4 = 3 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[],[],[],[[]]]
=> 2 = 1 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 2 = 1 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[],[],[[],[]]]
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 2 = 1 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 2 = 1 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 2 = 1 + 1
Description
The depth minus 1 of an ordered tree.
The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
Matching statistic: St000094
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [[]]
=> 2 = 0 + 2
[1,2] => [.,[.,.]]
=> [[],[]]
=> 2 = 0 + 2
[2,1] => [[.,.],.]
=> [[[]]]
=> 3 = 1 + 2
[1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 2 = 0 + 2
[1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 3 = 1 + 2
[2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 3 = 1 + 2
[2,3,1] => [[.,.],[.,.]]
=> [[[]],[]]
=> 3 = 1 + 2
[3,1,2] => [[.,[.,.]],.]
=> [[[],[]]]
=> 3 = 1 + 2
[3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 4 = 2 + 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 2 = 0 + 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 3 = 1 + 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 3 = 1 + 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 3 = 1 + 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 3 = 1 + 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 4 = 2 + 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3 = 1 + 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3 = 1 + 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3 = 1 + 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3 = 1 + 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3 = 1 + 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3 = 1 + 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3 = 1 + 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3 = 1 + 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4 = 2 + 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4 = 2 + 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3 = 1 + 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4 = 2 + 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 3 = 1 + 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> 4 = 2 + 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 4 = 2 + 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 4 = 2 + 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[[],[]]]]
=> 4 = 2 + 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 5 = 3 + 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 2 = 0 + 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[],[],[],[[]]]
=> 3 = 1 + 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 3 = 1 + 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 3 = 1 + 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[],[],[[],[]]]
=> 3 = 1 + 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> 4 = 2 + 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3 = 1 + 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3 = 1 + 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3 = 1 + 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3 = 1 + 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3 = 1 + 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3 = 1 + 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3 = 1 + 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3 = 1 + 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 4 = 2 + 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 4 = 2 + 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3 = 1 + 2
Description
The depth of an ordered tree.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 1
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => 1
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,4,3,2] => 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,4,3,2] => 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,1,3,2] => 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,1,4,3] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,1,3,2] => 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,1,3,2] => 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,1,4,3] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,1,4,3] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,5,2,4,3] => 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,5,2,4,3] => 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,5,2,4,3] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000080
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 0
[1,2] => [.,[.,.]]
=> [2,1] => ([],2)
=> 0
[2,1] => [[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 1
Description
The rank of the poset.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 0
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 0
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,5,4,3,1] => 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,5,4,3,1] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,5,4,2,1] => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,5,4,2] => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,5,4,2,1] => 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,5,4,2] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,5,4,1] => 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,1,5,4] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 1
Description
The number of ascents of a permutation.
The following 110 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000536The pathwidth of a graph. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000010The length of the partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000527The width of the poset. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the 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:
St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001717The largest size of an interval in a poset. St001963The tree-depth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000730The maximal arc length of a set partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001432The order dimension of the partition. St000783The side length of the largest staircase partition fitting into a partition. St000260The radius of a connected graph. St000668The least common multiple of the parts of the partition. St000259The diameter of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000444The length of the maximal rise of a Dyck path. St000939The number of characters of the symmetric group whose value on the partition is positive. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000460The hook length of the last cell along the main diagonal of an integer partition. St001330The hat guessing number of a graph. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St001645The pebbling number of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001060The distinguishing index of a graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001389The number of partitions of the same length below the given integer partition. St001933The largest multiplicity of a part in an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St000143The largest repeated part of a partition. St001877Number of indecomposable injective modules with projective dimension 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001589The nesting number of a perfect matching. St001128The exponens consonantiae of a partition. St000284The Plancherel distribution on integer partitions. St000681The Grundy value of Chomp on Ferrers diagrams. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000264The girth of a graph, which is not a tree. St001587Half of the largest even part of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001875The number of simple modules with projective dimension at most 1. St001896The number of right descents of a signed permutations. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001946The number of descents in a parking function. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice.
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