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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St000542
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Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => 2
[-,+,+] => [2,3,1] => 2
[-,-,+] => [3,1,2] => 2
[-,+,-] => [2,1,3] => 2
[-,3,2] => [2,1,3] => 2
[3,+,1] => [2,1,3] => 2
[-,+,+,+] => [2,3,4,1] => 2
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => 2
[-,-,-,+] => [4,1,2,3] => 2
[-,-,+,-] => [3,1,2,4] => 2
[-,+,-,-] => [2,1,3,4] => 2
[-,+,4,3] => [2,3,1,4] => 2
[-,-,4,3] => [3,1,2,4] => 2
[-,3,2,+] => [2,4,1,3] => 2
[-,3,2,-] => [2,1,3,4] => 2
[-,3,4,2] => [2,1,3,4] => 2
[-,4,2,3] => [2,3,1,4] => 2
[-,4,+,2] => [3,2,1,4] => 3
[-,4,-,2] => [2,1,4,3] => 2
[2,4,+,1] => [3,1,2,4] => 2
[3,+,1,+] => [2,1,4,3] => 2
[3,+,1,-] => [2,1,3,4] => 2
[3,+,4,1] => [2,1,3,4] => 2
[3,4,2,1] => [2,1,3,4] => 2
[4,+,1,3] => [2,1,3,4] => 2
[4,+,+,1] => [2,3,1,4] => 2
[4,-,+,1] => [3,1,4,2] => 2
[4,+,-,1] => [2,1,4,3] => 2
[4,3,2,1] => [2,1,4,3] => 2
[-,+,+,+,+] => [2,3,4,5,1] => 2
[-,-,+,+,+] => [3,4,5,1,2] => 2
[-,+,-,+,+] => [2,4,5,1,3] => 2
[-,+,+,-,+] => [2,3,5,1,4] => 2
[-,+,+,+,-] => [2,3,4,1,5] => 2
[-,-,-,+,+] => [4,5,1,2,3] => 2
[-,-,+,-,+] => [3,5,1,2,4] => 2
[-,-,+,+,-] => [3,4,1,2,5] => 2
[-,+,-,-,+] => [2,5,1,3,4] => 2
[-,+,-,+,-] => [2,4,1,3,5] => 2
[-,+,+,-,-] => [2,3,1,4,5] => 2
[-,-,-,-,+] => [5,1,2,3,4] => 2
[-,-,-,+,-] => [4,1,2,3,5] => 2
[-,-,+,-,-] => [3,1,2,4,5] => 2
[-,+,-,-,-] => [2,1,3,4,5] => 2
[-,+,+,5,4] => [2,3,4,1,5] => 2
[-,-,+,5,4] => [3,4,1,2,5] => 2
[-,+,-,5,4] => [2,4,1,3,5] => 2
[-,-,-,5,4] => [4,1,2,3,5] => 2
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000541
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(load all 4 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => 1 = 2 - 1
[-,+,+] => [2,3,1] => 1 = 2 - 1
[-,-,+] => [3,1,2] => 1 = 2 - 1
[-,+,-] => [2,1,3] => 1 = 2 - 1
[-,3,2] => [2,1,3] => 1 = 2 - 1
[3,+,1] => [2,1,3] => 1 = 2 - 1
[-,+,+,+] => [2,3,4,1] => 1 = 2 - 1
[-,-,+,+] => [3,4,1,2] => 1 = 2 - 1
[-,+,-,+] => [2,4,1,3] => 1 = 2 - 1
[-,+,+,-] => [2,3,1,4] => 1 = 2 - 1
[-,-,-,+] => [4,1,2,3] => 1 = 2 - 1
[-,-,+,-] => [3,1,2,4] => 1 = 2 - 1
[-,+,-,-] => [2,1,3,4] => 1 = 2 - 1
[-,+,4,3] => [2,3,1,4] => 1 = 2 - 1
[-,-,4,3] => [3,1,2,4] => 1 = 2 - 1
[-,3,2,+] => [2,4,1,3] => 1 = 2 - 1
[-,3,2,-] => [2,1,3,4] => 1 = 2 - 1
[-,3,4,2] => [2,1,3,4] => 1 = 2 - 1
[-,4,2,3] => [2,3,1,4] => 1 = 2 - 1
[-,4,+,2] => [3,2,1,4] => 2 = 3 - 1
[-,4,-,2] => [2,1,4,3] => 1 = 2 - 1
[2,4,+,1] => [3,1,2,4] => 1 = 2 - 1
[3,+,1,+] => [2,1,4,3] => 1 = 2 - 1
[3,+,1,-] => [2,1,3,4] => 1 = 2 - 1
[3,+,4,1] => [2,1,3,4] => 1 = 2 - 1
[3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[4,+,1,3] => [2,1,3,4] => 1 = 2 - 1
[4,+,+,1] => [2,3,1,4] => 1 = 2 - 1
[4,-,+,1] => [3,1,4,2] => 1 = 2 - 1
[4,+,-,1] => [2,1,4,3] => 1 = 2 - 1
[4,3,2,1] => [2,1,4,3] => 1 = 2 - 1
[-,+,+,+,+] => [2,3,4,5,1] => 1 = 2 - 1
[-,-,+,+,+] => [3,4,5,1,2] => 1 = 2 - 1
[-,+,-,+,+] => [2,4,5,1,3] => 1 = 2 - 1
[-,+,+,-,+] => [2,3,5,1,4] => 1 = 2 - 1
[-,+,+,+,-] => [2,3,4,1,5] => 1 = 2 - 1
[-,-,-,+,+] => [4,5,1,2,3] => 1 = 2 - 1
[-,-,+,-,+] => [3,5,1,2,4] => 1 = 2 - 1
[-,-,+,+,-] => [3,4,1,2,5] => 1 = 2 - 1
[-,+,-,-,+] => [2,5,1,3,4] => 1 = 2 - 1
[-,+,-,+,-] => [2,4,1,3,5] => 1 = 2 - 1
[-,+,+,-,-] => [2,3,1,4,5] => 1 = 2 - 1
[-,-,-,-,+] => [5,1,2,3,4] => 1 = 2 - 1
[-,-,-,+,-] => [4,1,2,3,5] => 1 = 2 - 1
[-,-,+,-,-] => [3,1,2,4,5] => 1 = 2 - 1
[-,+,-,-,-] => [2,1,3,4,5] => 1 = 2 - 1
[-,+,+,5,4] => [2,3,4,1,5] => 1 = 2 - 1
[-,-,+,5,4] => [3,4,1,2,5] => 1 = 2 - 1
[-,+,-,5,4] => [2,4,1,3,5] => 1 = 2 - 1
[-,-,-,5,4] => [4,1,2,3,5] => 1 = 2 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000061
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(load all 4 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000061: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000061: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [[.,.],.]
=> 2
[-,+,+] => [2,3,1] => [[.,[.,.]],.]
=> 2
[-,-,+] => [3,1,2] => [[.,.],[.,.]]
=> 2
[-,+,-] => [2,1,3] => [[.,.],[.,.]]
=> 2
[-,3,2] => [2,1,3] => [[.,.],[.,.]]
=> 2
[3,+,1] => [2,1,3] => [[.,.],[.,.]]
=> 2
[-,+,+,+] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 2
[-,-,+,+] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[-,+,-,+] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[-,+,+,-] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[-,-,-,+] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[-,-,+,-] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[-,+,-,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[-,+,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[-,-,4,3] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[-,3,2,+] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[-,3,2,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[-,3,4,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[-,4,2,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[-,4,+,2] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 3
[-,4,-,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[2,4,+,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,+,1,+] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[3,+,1,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[3,+,4,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[4,+,1,3] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[4,+,+,1] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[4,-,+,1] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[4,+,-,1] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[4,3,2,1] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[-,+,+,+,+] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 2
[-,-,+,+,+] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> 2
[-,+,-,+,+] => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> 2
[-,+,+,-,+] => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> 2
[-,+,+,+,-] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 2
[-,-,-,+,+] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> 2
[-,-,+,-,+] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> 2
[-,-,+,+,-] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[-,+,-,-,+] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> 2
[-,+,-,+,-] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[-,+,+,-,-] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[-,-,-,-,+] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 2
[-,-,-,+,-] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[-,-,+,-,-] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[-,+,-,-,-] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[-,+,+,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 2
[-,-,+,5,4] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[-,+,-,5,4] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[-,-,-,5,4] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
Description
The number of nodes on the left branch of a binary tree.
Also corresponds to [[/StatisticsDatabase/St000011/|ST000011]] after applying the [[/BinaryTrees#Maps|Tamari bijection]] between binary trees and Dyck path.
Matching statistic: St000068
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => ([],2)
=> 2
[-,+,+] => [2,3,1] => ([(1,2)],3)
=> 2
[-,-,+] => [3,1,2] => ([(1,2)],3)
=> 2
[-,+,-] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[-,3,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[3,+,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[-,+,+,+] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[-,-,+,+] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[-,+,+,-] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[-,-,-,+] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[-,-,+,-] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[-,+,-,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[-,+,4,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[-,-,4,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[-,3,2,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[-,3,4,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[-,4,2,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[-,4,+,2] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[-,4,-,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,+,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,+,1,+] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,+,1,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[3,+,4,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[4,+,1,3] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[4,+,+,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[4,-,+,1] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[4,+,-,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[4,3,2,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[-,+,+,+,+] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 2
[-,-,+,+,+] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 2
[-,+,-,+,+] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[-,+,+,-,+] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[-,+,+,+,-] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[-,-,-,+,+] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 2
[-,-,+,-,+] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[-,-,+,+,-] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[-,+,-,-,+] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[-,+,-,+,-] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[-,+,+,-,-] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[-,-,-,-,+] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 2
[-,-,-,+,-] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[-,-,+,-,-] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[-,+,-,-,-] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[-,+,+,5,4] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[-,-,+,5,4] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[-,+,-,5,4] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[-,-,-,5,4] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
Description
The number of minimal elements in a poset.
Matching statistic: St000314
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [1,2] => 2
[-,+,+] => [2,3,1] => [2,1,3] => 2
[-,-,+] => [3,1,2] => [1,3,2] => 2
[-,+,-] => [2,1,3] => [2,3,1] => 2
[-,3,2] => [2,1,3] => [2,3,1] => 2
[3,+,1] => [2,1,3] => [2,3,1] => 2
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => 2
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 2
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => 2
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => 2
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => 2
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => 2
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => 2
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => 2
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => 2
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => 2
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => 2
[-,4,2,3] => [2,3,1,4] => [3,2,4,1] => 2
[-,4,+,2] => [3,2,1,4] => [2,3,4,1] => 3
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => 2
[2,4,+,1] => [3,1,2,4] => [2,4,3,1] => 2
[3,+,1,+] => [2,1,4,3] => [3,4,1,2] => 2
[3,+,1,-] => [2,1,3,4] => [3,4,2,1] => 2
[3,+,4,1] => [2,1,3,4] => [3,4,2,1] => 2
[3,4,2,1] => [2,1,3,4] => [3,4,2,1] => 2
[4,+,1,3] => [2,1,3,4] => [3,4,2,1] => 2
[4,+,+,1] => [2,3,1,4] => [3,2,4,1] => 2
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => 2
[4,+,-,1] => [2,1,4,3] => [3,4,1,2] => 2
[4,3,2,1] => [2,1,4,3] => [3,4,1,2] => 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => 2
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => 2
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => 2
[-,+,+,+,-] => [2,3,4,1,5] => [4,3,2,5,1] => 2
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => 2
[-,-,+,+,-] => [3,4,1,2,5] => [3,2,5,4,1] => 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => 2
[-,+,-,+,-] => [2,4,1,3,5] => [4,2,5,3,1] => 2
[-,+,+,-,-] => [2,3,1,4,5] => [4,3,5,2,1] => 2
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[-,-,-,+,-] => [4,1,2,3,5] => [2,5,4,3,1] => 2
[-,-,+,-,-] => [3,1,2,4,5] => [3,5,4,2,1] => 2
[-,+,-,-,-] => [2,1,3,4,5] => [4,5,3,2,1] => 2
[-,+,+,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2
[-,-,+,5,4] => [3,4,1,2,5] => [3,2,5,4,1] => 2
[-,+,-,5,4] => [2,4,1,3,5] => [4,2,5,3,1] => 2
[-,-,-,5,4] => [4,1,2,3,5] => [2,5,4,3,1] => 2
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000991
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [1,2] => 2
[-,+,+] => [2,3,1] => [1,3,2] => 2
[-,-,+] => [3,1,2] => [2,1,3] => 2
[-,+,-] => [2,1,3] => [3,1,2] => 2
[-,3,2] => [2,1,3] => [3,1,2] => 2
[3,+,1] => [2,1,3] => [3,1,2] => 2
[-,+,+,+] => [2,3,4,1] => [1,4,3,2] => 2
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 2
[-,+,+,-] => [2,3,1,4] => [4,1,3,2] => 2
[-,-,-,+] => [4,1,2,3] => [3,2,1,4] => 2
[-,-,+,-] => [3,1,2,4] => [4,2,1,3] => 2
[-,+,-,-] => [2,1,3,4] => [4,3,1,2] => 2
[-,+,4,3] => [2,3,1,4] => [4,1,3,2] => 2
[-,-,4,3] => [3,1,2,4] => [4,2,1,3] => 2
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => 2
[-,3,2,-] => [2,1,3,4] => [4,3,1,2] => 2
[-,3,4,2] => [2,1,3,4] => [4,3,1,2] => 2
[-,4,2,3] => [2,3,1,4] => [4,1,3,2] => 2
[-,4,+,2] => [3,2,1,4] => [4,1,2,3] => 3
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => 2
[2,4,+,1] => [3,1,2,4] => [4,2,1,3] => 2
[3,+,1,+] => [2,1,4,3] => [3,4,1,2] => 2
[3,+,1,-] => [2,1,3,4] => [4,3,1,2] => 2
[3,+,4,1] => [2,1,3,4] => [4,3,1,2] => 2
[3,4,2,1] => [2,1,3,4] => [4,3,1,2] => 2
[4,+,1,3] => [2,1,3,4] => [4,3,1,2] => 2
[4,+,+,1] => [2,3,1,4] => [4,1,3,2] => 2
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => 2
[4,+,-,1] => [2,1,4,3] => [3,4,1,2] => 2
[4,3,2,1] => [2,1,4,3] => [3,4,1,2] => 2
[-,+,+,+,+] => [2,3,4,5,1] => [1,5,4,3,2] => 2
[-,-,+,+,+] => [3,4,5,1,2] => [2,1,5,4,3] => 2
[-,+,-,+,+] => [2,4,5,1,3] => [3,1,5,4,2] => 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,1,5,3,2] => 2
[-,+,+,+,-] => [2,3,4,1,5] => [5,1,4,3,2] => 2
[-,-,-,+,+] => [4,5,1,2,3] => [3,2,1,5,4] => 2
[-,-,+,-,+] => [3,5,1,2,4] => [4,2,1,5,3] => 2
[-,-,+,+,-] => [3,4,1,2,5] => [5,2,1,4,3] => 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,3,1,5,2] => 2
[-,+,-,+,-] => [2,4,1,3,5] => [5,3,1,4,2] => 2
[-,+,+,-,-] => [2,3,1,4,5] => [5,4,1,3,2] => 2
[-,-,-,-,+] => [5,1,2,3,4] => [4,3,2,1,5] => 2
[-,-,-,+,-] => [4,1,2,3,5] => [5,3,2,1,4] => 2
[-,-,+,-,-] => [3,1,2,4,5] => [5,4,2,1,3] => 2
[-,+,-,-,-] => [2,1,3,4,5] => [5,4,3,1,2] => 2
[-,+,+,5,4] => [2,3,4,1,5] => [5,1,4,3,2] => 2
[-,-,+,5,4] => [3,4,1,2,5] => [5,2,1,4,3] => 2
[-,+,-,5,4] => [2,4,1,3,5] => [5,3,1,4,2] => 2
[-,-,-,5,4] => [4,1,2,3,5] => [5,3,2,1,4] => 2
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000007
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [1,2] => [2,1] => 2
[-,+,+] => [2,3,1] => [1,3,2] => [3,1,2] => 2
[-,-,+] => [3,1,2] => [2,1,3] => [2,3,1] => 2
[-,+,-] => [2,1,3] => [3,1,2] => [1,3,2] => 2
[-,3,2] => [2,1,3] => [3,1,2] => [1,3,2] => 2
[3,+,1] => [2,1,3] => [3,1,2] => [1,3,2] => 2
[-,+,+,+] => [2,3,4,1] => [1,4,3,2] => [4,1,2,3] => 2
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 2
[-,-,-,+] => [4,1,2,3] => [3,2,1,4] => [2,3,4,1] => 2
[-,-,+,-] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 2
[-,+,-,-] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 2
[-,+,4,3] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 2
[-,-,4,3] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 2
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[-,3,2,-] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 2
[-,3,4,2] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 2
[-,4,2,3] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 2
[-,4,+,2] => [3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 3
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 2
[2,4,+,1] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 2
[3,+,1,+] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 2
[3,+,1,-] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 2
[3,+,4,1] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 2
[3,4,2,1] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 2
[4,+,1,3] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 2
[4,+,+,1] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 2
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[4,+,-,1] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 2
[4,3,2,1] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 2
[-,+,+,+,+] => [2,3,4,5,1] => [1,5,4,3,2] => [5,1,2,3,4] => 2
[-,-,+,+,+] => [3,4,5,1,2] => [2,1,5,4,3] => [4,5,1,2,3] => 2
[-,+,-,+,+] => [2,4,5,1,3] => [3,1,5,4,2] => [3,5,1,2,4] => 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,1,5,3,2] => [2,5,1,3,4] => 2
[-,+,+,+,-] => [2,3,4,1,5] => [5,1,4,3,2] => [1,5,2,3,4] => 2
[-,-,-,+,+] => [4,5,1,2,3] => [3,2,1,5,4] => [3,4,5,1,2] => 2
[-,-,+,-,+] => [3,5,1,2,4] => [4,2,1,5,3] => [2,4,5,1,3] => 2
[-,-,+,+,-] => [3,4,1,2,5] => [5,2,1,4,3] => [1,4,5,2,3] => 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,3,1,5,2] => [2,3,5,1,4] => 2
[-,+,-,+,-] => [2,4,1,3,5] => [5,3,1,4,2] => [1,3,5,2,4] => 2
[-,+,+,-,-] => [2,3,1,4,5] => [5,4,1,3,2] => [1,2,5,3,4] => 2
[-,-,-,-,+] => [5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,5,1] => 2
[-,-,-,+,-] => [4,1,2,3,5] => [5,3,2,1,4] => [1,3,4,5,2] => 2
[-,-,+,-,-] => [3,1,2,4,5] => [5,4,2,1,3] => [1,2,4,5,3] => 2
[-,+,-,-,-] => [2,1,3,4,5] => [5,4,3,1,2] => [1,2,3,5,4] => 2
[-,+,+,5,4] => [2,3,4,1,5] => [5,1,4,3,2] => [1,5,2,3,4] => 2
[-,-,+,5,4] => [3,4,1,2,5] => [5,2,1,4,3] => [1,4,5,2,3] => 2
[-,+,-,5,4] => [2,4,1,3,5] => [5,3,1,4,2] => [1,3,5,2,4] => 2
[-,-,-,5,4] => [4,1,2,3,5] => [5,3,2,1,4] => [1,3,4,5,2] => 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000015
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [1,2] => [1,0,1,0]
=> 2
[-,+,+] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[-,-,+] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[-,+,-] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[-,3,2] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,+,1] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[-,4,2,3] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[-,4,+,2] => [3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,+,1] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,+,1,+] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,+,1,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[3,+,4,1] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,+,1,3] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,+,+,1] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[4,+,-,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[-,+,+,+,-] => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[-,-,+,+,-] => [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[-,+,-,+,-] => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[-,+,+,-,-] => [2,3,1,4,5] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[-,-,-,+,-] => [4,1,2,3,5] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[-,-,+,-,-] => [3,1,2,4,5] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[-,+,-,-,-] => [2,1,3,4,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[-,+,+,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[-,-,+,5,4] => [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[-,+,-,5,4] => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[-,-,-,5,4] => [4,1,2,3,5] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The number of peaks of a Dyck path.
Matching statistic: St000031
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [1,2] => [1,2] => 2
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 2
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[-,+,-] => [2,1,3] => [2,3,1] => [3,2,1] => 2
[-,3,2] => [2,1,3] => [2,3,1] => [3,2,1] => 2
[3,+,1] => [2,1,3] => [2,3,1] => [3,2,1] => 2
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => [3,1,2,4] => 2
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 2
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 2
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => [4,2,1,3] => 2
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 2
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 2
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => [4,2,1,3] => 2
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 2
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 2
[-,4,2,3] => [2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 2
[-,4,+,2] => [3,2,1,4] => [2,3,4,1] => [4,2,3,1] => 3
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => [2,4,3,1] => 2
[2,4,+,1] => [3,1,2,4] => [2,4,3,1] => [4,2,1,3] => 2
[3,+,1,+] => [2,1,4,3] => [3,4,1,2] => [2,4,3,1] => 2
[3,+,1,-] => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 2
[3,+,4,1] => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 2
[3,4,2,1] => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 2
[4,+,1,3] => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 2
[4,+,+,1] => [2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 2
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => [3,2,4,1] => 2
[4,+,-,1] => [2,1,4,3] => [3,4,1,2] => [2,4,3,1] => 2
[4,3,2,1] => [2,1,4,3] => [3,4,1,2] => [2,4,3,1] => 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => [4,1,2,3,5] => 2
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => [3,1,2,5,4] => 2
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => [4,1,5,2,3] => 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => [4,5,1,3,2] => 2
[-,+,+,+,-] => [2,3,4,1,5] => [4,3,2,5,1] => [5,4,2,3,1] => 2
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,3,4] => 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [3,5,1,2,4] => 2
[-,-,+,+,-] => [3,4,1,2,5] => [3,2,5,4,1] => [5,3,2,1,4] => 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => [4,5,2,1,3] => 2
[-,+,-,+,-] => [2,4,1,3,5] => [4,2,5,3,1] => [5,4,1,2,3] => 2
[-,+,+,-,-] => [2,3,1,4,5] => [4,3,5,2,1] => [5,1,4,3,2] => 2
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => 2
[-,-,-,+,-] => [4,1,2,3,5] => [2,5,4,3,1] => [5,2,1,3,4] => 2
[-,-,+,-,-] => [3,1,2,4,5] => [3,5,4,2,1] => [5,1,3,2,4] => 2
[-,+,-,-,-] => [2,1,3,4,5] => [4,5,3,2,1] => [5,1,2,4,3] => 2
[-,+,+,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => [5,4,2,3,1] => 2
[-,-,+,5,4] => [3,4,1,2,5] => [3,2,5,4,1] => [5,3,2,1,4] => 2
[-,+,-,5,4] => [2,4,1,3,5] => [4,2,5,3,1] => [5,4,1,2,3] => 2
[-,-,-,5,4] => [4,1,2,3,5] => [2,5,4,3,1] => [5,2,1,3,4] => 2
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000069
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => ([],2)
=> ([],2)
=> 2
[-,+,+] => [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[-,-,+] => [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[-,+,-] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[-,3,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[3,+,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[-,+,+,+] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,2),(2,3)],4)
=> 2
[-,-,+,+] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[-,+,+,-] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[-,-,-,+] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,2),(2,3)],4)
=> 2
[-,-,+,-] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[-,+,-,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[-,+,4,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[-,-,4,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[-,3,2,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[-,3,4,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[-,4,2,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[-,4,+,2] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[-,4,-,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,+,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[3,+,1,+] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,+,1,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[3,+,4,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[4,+,1,3] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[4,+,+,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[4,-,+,1] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[4,+,-,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[4,3,2,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[-,+,+,+,+] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(3,2),(4,3)],5)
=> 2
[-,-,+,+,+] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(1,4),(4,2)],5)
=> 2
[-,+,-,+,+] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[-,+,+,-,+] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[-,+,+,+,-] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[-,-,-,+,+] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(1,4),(4,2)],5)
=> 2
[-,-,+,-,+] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[-,-,+,+,-] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[-,+,-,-,+] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[-,+,-,+,-] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[-,+,+,-,-] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[-,-,-,-,+] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(3,2),(4,3)],5)
=> 2
[-,-,-,+,-] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[-,-,+,-,-] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[-,+,-,-,-] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[-,+,+,5,4] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[-,-,+,5,4] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[-,+,-,5,4] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[-,-,-,5,4] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
Description
The number of maximal elements of a poset.
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000213The number of weak exceedances (also weak excedences) of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000702The number of weak deficiencies of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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 $eA$. St000053The number of valleys of the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001625The Möbius invariant of a lattice. St001866The nesting alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001964The interval resolution global dimension of a poset.
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