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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000542
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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 σi in the one-line notation of a permutation σ is a left-to-right-minimum if there does not exist a j < i such that σj<σi.
Matching statistic: St000541
(load all 4 compositions to match this statistic)
(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 π of length n, this is the number of indices 2≤j≤n such that for all 1≤i<j, the pair (i,j) is an inversion of π.
Matching statistic: St000061
(load all 4 compositions to match this statistic)
(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: 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 σi in the one-line notation of a permutation σ is a '''left-to-right-maximum''' if there does not exist a j<i such that σj>σi.
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: 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.
Matching statistic: St000213
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [2,1] => [1,2] => 2
[-,+,+] => [2,3,1] => [2,3,1] => [2,1,3] => 2
[-,-,+] => [3,1,2] => [3,1,2] => [1,3,2] => 2
[-,+,-] => [2,1,3] => [2,1,3] => [2,3,1] => 2
[-,3,2] => [2,1,3] => [2,1,3] => [2,3,1] => 2
[3,+,1] => [2,1,3] => [2,1,3] => [2,3,1] => 2
[-,+,+,+] => [2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 2
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[-,+,+,-] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[-,-,-,+] => [4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 2
[-,-,+,-] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[-,+,-,-] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[-,+,4,3] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[-,-,4,3] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[-,3,2,-] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[-,3,4,2] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[-,4,2,3] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[-,4,+,2] => [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[-,4,-,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,4,+,1] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[3,+,1,+] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[3,+,1,-] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[3,+,4,1] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[4,+,1,3] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[4,+,+,1] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[4,-,+,1] => [3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[4,+,-,1] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[4,3,2,1] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[-,+,+,+,+] => [2,3,4,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 2
[-,-,+,+,+] => [3,4,5,1,2] => [3,5,4,1,2] => [3,1,2,5,4] => 2
[-,+,-,+,+] => [2,4,5,1,3] => [2,5,4,1,3] => [4,1,2,5,3] => 2
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,4,1,3] => [4,1,2,5,3] => 2
[-,+,+,+,-] => [2,3,4,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => 2
[-,-,-,+,+] => [4,5,1,2,3] => [4,5,1,3,2] => [2,1,5,3,4] => 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,5,1,4,2] => [3,1,5,2,4] => 2
[-,-,+,+,-] => [3,4,1,2,5] => [3,5,1,4,2] => [3,1,5,2,4] => 2
[-,+,-,-,+] => [2,5,1,3,4] => [2,5,1,4,3] => [4,1,5,2,3] => 2
[-,+,-,+,-] => [2,4,1,3,5] => [2,5,1,4,3] => [4,1,5,2,3] => 2
[-,+,+,-,-] => [2,3,1,4,5] => [2,5,1,4,3] => [4,1,5,2,3] => 2
[-,-,-,-,+] => [5,1,2,3,4] => [5,1,4,3,2] => [1,5,2,3,4] => 2
[-,-,-,+,-] => [4,1,2,3,5] => [4,1,5,3,2] => [2,5,1,3,4] => 2
[-,-,+,-,-] => [3,1,2,4,5] => [3,1,5,4,2] => [3,5,1,2,4] => 2
[-,+,-,-,-] => [2,1,3,4,5] => [2,1,5,4,3] => [4,5,1,2,3] => 2
[-,+,+,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => 2
[-,-,+,5,4] => [3,4,1,2,5] => [3,5,1,4,2] => [3,1,5,2,4] => 2
[-,+,-,5,4] => [2,4,1,3,5] => [2,5,1,4,3] => [4,1,5,2,3] => 2
[-,-,-,5,4] => [4,1,2,3,5] => [4,1,5,3,2] => [2,5,1,3,4] => 2
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
wex(σ)=#{i:σ(i)≥i}.
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of σ.
Matching statistic: St000702
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [2,1] => [1,2] => 2
[-,+,+] => [2,3,1] => [2,3,1] => [1,3,2] => 2
[-,-,+] => [3,1,2] => [3,1,2] => [2,1,3] => 2
[-,+,-] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[-,3,2] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[3,+,1] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[-,+,+,+] => [2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 2
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[-,+,+,-] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[-,-,-,+] => [4,1,2,3] => [4,1,3,2] => [2,3,1,4] => 2
[-,-,+,-] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[-,+,-,-] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[-,+,4,3] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[-,-,4,3] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[-,3,2,-] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[-,3,4,2] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[-,4,2,3] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[-,4,+,2] => [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[-,4,-,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,4,+,1] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[3,+,1,+] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[3,+,1,-] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[3,+,4,1] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[4,+,1,3] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[4,+,+,1] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[4,-,+,1] => [3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[4,+,-,1] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[4,3,2,1] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[-,+,+,+,+] => [2,3,4,5,1] => [2,5,4,3,1] => [1,3,4,5,2] => 2
[-,-,+,+,+] => [3,4,5,1,2] => [3,5,4,1,2] => [2,1,4,5,3] => 2
[-,+,-,+,+] => [2,4,5,1,3] => [2,5,4,1,3] => [3,1,4,5,2] => 2
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,4,1,3] => [3,1,4,5,2] => 2
[-,+,+,+,-] => [2,3,4,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => 2
[-,-,-,+,+] => [4,5,1,2,3] => [4,5,1,3,2] => [2,3,1,5,4] => 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,5,1,4,2] => [2,4,1,5,3] => 2
[-,-,+,+,-] => [3,4,1,2,5] => [3,5,1,4,2] => [2,4,1,5,3] => 2
[-,+,-,-,+] => [2,5,1,3,4] => [2,5,1,4,3] => [3,4,1,5,2] => 2
[-,+,-,+,-] => [2,4,1,3,5] => [2,5,1,4,3] => [3,4,1,5,2] => 2
[-,+,+,-,-] => [2,3,1,4,5] => [2,5,1,4,3] => [3,4,1,5,2] => 2
[-,-,-,-,+] => [5,1,2,3,4] => [5,1,4,3,2] => [2,3,4,1,5] => 2
[-,-,-,+,-] => [4,1,2,3,5] => [4,1,5,3,2] => [2,3,5,1,4] => 2
[-,-,+,-,-] => [3,1,2,4,5] => [3,1,5,4,2] => [2,4,5,1,3] => 2
[-,+,-,-,-] => [2,1,3,4,5] => [2,1,5,4,3] => [3,4,5,1,2] => 2
[-,+,+,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => 2
[-,-,+,5,4] => [3,4,1,2,5] => [3,5,1,4,2] => [2,4,1,5,3] => 2
[-,+,-,5,4] => [2,4,1,3,5] => [2,5,1,4,3] => [3,4,1,5,2] => 2
[-,-,-,5,4] => [4,1,2,3,5] => [4,1,5,3,2] => [2,3,5,1,4] => 2
Description
The number of weak deficiencies of a permutation.
This is defined as
wdec(σ)=#{i:σ(i)≤i}.
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. 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. St000031The number of cycles in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St001330The hat guessing number of a graph. 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. 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. St001964The interval resolution global dimension of a poset. St001866The nesting alignments of a signed permutation. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. 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.
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