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Your data matches 295 different statistics following compositions of up to 3 maps.
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Matching statistic: St001258
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St001258: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> 2 = 3 - 1
[1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.
For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
Matching statistic: St000725
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [1,2] => 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [3,2,1] => 3 = 4 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,2,1,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,4,3,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,4,2,1] => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,3,1,4] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,4,1,3] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,2,3,4,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,2,3,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,2,3,5,1] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2,1,4,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,3,4] => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2,5,1,4] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,3,4,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,5,3,4,1] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,5,3,1,4] => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,2,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,5,2,4,1] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,1,2,3,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,5,2,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,4,2,1,5] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,5,2,1,4] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,5,2,1,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,4,2,5,1] => 4 = 5 - 1
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000991
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,3,4] => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,5,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 4 = 5 - 1
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000007
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => [2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => [3,1,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [3,2,1] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [4,1,3,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [5,1,4,3,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [5,3,2,1,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => [5,3,1,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => [5,2,1,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => [5,1,4,3,2] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => [5,2,1,4,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => [5,1,4,3,2] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [5,4,3,1,2] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => [5,4,2,3,1] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,5,3] => [5,4,2,1,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [5,4,1,3,2] => 4 = 5 - 1
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: St000062
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [1,5,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => [1,5,3,2,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => [1,5,2,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => [1,5,2,4,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,4,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 4 = 5 - 1
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000203
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [.,.]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => [.,[.,.]]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 4 = 5 - 1
Description
The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St000213
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => [1,3,5,4,2] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [1,4,5,2,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [1,5,2,4,3] => [1,3,5,4,2] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 3 - 1
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000308
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [1,5,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => [1,5,3,2,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => [1,5,2,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => [1,5,2,4,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,4,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 4 = 5 - 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: St000314
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => [1,3,5,4,2] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 5 - 1
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: St000542
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => [2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,3,1] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => [2,3,1] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [3,2,1] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => [3,2,5,4,1] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => [3,4,2,5,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => [5,3,4,2,1] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,5,3] => [3,5,4,2,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 4 = 5 - 1
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 $\sigma_j < \sigma_i$.
The following 285 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000636The hull number of a graph. St000831The number of indices that are either descents or recoils. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St000155The number of exceedances (also excedences) of a permutation. St000245The number of ascents of a permutation. St000331The number of upper interactions of a Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000702The number of weak deficiencies of a permutation. St000619The number of cyclic descents of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000836The number of descents of distance 2 of a permutation. St001388The number of non-attacking neighbors of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001864The number of excedances of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000288The number of ones in a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000259The diameter of a connected graph. St000291The number of descents of a binary word. St000392The length of the longest run of ones in a binary word. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000982The length of the longest constant subword. St001863The number of weak excedances of a signed permutation. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000381The largest part of an integer composition. St000013The height of a Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001488The number of corners of a skew partition. St000389The number of runs of ones of odd length in a binary word. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001720The minimal length of a chain of small intervals in a lattice. St000455The second largest eigenvalue of a graph if it is integral. St000628The balance of a binary word. St000035The number of left outer peaks of a permutation. St000648The number of 2-excedences of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000260The radius of a connected graph. St000201The number of leaf nodes in a binary tree. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000097The order of the largest clique of the graph. St000120The number of left tunnels of a Dyck path. St000904The maximal number of repetitions of an integer composition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001424The number of distinct squares in a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001581The achromatic number of a graph. St001668The number of points of the poset minus the width of the poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000039The number of crossings of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001712The number of natural descents of a standard Young tableau. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000386The number of factors DDU in a Dyck path. St000264The girth of a graph, which is not a tree. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000640The rank of the largest boolean interval in a poset. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001626The number of maximal proper sublattices of a lattice. St000568The hook number of a binary tree. St000352The Elizalde-Pak rank of a permutation. St000662The staircase size of the code of a permutation. St000834The number of right outer peaks of a permutation. St001896The number of right descents of a signed permutations. St001877Number of indecomposable injective modules with projective dimension 2. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St000528The height of a poset. St000080The rank of the poset. St000098The chromatic number of a graph. St000522The number of 1-protected nodes of a rooted tree. St001096The size of the overlap set of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000519The largest length of a factor maximising the subword complexity. St000839The largest opener of a set partition. St000922The minimal number such that all substrings of this length are unique. St000153The number of adjacent cycles of a permutation. St000492The rob statistic of a set partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000647The number of big descents of a permutation. St000670The reversal length of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. St001394The genus of a permutation. St000022The number of fixed points of a permutation. St000356The number of occurrences of the pattern 13-2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000031The number of cycles in the cycle decomposition of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001623The number of doubly irreducible elements of a lattice. St001645The pebbling number of a connected graph. St001893The flag descent of a signed permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St001115The number of even descents of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000451The length of the longest pattern of the form k 1 2. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000374The number of exclusive right-to-left minima of a permutation. St000493The los statistic of a set partition. St000891The number of distinct diagonal sums of a permutation matrix. St001050The number of terminal closers of a set partition. St001075The minimal size of a block of a set partition. St000028The number of stack-sorts needed to sort a permutation. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000422The energy of a graph, if it is integral. St000099The number of valleys of a permutation, including the boundary. St001517The length of a longest pair of twins in a permutation. St000023The number of inner peaks of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000779The tier of a permutation. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St000485The length of the longest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000844The size of the largest block in the direct sum decomposition of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000021The number of descents of a permutation. St000092The number of outer peaks of a permutation. St000163The size of the orbit of the set partition under rotation. St000166The depth minus 1 of an ordered tree. St000172The Grundy number of a graph. St000209Maximum difference of elements in cycles. St000230Sum of the minimal elements of the blocks of a set partition. St000238The number of indices that are not small weak excedances. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000495The number of inversions of distance at most 2 of a permutation. St000730The maximal arc length of a set partition. St000822The Hadwiger number of the graph. St000956The maximal displacement of a permutation. St001029The size of the core of a graph. St001114The number of odd descents of a permutation. St001116The game chromatic number 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:
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001557The number of inversions of the second entry of a permutation. St001580The acyclic chromatic number of a graph. St001642The Prague dimension of a graph. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001670The connected partition number of a graph. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001716The 1-improper chromatic number of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001769The reflection length of a signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001928The number of non-overlapping descents in a permutation. St001963The tree-depth of a graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000221The number of strong fixed points of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000254The nesting number of a set partition. St000272The treewidth of a graph. St000353The number of inner valleys of a permutation. St000360The number of occurrences of the pattern 32-1. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. 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. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001470The cyclic holeyness of a permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001556The number of inversions of the third entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001781The interlacing number of a set partition. St001792The arboricity of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001812The biclique partition number of a graph. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St001946The number of descents in a parking function. St001948The number of augmented double ascents of a permutation. St001621The number of atoms of a lattice. St001637The number of (upper) dissectors of a poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000527The width of the poset. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000632The jump number of the poset. St000824The sum of the number of descents and the number of recoils of a permutation. St001516The number of cyclic bonds of a permutation. St001875The number of simple modules with projective dimension at most 1. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000654The first descent of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000486The number of cycles of length at least 3 of a permutation. St000649The number of 3-excedences of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001782The order of rowmotion on the set of order ideals of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000307The number of rowmotion orbits of a poset. St001555The order of a signed permutation. St001644The dimension of a graph. St001638The book thickness of a graph.
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