searching the database
Your data matches 75 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001088
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St001088: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 3
Description
Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
Matching statistic: St000007
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [1,2] => 1
[1,0,1,0]
=> [3,1,2] => [3,2,1] => [1,2,3] => 1
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => [1,3,4,5,2] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [1,2,4,5,3] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [1,4,5,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [1,2,3,5,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [1,3,5,4,2] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => [1,4,3,5,2] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => [1,3,2,5,4] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => [1,4,3,2,5] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => [1,3,4,5,6,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => [1,2,4,5,6,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => [1,3,2,4,5,6] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => [1,4,5,6,3,2] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => [1,2,3,5,6,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => [1,3,5,6,4,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => [2,4,1,3,5,6] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => [1,4,3,5,6,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => [1,2,5,6,4,3] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => [1,3,2,5,6,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => [1,4,3,2,5,6] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => [1,5,6,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => [1,3,4,6,5,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => [1,2,4,6,5,3] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => [1,3,2,4,6,5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => [1,4,6,5,3,2] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => [1,3,5,4,6,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => [3,5,1,2,4,6] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => [2,4,6,1,3,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => [3,5,1,4,6,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => [1,2,5,4,6,3] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => [2,5,3,1,4,6] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => [1,5,4,6,3,2] => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7,5,4,3,2,1,6] => [1,3,4,5,6,7,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [7,6,4,3,2,1,5] => [1,2,4,5,6,7,3] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [7,4,3,2,1,5,6] => [1,4,5,6,7,3,2] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [7,6,5,3,2,1,4] => [1,2,3,5,6,7,4] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [7,5,3,2,1,4,6] => [1,3,5,6,7,4,2] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [7,4,5,3,2,1,6] => [1,4,3,5,6,7,2] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7,6,3,2,1,4,5] => [1,2,5,6,7,4,3] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [7,5,6,3,2,1,4] => [1,3,2,5,6,7,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [7,4,5,6,3,2,1] => [1,4,3,2,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [7,5,4,2,1,3,6] => [1,3,4,6,7,5,2] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [7,6,4,2,1,3,5] => [1,2,4,6,7,5,3] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [7,5,6,4,2,1,3] => [1,3,2,4,6,7,5] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [7,5,3,4,2,1,6] => [1,3,5,4,6,7,2] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [6,4,2,1,7,5,3] => [2,4,6,7,1,3,5] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,3,7,4,2,1,6] => [3,5,1,4,6,7,2] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [7,6,3,4,2,1,5] => [1,2,5,4,6,7,3] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,5,6,3,4,2,1] => [1,3,2,5,4,6,7] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [6,3,5,7,4,2,1] => [2,5,3,1,4,6,7] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [7,3,4,2,1,5,6] => [1,5,4,6,7,3,2] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [7,6,5,2,1,3,4] => [1,2,3,6,7,5,4] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [7,5,2,1,3,4,6] => [1,3,6,7,5,4,2] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [7,6,4,5,2,1,3] => [1,2,4,3,6,7,5] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [6,4,7,5,2,1,3] => [2,4,1,3,6,7,5] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [7,4,5,2,1,3,6] => [1,4,3,6,7,5,2] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [6,3,4,7,5,2,1] => [2,5,4,1,3,6,7] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [6,3,7,4,5,2,1] => [2,5,1,4,3,6,7] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [7,3,4,5,2,1,6] => [1,5,4,3,6,7,2] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [7,6,2,1,3,4,5] => [1,2,6,7,5,4,3] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [7,5,6,2,1,3,4] => [1,3,2,6,7,5,4] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [7,4,5,6,2,1,3] => [1,4,3,2,6,7,5] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [7,5,4,3,1,2,6] => [1,3,4,5,7,6,2] => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [7,6,4,3,1,2,5] => [1,2,4,5,7,6,3] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [7,5,6,4,3,1,2] => [1,3,2,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [7,4,3,1,2,5,6] => [1,4,5,7,6,3,2] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => [1,2,3,5,7,6,4] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => [1,3,5,7,6,4,2] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [7,6,4,5,3,1,2] => [1,2,4,3,5,7,6] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [6,4,7,5,3,1,2] => [2,4,1,3,5,7,6] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [7,4,5,3,1,2,6] => [1,4,3,5,7,6,2] => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [7,6,3,1,2,4,5] => [1,2,5,7,6,4,3] => ? = 4
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [7,5,6,3,1,2,4] => [1,3,2,5,7,6,4] => ? = 3
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [7,4,5,6,3,1,2] => [1,4,3,2,5,7,6] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [7,5,4,2,3,1,6] => [1,3,4,6,5,7,2] => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [7,6,4,2,3,1,5] => [1,2,4,6,5,7,3] => ? = 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: St000996
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0]
=> [3,1,2] => [3,2,1] => [3,2,1] => 1
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [2,3,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [3,2,4,1] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => [3,4,2,1] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => [4,3,2,5,1] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => [5,4,2,3,1] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [3,2,4,5,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [4,5,3,2,1] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [3,4,2,5,1] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => [5,3,4,2,1] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => [5,2,4,1,3] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => [4,2,3,5,1] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [3,4,5,2,1] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => [4,5,2,3,1] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => [5,2,3,4,1] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => [5,4,3,2,6,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => [5,4,3,6,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => [6,5,4,2,3,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => [4,3,2,5,6,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => [5,4,6,3,2,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => [4,3,5,2,6,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => [6,5,2,4,1,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => [5,4,2,3,6,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => [4,3,5,6,2,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => [5,4,6,2,3,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => [6,5,2,3,4,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => [3,2,4,5,6,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => [5,6,4,3,2,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => [4,5,3,2,6,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => [4,5,3,6,2,1] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => [5,6,4,2,3,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => [6,4,5,3,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => [5,3,4,2,6,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => [6,2,5,1,4,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => [5,2,4,1,6,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => [5,3,4,6,2,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => [6,2,5,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => [4,2,3,5,6,1] => 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7,5,4,3,2,1,6] => [6,5,4,3,2,7,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [7,6,4,3,2,1,5] => [6,5,4,3,7,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,5,6,4,3,2,1] => [7,6,5,4,2,3,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [7,4,3,2,1,5,6] => [5,4,3,2,6,7,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [7,6,5,3,2,1,4] => [6,5,4,7,3,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [7,5,3,2,1,4,6] => [5,4,3,6,2,7,1] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,6,4,5,3,2,1] => [7,6,5,3,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [6,4,7,5,3,2,1] => [7,6,5,2,4,1,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [7,4,5,3,2,1,6] => [6,5,4,2,3,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7,6,3,2,1,4,5] => [5,4,3,6,7,2,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [7,5,6,3,2,1,4] => [6,5,4,7,2,3,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [7,4,5,6,3,2,1] => [7,6,5,2,3,4,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [7,6,5,4,2,1,3] => [6,5,7,4,3,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [7,5,4,2,1,3,6] => [5,4,6,3,2,7,1] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [7,6,4,2,1,3,5] => [5,4,6,3,7,2,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [7,5,6,4,2,1,3] => [6,5,7,4,2,3,1] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,6,5,3,4,2,1] => [7,6,4,5,3,2,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [7,5,3,4,2,1,6] => [6,5,3,4,2,7,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,3,7,6,4,2,1] => [7,6,2,5,1,4,3] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [6,4,2,1,7,5,3] => [4,3,7,2,6,1,5] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,3,7,4,2,1,6] => [6,5,2,4,1,7,3] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [7,6,3,4,2,1,5] => [6,5,3,4,7,2,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,5,6,3,4,2,1] => [7,6,4,5,2,3,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [6,3,5,7,4,2,1] => [7,6,2,5,3,1,4] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [7,3,4,2,1,5,6] => [5,4,2,3,6,7,1] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [7,6,5,2,1,3,4] => [5,4,6,7,3,2,1] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [7,5,2,1,3,4,6] => [4,3,5,6,2,7,1] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [7,6,4,5,2,1,3] => [6,5,7,3,4,2,1] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [6,4,7,5,2,1,3] => [6,5,7,2,4,1,3] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [7,4,5,2,1,3,6] => [5,4,6,2,3,7,1] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,6,3,4,5,2,1] => [7,6,3,4,5,2,1] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [6,3,4,7,5,2,1] => [7,6,2,3,5,1,4] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [6,3,7,4,5,2,1] => [7,6,2,4,5,1,3] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [7,3,4,5,2,1,6] => [6,5,2,3,4,7,1] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [7,6,2,1,3,4,5] => [4,3,5,6,7,2,1] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [7,5,6,2,1,3,4] => [5,4,6,7,2,3,1] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [7,4,5,6,2,1,3] => [6,5,7,2,3,4,1] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,3,4,5,6,2,1] => [7,6,2,3,4,5,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [7,5,4,3,1,2,6] => [5,6,4,3,2,7,1] => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [7,6,4,3,1,2,5] => [5,6,4,3,7,2,1] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [7,5,6,4,3,1,2] => [6,7,5,4,2,3,1] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [7,4,3,1,2,5,6] => [4,5,3,2,6,7,1] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => [5,6,4,7,3,2,1] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => [4,5,3,6,2,7,1] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [7,6,4,5,3,1,2] => [6,7,5,3,4,2,1] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [6,4,7,5,3,1,2] => [6,7,5,2,4,1,3] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [7,4,5,3,1,2,6] => [5,6,4,2,3,7,1] => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [7,6,3,1,2,4,5] => [4,5,3,6,7,2,1] => ? = 4
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [7,5,6,3,1,2,4] => [5,6,4,7,2,3,1] => ? = 3
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000374
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [3,1,2] => [2,3,1] => 1
[1,1,0,0]
=> [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,4,2,1] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [2,4,5,3,1] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,5,2,1,4] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,1,5,3,2] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,4,6,1,5] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,3,5,1,6,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,3,5,6,4,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,3,6,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [2,4,1,5,6,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [2,4,1,6,3,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [2,4,5,3,6,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [2,4,5,6,3,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,4,6,3,1,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [2,5,1,3,6,4] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [2,5,1,6,4,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [2,5,6,3,4,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [2,6,1,3,4,5] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,1,4,5,6,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,1,4,6,2,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [3,1,5,2,6,4] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [3,1,5,6,4,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,1,6,2,4,5] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,4,2,6,1,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [3,4,6,2,1,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,5,2,1,6,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,5,2,6,4,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,5,6,2,4,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,6,2,1,4,5] => 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [2,3,4,6,7,5,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [2,3,5,1,6,7,4] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [2,3,5,1,7,4,6] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [2,3,5,6,4,7,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [2,3,5,6,7,4,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [2,3,5,7,4,1,6] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [2,3,6,1,4,7,5] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [2,3,6,1,7,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [2,3,6,7,4,5,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [2,3,7,1,4,5,6] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [2,4,1,5,6,7,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [2,4,1,5,7,3,6] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [2,4,1,6,3,7,5] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [2,4,1,6,7,5,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [2,4,1,7,3,5,6] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [2,4,5,3,6,7,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [2,4,5,3,7,1,6] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [2,4,5,6,3,7,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [2,4,5,6,7,1,3] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [2,4,5,7,3,1,6] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [2,4,6,3,1,7,5] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [2,4,6,3,7,5,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [2,4,6,7,3,5,1] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [2,4,7,3,1,5,6] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [2,5,1,3,6,7,4] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [2,5,1,3,7,4,6] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [2,5,1,6,4,7,3] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [2,5,1,6,7,4,3] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [2,5,1,7,4,3,6] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [2,5,6,3,4,7,1] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [2,5,6,3,7,4,1] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [2,5,6,7,4,3,1] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [2,5,7,3,4,1,6] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [2,6,1,3,4,7,5] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [2,6,1,3,7,5,4] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [2,6,1,7,4,5,3] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [2,6,7,3,4,5,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [3,1,4,5,7,2,6] => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [3,1,4,6,2,7,5] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [3,1,4,6,7,5,2] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [3,1,4,7,2,5,6] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [3,1,5,2,6,7,4] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [3,1,5,2,7,4,6] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [3,1,5,6,4,7,2] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [3,1,5,6,7,4,2] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [3,1,5,7,4,2,6] => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [3,1,6,2,4,7,5] => ? = 4
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [3,1,6,2,7,5,4] => ? = 3
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [3,1,6,7,4,5,2] => ? = 2
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000031
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [1,3,5,4,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,4,3,5,2] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [1,4,5,3,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,4,2,5,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,3,5,2,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [1,5,2,4,3] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,3,4] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,3,4,5,6,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,3,4,6,5,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,3,5,4,6,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,3,5,6,4,2] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,3,6,4,5,2] => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,4,3,5,6,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,4,3,6,5,2] => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,4,5,3,6,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,4,2,6,3,5] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,4,6,3,5,2] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,5,3,4,6,2] => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,5,3,6,4,2] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,5,6,3,4,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,4,5,6,3] => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,4,6,5,3] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,5,4,6,3] => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,5,6,4,3] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,4,2,5,6,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,4,2,6,5,3] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,3,5,2,6,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,3,5,6,2,4] => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,3,6,2,5,4] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,5,2,4,6,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,5,2,6,4,3] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,3,6,2,4,5] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,6,7,5,4,3,2] => [1,3,4,5,7,6,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [1,5,7,6,4,3,2] => [1,3,4,6,5,7,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,7,5,6,4,3,2] => [1,3,4,6,7,5,2] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,5,6,7,4,3,2] => [1,3,4,7,5,6,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => [1,3,5,4,6,7,2] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [1,4,6,7,5,3,2] => [1,3,5,4,7,6,2] => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,7,6,4,5,3,2] => [1,3,5,6,4,7,2] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,7,5,3,2,4,6] => [1,3,5,2,7,4,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [1,6,7,4,5,3,2] => [1,3,5,7,4,6,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [1,4,5,7,6,3,2] => [1,3,6,4,5,7,2] => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [1,4,7,5,6,3,2] => [1,3,6,4,7,5,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [1,7,4,5,6,3,2] => [1,3,6,7,4,5,2] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [1,4,5,6,7,3,2] => [1,3,7,4,5,6,2] => ? = 4 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => [1,4,3,5,6,7,2] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [1,3,6,7,5,4,2] => [1,4,3,5,7,6,2] => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [1,3,5,7,6,4,2] => [1,4,3,6,5,7,2] => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [1,3,7,5,6,4,2] => [1,4,3,6,7,5,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => [1,4,3,7,5,6,2] => ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [1,7,6,5,3,4,2] => [1,4,5,3,6,7,2] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [1,6,7,5,3,4,2] => [1,4,5,3,7,6,2] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => [1,4,2,6,3,7,5] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [1,6,4,2,3,7,5] => [1,4,2,6,7,3,5] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [1,4,2,7,3,6,5] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [1,5,7,6,3,4,2] => [1,4,6,3,5,7,2] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [1,7,5,6,3,4,2] => [1,4,6,3,7,5,2] => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [1,7,4,2,3,5,6] => [1,4,2,7,3,5,6] => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [1,5,6,7,3,4,2] => [1,4,7,3,5,6,2] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [1,3,4,7,6,5,2] => [1,5,3,4,6,7,2] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [1,3,4,6,7,5,2] => [1,5,3,4,7,6,2] => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [1,3,7,6,4,5,2] => [1,5,3,6,4,7,2] => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,5,3,2,7,4,6] => ? = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [1,3,6,7,4,5,2] => [1,5,3,7,4,6,2] => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [1,7,6,3,4,5,2] => [1,5,6,3,4,7,2] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => [1,5,2,3,7,4,6] => ? = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [1,5,2,7,4,3,6] => ? = 1 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [1,6,7,3,4,5,2] => [1,5,7,3,4,6,2] => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [1,3,4,5,7,6,2] => [1,6,3,4,5,7,2] => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [1,3,4,7,5,6,2] => [1,6,3,4,7,5,2] => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [1,3,7,4,5,6,2] => [1,6,3,7,4,5,2] => ? = 2 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,7,3,4,5,6,2] => [1,6,7,3,4,5,2] => ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 5 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [1,2,7,6,5,4,3] => [1,2,4,5,6,7,3] => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [1,2,6,7,5,4,3] => [1,2,4,5,7,6,3] => ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [1,2,5,7,6,4,3] => [1,2,4,6,5,7,3] => ? = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [1,2,7,5,6,4,3] => [1,2,4,6,7,5,3] => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => [1,2,4,7,5,6,3] => ? = 4 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [1,2,4,7,6,5,3] => [1,2,5,4,6,7,3] => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [1,2,7,6,4,5,3] => [1,2,5,6,4,7,3] => ? = 2 + 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000308
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [4,3,2,6,5,7,1] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,3,2,5,7,6,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,7,6,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [7,6,4,3,5,2,1] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => ? = 4
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => ? = 3
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: St000542
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [2,1] => [1,2] => 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1,0,0]
=> [2,3,1] => [2,1,3] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,2,4,3] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [2,3,5,1,4] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,1,2,5,3] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,6,5,1,4,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,5,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,6,1,5,3,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,3,5,4,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,2,5,4,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,6,3,5,1,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,6,3,1,5,2] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,1,2,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,2,5,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,6,3,2,1,5] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,6,4,1,3] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,1,6,2,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [5,3,6,2,1,4] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,4,6,5,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [2,3,6,5,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,1,5,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,4,6,2,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,3,6,2,5,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,4,6,2,1,5] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [3,7,6,5,1,4,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,7,6,5,2,4,3] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [3,7,6,5,2,1,4] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,7,6,1,5,3,2] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,7,6,2,5,1,3] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,7,6,3,5,4,2] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,7,6,2,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [2,7,6,3,5,1,4] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,7,6,3,1,5,2] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,7,6,1,2,5,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [1,7,6,3,2,5,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,7,6,3,2,1,5] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [5,7,1,6,4,3,2] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [5,7,2,6,4,1,3] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [5,7,3,6,1,4,2] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [5,7,1,6,2,4,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [5,7,3,6,2,1,4] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [1,7,4,6,5,3,2] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [2,7,4,6,5,1,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,7,3,6,5,4,2] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [2,7,3,6,5,1,4] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [3,7,4,6,1,5,2] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [1,7,4,6,2,5,3] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [1,7,3,6,2,5,4] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [3,7,4,6,2,1,5] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [5,7,4,1,6,3,2] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [5,7,4,2,6,1,3] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [5,7,1,3,6,4,2] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,7,1,2,6,4,3] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [5,7,2,3,6,1,4] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [1,7,4,3,6,5,2] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [1,7,4,2,6,5,3] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,2,3,6,5,4] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [2,7,4,3,6,1,5] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [5,7,4,3,1,6,2] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [5,7,4,1,2,6,3] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [5,7,1,3,2,6,4] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,7,4,3,2,6,5] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [5,7,4,3,2,1,6] => ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [6,1,7,5,4,3,2] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [6,2,7,5,4,1,3] => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [6,3,7,5,1,4,2] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [6,1,7,5,2,4,3] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [6,3,7,5,2,1,4] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [6,4,7,1,5,3,2] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [6,4,7,2,5,1,3] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [6,1,7,3,5,4,2] => ? = 2
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000541
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Mp00069: Permutations —complement⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,3,1] => [2,1,3] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,2,4,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [2,3,5,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,1,2,5,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,6,5,1,4,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,5,2,4,3] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,6,1,5,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,3,5,4,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,2,5,4,3] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,6,3,5,1,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,6,3,1,5,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,1,2,5,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,6,3,2,1,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,6,4,1,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,1,6,2,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [5,3,6,2,1,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,4,6,5,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [2,3,6,5,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,1,5,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,4,6,2,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,3,6,2,5,4] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,4,6,2,1,5] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [2,7,6,5,4,1,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [3,7,6,5,1,4,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,7,6,5,2,4,3] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [3,7,6,5,2,1,4] => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,7,6,1,5,3,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,7,6,2,5,1,3] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,7,6,3,5,4,2] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,7,6,2,5,4,3] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [2,7,6,3,5,1,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,7,6,3,1,5,2] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,7,6,1,2,5,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [1,7,6,3,2,5,4] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,7,6,3,2,1,5] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [5,7,1,6,4,3,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [5,7,2,6,4,1,3] => ? = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [5,7,3,6,1,4,2] => ? = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [5,7,1,6,2,4,3] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [5,7,3,6,2,1,4] => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [1,7,4,6,5,3,2] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [2,7,4,6,5,1,3] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,7,3,6,5,4,2] => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [2,7,1,6,5,4,3] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [2,7,3,6,5,1,4] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [3,7,4,6,1,5,2] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [1,7,4,6,2,5,3] => ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [1,7,3,6,2,5,4] => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [3,7,4,6,2,1,5] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [5,7,4,1,6,3,2] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [5,7,4,2,6,1,3] => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [5,7,1,3,6,4,2] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,7,1,2,6,4,3] => ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [5,7,2,3,6,1,4] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [1,7,4,3,6,5,2] => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [1,7,4,2,6,5,3] => ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,2,3,6,5,4] => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [2,7,4,3,6,1,5] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [5,7,4,3,1,6,2] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [5,7,4,1,2,6,3] => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [5,7,1,3,2,6,4] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,7,4,3,2,6,5] => ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [5,7,4,3,2,1,6] => ? = 5 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [6,1,7,5,4,3,2] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [6,2,7,5,4,1,3] => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [6,3,7,5,1,4,2] => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [6,1,7,5,2,4,3] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [6,3,7,5,2,1,4] => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [6,4,7,1,5,3,2] => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [6,4,7,2,5,1,3] => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [6,1,7,3,5,4,2] => ? = 2 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000314
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
St000314: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [2,1] => 1
[1,0,1,0]
=> [3,1,2] => 1
[1,1,0,0]
=> [2,3,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => ? = 2
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000015
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
Description
The number of peaks of a Dyck path.
The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000991The number of right-to-left minima of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000331The number of upper interactions of a Dyck path. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000654The first descent of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000021The number of descents of a permutation. St000039The number of crossings of a permutation. St000053The number of valleys of the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000236The number of cyclical small weak excedances. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000338The number of pixed points 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. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001330The hat guessing number of a graph. St001889The size of the connectivity set of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000702The number of weak deficiencies of a permutation. St000710The number of big deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000894The trace of an alternating sign matrix. St000989The number of final rises of a permutation. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000075The orbit size of a standard tableau under promotion.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!