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Matching statistic: St000439
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2
[1,2] => [2,1] => [1,1,0,0]
=> 3
[2,1] => [1,2] => [1,0,1,0]
=> 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 4
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 3
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 4
Description
The position of the first down step of a Dyck path.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 100%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 100%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> [1] => 1 = 2 - 1
[1,2] => [2,1] => [1,1,0,0]
=> [2] => 2 = 3 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,1] => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [3] => 3 = 4 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [3] => 3 = 4 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 5 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3 = 4 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 5 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => 3 = 4 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 5 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3 = 4 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 5 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => 3 = 4 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 5 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 5 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 4 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5 = 6 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4 = 5 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5 = 6 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3 = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4 = 5 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3 = 4 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5 = 6 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4 = 5 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5 = 6 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4 = 5 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5 = 6 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3 = 4 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5 = 6 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2 = 3 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3 = 4 - 1
[2,1,3,4,5,6,7,8,9,10,11] => [11,10,9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[10,9,8,7,6,5,4,3,2,1,11] => [11,1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[1,11,10,9,8,7,6,5,4,3,2] => [2,3,4,5,6,7,8,9,10,11,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1,1] => ? = 3 - 1
[10,1,2,3,4,5,6,7,8,9,11] => [11,9,8,7,6,5,4,3,2,1,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[11,10,9,8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8,9,10,11] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => ? = 3 - 1
[2,3,4,5,6,7,8,9,10,1,11] => [11,1,10,9,8,7,6,5,4,3,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[9,8,10,7,6,5,4,3,2,1,11] => [11,1,2,3,4,5,6,7,10,8,9] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[9,8,7,10,6,5,4,3,2,1,11] => [11,1,2,3,4,5,6,10,7,8,9] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[9,8,7,6,5,10,4,3,2,1,11] => [11,1,2,3,4,10,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[9,8,7,6,5,4,10,3,2,1,11] => [11,1,2,3,10,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[9,8,7,6,5,4,3,2,10,1,11] => [11,1,10,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[9,8,7,6,5,4,3,2,1,10,11] => [11,10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[1,2,11,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,11,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1,1,1,1] => ? = 4 - 1
[2,1,11,10,9,8,7,6,5,4,3] => [3,4,5,6,7,8,9,10,11,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1,1,1,1] => ? = 4 - 1
[3,4,5,6,7,8,9,10,11,1,2] => [2,1,11,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,9] => ? = 3 - 1
[1,2,3,4,5,6,7,8,9,10,11] => [11,10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[10,11,1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1,11,10] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,2] => ? = 10 - 1
[1,2,3,4,5,6,7,8,10,9,11] => [11,9,10,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[9,11,10,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,10,11,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,2,1] => ? = 2 - 1
[1,10,9,8,7,6,5,4,3,2,11] => [11,2,3,4,5,6,7,8,9,10,1] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[10,9,8,7,6,5,4,2,1,3,11] => [11,3,1,2,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[10,9,8,7,6,5,3,2,1,4,11] => [11,4,1,2,3,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[10,9,8,7,5,4,3,2,1,6,11] => [11,6,1,2,3,4,5,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[10,9,8,6,5,4,3,2,1,7,11] => [11,7,1,2,3,4,5,6,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[10,8,7,6,5,4,3,2,1,9,11] => [11,9,1,2,3,4,5,6,7,8,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[8,10,9,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,9,10,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,1,1,1,1,2,1] => ? = 3 - 1
[1,2,3,4,6,8,10,5,7,9,11] => [11,9,7,5,10,8,6,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[1,2,4,6,8,10,3,5,7,9,11] => [11,9,7,5,3,10,8,6,4,2,1] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[2,1,10,9,8,7,6,5,4,3,11] => [11,3,4,5,6,7,8,9,10,1,2] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[10,9,8,7,6,5,4,3,1,2,11] => [11,2,1,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[1,2,10,9,8,7,6,5,4,3,11] => [11,3,4,5,6,7,8,9,10,2,1] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
[1,9,8,7,6,5,4,3,2,10,11] => [11,10,2,3,4,5,6,7,8,9,1] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 12 - 1
Description
The first part of an integer composition.
Matching statistic: St000011
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,2] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[8,5,6,7,4,2,3,1] => [1,3,2,4,7,6,5,8] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
[8,6,7,4,5,2,3,1] => [1,3,2,5,4,7,6,8] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 2 - 1
[7,6,8,4,5,2,3,1] => [1,3,2,5,4,8,6,7] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 2 - 1
[6,7,8,4,5,2,3,1] => [1,3,2,5,4,8,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 2 - 1
[8,7,5,4,6,2,3,1] => [1,3,2,6,4,5,7,8] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[8,7,4,5,6,2,3,1] => [1,3,2,6,5,4,7,8] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[7,8,4,5,6,2,3,1] => [1,3,2,6,5,4,8,7] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[7,8,6,5,3,2,4,1] => [1,4,2,3,5,6,8,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
[8,6,7,5,3,2,4,1] => [1,4,2,3,5,7,6,8] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
[8,7,5,6,3,2,4,1] => [1,4,2,3,6,5,7,8] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[7,8,5,6,3,2,4,1] => [1,4,2,3,6,5,8,7] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[7,6,5,8,3,2,4,1] => [1,4,2,3,8,5,6,7] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[6,7,5,8,3,2,4,1] => [1,4,2,3,8,5,7,6] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[7,5,6,8,3,2,4,1] => [1,4,2,3,8,6,5,7] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[6,5,7,8,3,2,4,1] => [1,4,2,3,8,7,5,6] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[5,6,7,8,3,2,4,1] => [1,4,2,3,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[8,6,7,5,2,3,4,1] => [1,4,3,2,5,7,6,8] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[7,6,5,8,2,3,4,1] => [1,4,3,2,8,5,6,7] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[7,5,6,8,2,3,4,1] => [1,4,3,2,8,6,5,7] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[8,6,7,4,3,2,5,1] => [1,5,2,3,4,7,6,8] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[8,6,7,3,4,2,5,1] => [1,5,2,4,3,7,6,8] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[8,6,7,4,2,3,5,1] => [1,5,3,2,4,7,6,8] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[8,6,7,3,2,4,5,1] => [1,5,4,2,3,7,6,8] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[8,6,7,2,3,4,5,1] => [1,5,4,3,2,7,6,8] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[7,8,5,6,4,3,1,2] => [2,1,3,4,6,5,8,7] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[8,6,5,7,4,3,1,2] => [2,1,3,4,7,5,6,8] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[7,6,5,8,4,3,1,2] => [2,1,3,4,8,5,6,7] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[6,7,5,8,4,3,1,2] => [2,1,3,4,8,5,7,6] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[7,5,6,8,4,3,1,2] => [2,1,3,4,8,6,5,7] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[6,5,7,8,4,3,1,2] => [2,1,3,4,8,7,5,6] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[5,6,7,8,4,3,1,2] => [2,1,3,4,8,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[7,8,6,4,5,3,1,2] => [2,1,3,5,4,6,8,7] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 3 - 1
[6,7,8,4,5,3,1,2] => [2,1,3,5,4,8,7,6] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 3 - 1
[8,7,5,4,6,3,1,2] => [2,1,3,6,4,5,7,8] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 3 - 1
[7,8,5,4,6,3,1,2] => [2,1,3,6,4,5,8,7] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 3 - 1
[7,8,4,5,6,3,1,2] => [2,1,3,6,5,4,8,7] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 3 - 1
[7,8,6,5,3,4,1,2] => [2,1,4,3,5,6,8,7] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 3 - 1
[7,6,8,5,3,4,1,2] => [2,1,4,3,5,8,6,7] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 3 - 1
[6,7,8,5,3,4,1,2] => [2,1,4,3,5,8,7,6] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 3 - 1
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[7,6,5,8,3,4,1,2] => [2,1,4,3,8,5,6,7] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[6,7,5,8,3,4,1,2] => [2,1,4,3,8,5,7,6] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[7,5,6,8,3,4,1,2] => [2,1,4,3,8,6,5,7] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[6,5,7,8,3,4,1,2] => [2,1,4,3,8,7,5,6] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[7,8,6,4,3,5,1,2] => [2,1,5,3,4,6,8,7] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[7,8,6,3,4,5,1,2] => [2,1,5,4,3,6,8,7] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> []
=> 0 = 2 - 2
[1,2] => [1,2] => [1,0,1,0]
=> [1]
=> 1 = 3 - 2
[2,1] => [2,1] => [1,1,0,0]
=> []
=> 0 = 2 - 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2 = 4 - 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1 = 3 - 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2 = 4 - 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0 = 2 - 2
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 3 - 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0 = 2 - 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 5 - 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 4 - 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3 = 5 - 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 3 - 2
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 4 - 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 3 - 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 5 - 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 4 - 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 5 - 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 2 - 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 4 - 2
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 2 - 2
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 5 - 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 3 - 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 5 - 2
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 2 - 2
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 3 - 2
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 2 - 2
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 4 - 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 3 - 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 4 - 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 2 - 2
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 3 - 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 2 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 6 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 5 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4 = 6 - 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 4 - 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 5 - 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 4 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4 = 6 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3 = 5 - 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 6 - 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 5 - 2
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4 = 6 - 2
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 4 - 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 6 - 2
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 3 - 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 4 - 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 8 - 2
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 7 - 2
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 8 - 2
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 6 - 2
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 7 - 2
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 6 - 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 8 - 2
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 7 - 2
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 8 - 2
[1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 7 - 2
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 8 - 2
[1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 6 - 2
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 8 - 2
[1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 7 - 2
[1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 6 - 2
[1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 7 - 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 8 - 2
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 7 - 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 8 - 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 6 - 2
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 7 - 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 6 - 2
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 8 - 2
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 7 - 2
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 8 - 2
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 7 - 2
[1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 8 - 2
[1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 7 - 2
[1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 8 - 2
[1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> ? = 7 - 2
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 8 - 2
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 7 - 2
[1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 8 - 2
[1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 6 - 2
[1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 7 - 2
[1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 6 - 2
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 8 - 2
[1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 7 - 2
[1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 8 - 2
[1,3,2,7,4,5,6] => [1,3,2,5,6,7,4] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> ? = 7 - 2
[1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 8 - 2
[1,4,2,3,5,6,7] => [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 8 - 2
[1,4,2,3,5,7,6] => [1,3,4,2,5,7,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 7 - 2
[1,4,2,3,6,5,7] => [1,3,4,2,6,5,7] => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 8 - 2
[1,4,2,3,7,5,6] => [1,3,4,2,6,7,5] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> ? = 7 - 2
[1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 8 - 2
[1,5,2,3,4,6,7] => [1,3,4,5,2,6,7] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 8 - 2
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 8 - 2
[2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 7 - 2
[2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 8 - 2
Description
The largest part of an integer partition.
Matching statistic: St000971
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> {{1}}
=> 1 = 2 - 1
[1,2] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2 = 3 - 1
[2,1] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 4 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 4 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3 = 4 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3 = 4 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3 = 4 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3 = 4 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 4 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 6 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 4 = 5 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 6 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3 = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 4 = 5 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3 = 4 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 6 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 4 = 5 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 6 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 4 = 5 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 6 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3 = 4 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 6 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3 = 4 - 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 2 - 1
[7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 2 - 1
[8,6,7,5,4,3,2,1] => [1,2,3,4,5,7,6,8] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 2 - 1
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 2 - 1
[6,7,8,5,4,3,2,1] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 2 - 1
[8,7,5,6,4,3,2,1] => [1,2,3,4,6,5,7,8] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3},{4},{5,6},{7},{8}}
=> ? = 2 - 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 2 - 1
[8,6,5,7,4,3,2,1] => [1,2,3,4,7,5,6,8] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 2 - 1
[8,5,6,7,4,3,2,1] => [1,2,3,4,7,6,5,8] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 2 - 1
[7,6,5,8,4,3,2,1] => [1,2,3,4,8,5,6,7] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 2 - 1
[6,7,5,8,4,3,2,1] => [1,2,3,4,8,5,7,6] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 2 - 1
[7,5,6,8,4,3,2,1] => [1,2,3,4,8,6,5,7] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 2 - 1
[6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 2 - 1
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 2 - 1
[8,7,6,4,5,3,2,1] => [1,2,3,5,4,6,7,8] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 2 - 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 2 - 1
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 2 - 1
[7,6,8,4,5,3,2,1] => [1,2,3,5,4,8,6,7] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 2 - 1
[6,7,8,4,5,3,2,1] => [1,2,3,5,4,8,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 2 - 1
[8,7,5,4,6,3,2,1] => [1,2,3,6,4,5,7,8] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[7,8,5,4,6,3,2,1] => [1,2,3,6,4,5,8,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2},{3},{4,5,6},{7,8}}
=> ? = 2 - 1
[8,7,4,5,6,3,2,1] => [1,2,3,6,5,4,7,8] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[7,8,4,5,6,3,2,1] => [1,2,3,6,5,4,8,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2},{3},{4,5,6},{7,8}}
=> ? = 2 - 1
[8,6,5,4,7,3,2,1] => [1,2,3,7,4,5,6,8] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 2 - 1
[8,5,6,4,7,3,2,1] => [1,2,3,7,4,6,5,8] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 2 - 1
[8,6,4,5,7,3,2,1] => [1,2,3,7,5,4,6,8] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 2 - 1
[8,5,4,6,7,3,2,1] => [1,2,3,7,6,4,5,8] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 2 - 1
[8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 2 - 1
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[6,7,5,4,8,3,2,1] => [1,2,3,8,4,5,7,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[7,5,6,4,8,3,2,1] => [1,2,3,8,4,6,5,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[5,6,7,4,8,3,2,1] => [1,2,3,8,4,7,6,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[7,6,4,5,8,3,2,1] => [1,2,3,8,5,4,6,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[6,7,4,5,8,3,2,1] => [1,2,3,8,5,4,7,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[7,5,4,6,8,3,2,1] => [1,2,3,8,6,4,5,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[7,4,5,6,8,3,2,1] => [1,2,3,8,6,5,4,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[6,5,4,7,8,3,2,1] => [1,2,3,8,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[5,6,4,7,8,3,2,1] => [1,2,3,8,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[6,4,5,7,8,3,2,1] => [1,2,3,8,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[5,4,6,7,8,3,2,1] => [1,2,3,8,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[4,5,6,7,8,3,2,1] => [1,2,3,8,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[8,7,6,5,3,4,2,1] => [1,2,4,3,5,6,7,8] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6},{7},{8}}
=> ? = 2 - 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 2 - 1
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5},{6,7},{8}}
=> ? = 2 - 1
[7,6,8,5,3,4,2,1] => [1,2,4,3,5,8,6,7] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4},{5},{6,7,8}}
=> ? = 2 - 1
[6,7,8,5,3,4,2,1] => [1,2,4,3,5,8,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4},{5},{6,7,8}}
=> ? = 2 - 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 2 - 1
[7,8,5,6,3,4,2,1] => [1,2,4,3,6,5,8,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1},{2},{3,4},{5,6},{7,8}}
=> ? = 2 - 1
[8,6,5,7,3,4,2,1] => [1,2,4,3,7,5,6,8] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3,4},{5,6,7},{8}}
=> ? = 2 - 1
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St000476
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 73%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 73%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 2 - 2
[1,2] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 3 - 2
[2,1] => [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 4 - 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 4 - 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 5 - 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 5 - 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 5 - 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 5 - 2
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 6 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 6 - 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 6 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 6 - 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 6 - 2
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 6 - 2
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,6,7,5,4,3,2,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,6,8,5,4,3,2,1] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,7,5,6,4,3,2,1] => [8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,8,5,6,4,3,2,1] => [8,7,6,5,3,4,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,6,5,7,4,3,2,1] => [8,7,6,5,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,5,6,7,4,3,2,1] => [8,7,6,5,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,6,5,8,4,3,2,1] => [8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,7,5,8,4,3,2,1] => [8,7,6,5,3,1,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,5,6,8,4,3,2,1] => [8,7,6,5,2,3,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,5,7,8,4,3,2,1] => [8,7,6,5,2,1,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,7,6,4,5,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,8,6,4,5,3,2,1] => [8,7,6,4,5,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,6,7,4,5,3,2,1] => [8,7,6,4,5,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,6,8,4,5,3,2,1] => [8,7,6,4,5,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,7,8,4,5,3,2,1] => [8,7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,7,5,4,6,3,2,1] => [8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,8,5,4,6,3,2,1] => [8,7,6,4,3,5,1,2] => ?
=> ?
=> ? = 2 - 2
[8,7,4,5,6,3,2,1] => [8,7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,8,4,5,6,3,2,1] => [8,7,6,3,4,5,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,6,5,4,7,3,2,1] => [8,7,6,4,3,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,5,6,4,7,3,2,1] => [8,7,6,4,2,3,5,1] => ?
=> ?
=> ? = 2 - 2
[8,6,4,5,7,3,2,1] => [8,7,6,3,4,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,5,4,6,7,3,2,1] => [8,7,6,3,2,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,4,5,6,7,3,2,1] => [8,7,6,2,3,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,6,5,4,8,3,2,1] => [8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,7,5,4,8,3,2,1] => [8,7,6,4,3,1,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,5,6,4,8,3,2,1] => [8,7,6,4,2,3,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,5,7,4,8,3,2,1] => [8,7,6,4,2,1,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[5,6,7,4,8,3,2,1] => [8,7,6,4,1,2,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,6,4,5,8,3,2,1] => [8,7,6,3,4,2,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,7,4,5,8,3,2,1] => [8,7,6,3,4,1,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,5,4,6,8,3,2,1] => [8,7,6,3,2,4,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,4,5,6,8,3,2,1] => [8,7,6,2,3,4,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,5,4,7,8,3,2,1] => [8,7,6,3,2,1,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[5,6,4,7,8,3,2,1] => [8,7,6,3,1,2,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,4,5,7,8,3,2,1] => [8,7,6,2,3,1,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[5,4,6,7,8,3,2,1] => [8,7,6,2,1,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,7,6,5,3,4,2,1] => [8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,8,6,5,3,4,2,1] => [8,7,5,6,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[8,6,7,5,3,4,2,1] => [8,7,5,6,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,6,8,5,3,4,2,1] => [8,7,5,6,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,7,8,5,3,4,2,1] => [8,7,5,6,4,1,2,3] => ?
=> ?
=> ? = 2 - 2
[8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[7,8,5,6,3,4,2,1] => [8,7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000069
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> ([],1)
=> 1 = 2 - 1
[1,2] => [2,1] => [1,1,0,0]
=> ([],2)
=> 2 = 3 - 1
[2,1] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ([],3)
=> 3 = 4 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ([],3)
=> 3 = 4 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 5 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3 = 4 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 5 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 3 = 4 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 5 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 5 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 3 = 4 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 5 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 5 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 3 = 4 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5 = 6 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 4 = 5 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5 = 6 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> 4 = 5 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5 = 6 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 4 = 5 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5 = 6 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> 4 = 5 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5 = 6 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5 = 6 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
[2,3,7,6,4,5,1] => [1,5,4,6,7,3,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,3,7,6,5,4,1] => [1,4,5,6,7,3,2] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,4,7,6,3,5,1] => [1,5,3,6,7,4,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,4,7,6,5,3,1] => [1,3,5,6,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 - 1
[2,6,7,3,4,5,1] => [1,5,4,3,7,6,2] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,6,7,4,3,5,1] => [1,5,3,4,7,6,2] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,6,7,5,3,4,1] => [1,4,3,5,7,6,2] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,7,3,6,4,5,1] => [1,5,4,6,3,7,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,7,4,6,3,5,1] => [1,5,3,6,4,7,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,7,4,6,5,3,1] => [1,3,5,6,4,7,2] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,6),(1,2),(1,3),(1,4),(2,6),(3,6),(4,5),(6,5)],7)
=> ? = 2 - 1
[2,7,5,6,3,4,1] => [1,4,3,6,5,7,2] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,7,5,6,4,3,1] => [1,3,4,6,5,7,2] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 2 - 1
[2,7,6,4,5,3,1] => [1,3,5,4,6,7,2] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(4,3),(5,4),(6,3)],7)
=> ? = 2 - 1
[3,2,7,6,4,5,1] => [1,5,4,6,7,2,3] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,2,7,6,5,4,1] => [1,4,5,6,7,2,3] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,4,7,6,2,5,1] => [1,5,2,6,7,4,3] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,4,7,6,5,1,2] => [2,1,5,6,7,4,3] => [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[3,6,7,2,4,5,1] => [1,5,4,2,7,6,3] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,6,7,4,2,5,1] => [1,5,2,4,7,6,3] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,6,7,5,2,4,1] => [1,4,2,5,7,6,3] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,7,2,6,4,5,1] => [1,5,4,6,2,7,3] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,7,4,5,6,1,2] => [2,1,6,5,4,7,3] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[3,7,4,6,2,5,1] => [1,5,2,6,4,7,3] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,7,5,4,6,1,2] => [2,1,6,4,5,7,3] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[3,7,5,6,2,4,1] => [1,4,2,6,5,7,3] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,7,5,6,4,1,2] => [2,1,4,6,5,7,3] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(1,6),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ? = 3 - 1
[3,7,5,6,4,2,1] => [1,2,4,6,5,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> ? = 2 - 1
[3,7,6,4,5,1,2] => [2,1,5,4,6,7,3] => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[3,7,6,5,4,1,2] => [2,1,4,5,6,7,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,5),(1,6),(2,5),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 3 - 1
[4,2,7,6,3,5,1] => [1,5,3,6,7,2,4] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[4,2,7,6,5,3,1] => [1,3,5,6,7,2,4] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 - 1
[4,3,7,6,2,5,1] => [1,5,2,6,7,3,4] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[4,3,7,6,5,1,2] => [2,1,5,6,7,3,4] => [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[4,5,7,6,1,2,3] => [3,2,1,6,7,5,4] => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 4 - 1
[4,5,7,6,2,1,3] => [3,1,2,6,7,5,4] => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 4 - 1
[4,6,7,2,3,5,1] => [1,5,3,2,7,6,4] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[4,6,7,3,2,5,1] => [1,5,2,3,7,6,4] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[4,7,2,6,3,5,1] => [1,5,3,6,2,7,4] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[4,7,2,6,5,3,1] => [1,3,5,6,2,7,4] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,6),(1,2),(1,3),(1,4),(2,6),(3,6),(4,5),(6,5)],7)
=> ? = 2 - 1
[4,7,3,5,6,1,2] => [2,1,6,5,3,7,4] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[4,7,3,6,2,5,1] => [1,5,2,6,3,7,4] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[4,7,5,3,6,1,2] => [2,1,6,3,5,7,4] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[4,7,5,6,1,2,3] => [3,2,1,6,5,7,4] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4 - 1
[4,7,5,6,1,3,2] => [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,5),(0,6),(1,3),(1,4),(3,5),(3,6),(4,5),(4,6),(6,2)],7)
=> ? = 3 - 1
[4,7,5,6,2,1,3] => [3,1,2,6,5,7,4] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4 - 1
[4,7,5,6,2,3,1] => [1,3,2,6,5,7,4] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 2 - 1
[4,7,5,6,3,1,2] => [2,1,3,6,5,7,4] => [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,6),(1,2),(1,3),(2,6),(3,6),(6,4),(6,5)],7)
=> ? = 3 - 1
[4,7,6,2,5,3,1] => [1,3,5,2,6,7,4] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(4,3),(5,4),(6,3)],7)
=> ? = 2 - 1
[4,7,6,3,5,1,2] => [2,1,5,3,6,7,4] => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[4,7,6,5,1,3,2] => [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
=> ? = 3 - 1
Description
The number of maximal elements of a poset.
Matching statistic: St000054
(load all 66 compositions to match this statistic)
(load all 66 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> [1] => 1 = 2 - 1
[1,2] => [2,1] => [1,1,0,0]
=> [2,1] => 2 = 3 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,2] => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 3 = 4 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 3 = 4 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4 = 5 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 4 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4 = 5 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 4 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4 = 5 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 4 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4 = 5 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 4 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4 = 5 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4 = 5 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 4 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5 = 6 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 4 = 5 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5 = 6 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 3 = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 4 = 5 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 3 = 4 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5 = 6 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 4 = 5 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5 = 6 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 4 = 5 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5 = 6 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 3 = 4 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5 = 6 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2 = 3 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 3 = 4 - 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 7 - 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 6 - 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 7 - 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 6 - 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 7 - 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 7 - 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => ? = 5 - 1
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 6 - 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => ? = 6 - 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,6,3,2,1] => ? = 5 - 1
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 7 - 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 6 - 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 7 - 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => ? = 5 - 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 6 - 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 5 - 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 7 - 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 6 - 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 7 - 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 6 - 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 7 - 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 7 - 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => ? = 4 - 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 6 - 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => ? = 6 - 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => ? = 4 - 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 7 - 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 6 - 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 7 - 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => ? = 4 - 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 6 - 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => ? = 4 - 1
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 7 - 1
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 7 - 1
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => ? = 5 - 1
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 7 - 1
[1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 7 - 1
[1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => ? = 4 - 1
[1,2,5,6,3,7,4] => [4,7,3,6,5,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3,7,6,5,2,1] => ? = 5 - 1
[1,2,5,6,7,4,3] => [3,4,7,6,5,2,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => ? = 4 - 1
[1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 7 - 1
[1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => ? = 5 - 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000141
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 2 - 2
[1,2] => [1,2] => [1,0,1,0]
=> [2,1] => 1 = 3 - 2
[2,1] => [2,1] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2 = 4 - 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 3 - 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 4 - 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 5 - 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 4 - 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 5 - 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 3 - 2
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 4 - 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 3 - 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 5 - 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 4 - 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 5 - 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 4 - 2
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 5 - 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 5 - 2
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 3 - 2
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2 = 4 - 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 4 - 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 3 - 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 6 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3 = 5 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 6 - 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 4 - 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3 = 5 - 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 4 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 6 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 5 - 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 6 - 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 3 - 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 5 - 2
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 3 - 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4 = 6 - 2
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 4 - 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 6 - 2
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 3 - 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 4 - 2
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 7 - 2
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 8 - 2
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 6 - 2
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 7 - 2
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 6 - 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ? = 8 - 2
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => ? = 7 - 2
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 8 - 2
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 5 - 2
[1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 7 - 2
[1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 5 - 2
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => ? = 8 - 2
[1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 6 - 2
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 8 - 2
[1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 5 - 2
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 6 - 2
[1,2,3,6,7,5,4] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 5 - 2
[1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 7 - 2
[1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 6 - 2
[1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 7 - 2
[1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 5 - 2
[1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 6 - 2
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 5 - 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ? = 8 - 2
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => ? = 7 - 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => ? = 8 - 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 6 - 2
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => ? = 7 - 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 6 - 2
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => ? = 8 - 2
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ? = 7 - 2
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 8 - 2
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 4 - 2
[1,2,4,5,7,3,6] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => ? = 7 - 2
[1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 4 - 2
[1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => ? = 8 - 2
[1,2,4,6,3,7,5] => [1,2,5,3,7,4,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 6 - 2
[1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 8 - 2
[1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 4 - 2
[1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => ? = 6 - 2
[1,2,4,6,7,5,3] => [1,2,7,3,6,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 4 - 2
[1,2,4,7,3,5,6] => [1,2,5,3,6,7,4] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => ? = 7 - 2
[1,2,4,7,3,6,5] => [1,2,5,3,7,6,4] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 6 - 2
[1,2,4,7,5,3,6] => [1,2,6,3,5,7,4] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => ? = 7 - 2
[1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 4 - 2
[1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => ? = 6 - 2
[1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 4 - 2
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => ? = 8 - 2
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => ? = 7 - 2
[1,2,5,3,6,4,7] => [1,2,4,6,3,5,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => ? = 8 - 2
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000066
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> [[1]]
=> 1 = 2 - 1
[1,2] => [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2 = 3 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 4 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 4 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 3 = 4 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 4 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 3 = 4 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 4 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 4 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 5 = 6 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> 4 = 5 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 5 = 6 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 3 = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> 4 = 5 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 3 = 4 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 5 = 6 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> 4 = 5 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 5 = 6 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> 4 = 5 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 5 = 6 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 3 = 4 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 5 = 6 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3 = 4 - 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 7 - 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6 - 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 7 - 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,1,-1,1,-1,1,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 6 - 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 7 - 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 5 - 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 7 - 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,-1,1],[1,-1,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 5 - 1
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6 - 1
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 5 - 1
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,0,1],[0,1,0,-1,0,1,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 6 - 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,0,1],[1,-1,1,-1,0,1,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 5 - 1
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 7 - 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,-1,1],[1,-1,1,0,-1,1,0],[0,1,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 6 - 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 7 - 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 5 - 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,1,0,-1,1,-1,1],[1,0,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 6 - 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[1,-1,1,-1,1,-1,1],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 5 - 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 7 - 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6 - 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 7 - 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,1,-1,1,-1,1,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 6 - 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 7 - 1
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,0,1],[1,-1,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 4 - 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 7 - 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[1,-1,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 4 - 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 6 - 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,0,1],[1,-1,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 4 - 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,0,1],[0,1,0,-1,0,1,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 6 - 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[1,-1,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 4 - 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 7 - 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,-1,1],[1,-1,1,0,-1,1,0],[0,1,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 6 - 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 7 - 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 4 - 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,1,0,-1,1,-1,1],[1,0,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 6 - 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[1,-1,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 4 - 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 8 - 1
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 7 - 1
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000505The biggest entry in the block containing the 1. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000297The number of leading ones in a binary word. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000653The last descent of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000989The number of final rises of a permutation. St001497The position of the largest weak excedence of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000133The "bounce" of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001480The number of simple summands of the module J^2/J^3. St000326The position of the first one in a binary word after appending a 1 at the end. St000391The sum of the positions of the ones in a binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St000682The Grundy value of Welter's game on a binary word. St000840The number of closers smaller than the largest opener in a perfect matching. St000200The row of the unique '1' in the last column of the alternating sign matrix. 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. St001434The number of negative sum pairs of a signed permutation. St001430The number of positive entries in a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation.
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