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Your data matches 64 different statistics following compositions of up to 3 maps.
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Matching statistic: St001033
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 2
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St000391
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 97%●distinct values known / distinct values provided: 95%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 97%●distinct values known / distinct values provided: 95%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => 01 => 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => 10 => 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => 01 => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,4,1] => 001 => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => 010 => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,4,1,3] => 010 => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 110 => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,4,1,2] => 010 => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => 100 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => 001 => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 101 => 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => 010 => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 010 => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => 001 => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 0001 => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => 0010 => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => 0010 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 1010 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => 0100 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => 0010 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => 0100 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 1010 => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,5,2,1,3] => 0110 => 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 1100 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => 0100 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [5,2,1,3,4] => 1100 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => 1010 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => 0010 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => 0100 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => 0100 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [5,3,1,2,4] => 1100 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => 1000 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => 0001 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => 0010 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,4,5,2] => 1001 => 5
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => 0101 => 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 1010 => 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,5,2,4] => 0010 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => 1010 => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,5,3,2,4] => 0110 => 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => 0100 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => 0100 => 2
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4,7,8] => [5,6,3,1,2,4,7,8] => ? => ? = 5
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,7,1,2,4,8] => [5,6,1,7,3,2,4,8] => ? => ? = 11
[1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,6,1,7,2,3,8] => [4,1,5,6,2,7,3,8] => ? => ? = 11
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,2,3,4] => [5,6,1,7,2,8,3,4] => ? => ? = 12
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,7,1,8,2,3,4] => [1,5,6,2,7,3,8,4] => ? => ? = 15
[]
=> [] => [] => ? => ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,7,8,1,2,3,9] => [4,5,6,1,7,2,8,3,9] => ? => ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,9,2,3,4] => [5,6,7,1,8,2,9,3,4] => ? => ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,5,6,7,8,2,3,4,9] => ? => ? => ? = 12
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,7,8,1,9,2,3,4] => [5,1,6,7,2,8,3,9,4] => ? => ? = 19
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,6,7,8,9,3,4,5] => [6,7,1,8,2,9,3,4,5] => ? => ? = 12
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,8,9,1,2,3,4,10] => [5,6,1,7,2,8,3,9,4,10] => ? => ? = 20
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,6,7,8,9,10,2,3,4,5] => [6,7,1,8,2,9,3,10,4,5] => ? => ? = 20
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000394
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 97%●distinct values known / distinct values provided: 90%
St000394: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 97%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 11
[1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 11
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 11
[1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 12
[]
=> []
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 12
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 19
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 20
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 16
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 12
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 20
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 20
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 24
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 24
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 96%●distinct values known / distinct values provided: 90%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 96%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [[1,3,5],[2],[4]]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,5,2,1,3] => [[1,2],[3,5],[4]]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [5,2,1,3,4] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,4,5,2] => [[1,3,4],[2,5]]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => [[1,2,4],[3,5]]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,5,3,2,4] => [[1,2,5],[3],[4]]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4,7,8] => [5,6,3,1,2,4,7,8] => ?
=> ? = 5
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,7,1,2,4,8] => [5,6,1,7,3,2,4,8] => ?
=> ? = 11
[1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,6,1,7,2,3,8] => [4,1,5,6,2,7,3,8] => ?
=> ? = 11
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,2,3,4] => [5,6,1,7,2,8,3,4] => ?
=> ? = 12
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,7,1,8,2,3,4] => [1,5,6,2,7,3,8,4] => ?
=> ? = 15
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,7,8,1,2,3,9] => [4,5,6,1,7,2,8,3,9] => ?
=> ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,9,2,3,4] => [5,6,7,1,8,2,9,3,4] => ?
=> ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,5,6,7,8,2,3,4,9] => ? => ?
=> ? = 12
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,7,8,1,9,2,3,4] => [5,1,6,7,2,8,3,9,4] => ?
=> ? = 19
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,9,1,2,3,4] => [5,6,1,7,2,8,3,9,4] => ?
=> ? = 20
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,6,7,8,9,3,4,5] => [6,7,1,8,2,9,3,4,5] => ?
=> ? = 12
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 7
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,8,9,1,2,3,4,10] => [5,6,1,7,2,8,3,9,4,10] => ?
=> ? = 20
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,6,7,8,9,10,2,3,4,5] => [6,7,1,8,2,9,3,10,4,5] => ?
=> ? = 20
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,9,10,1,2,3,4] => [5,6,7,1,8,2,9,3,10,4] => ?
=> ? = 24
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,7,8,9,10,1,2,3,4,6] => [7,1,8,2,9,3,10,5,4,6] => [[1,3,5,7,10],[2,4,6,8],[9]]
=> ? = 24
Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000008
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 81% ●values known / values provided: 96%●distinct values known / distinct values provided: 81%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 81% ●values known / values provided: 96%●distinct values known / distinct values provided: 81%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => [2,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => [1,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => [2,1] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,4,1,3] => [2,2] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,1,2] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,4,1,2] => [2,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => [3,1] => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [1,2,1] => 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => [2,2] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [2,2] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => [3,1] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [4,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [3,2] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [3,2] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,2,2] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [3,2] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [2,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,2,2] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,5,2,1,3] => [2,1,2] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,1,3] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [2,3] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [5,2,1,3,4] => [1,1,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => [1,2,2] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [3,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [2,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [5,3,1,2,4] => [1,1,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [4,1] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => [3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,4,5,2] => [1,3,1] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => [2,2,1] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [1,2,2] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,5,2,4] => [3,2] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [1,2,2] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,5,3,2,4] => [2,1,2] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [2,3] => 2
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,7,1,2,4,8] => [5,6,1,7,3,2,4,8] => ? => ? = 11
[]
=> [] => [] => [] => ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,7,8,1,2,3,9] => [4,5,6,1,7,2,8,3,9] => ? => ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,9,2,3,4] => [5,6,7,1,8,2,9,3,4] => ? => ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,5,6,7,8,2,3,4,9] => ? => ? => ? = 12
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,7,8,1,9,2,3,4] => [5,1,6,7,2,8,3,9,4] => ? => ? = 19
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,9,1,2,3,4] => [5,6,1,7,2,8,3,9,4] => [2,2,2,2,1] => ? = 20
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,8,1,2,3,4,9] => [5,1,6,2,7,3,8,4,9] => [1,2,2,2,2] => ? = 16
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,6,7,8,9,3,4,5] => [6,7,1,8,2,9,3,4,5] => ? => ? = 12
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,6,7,8,9,2,3,4,5] => [6,1,7,2,8,3,9,4,5] => [1,2,2,2,2] => ? = 16
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,9,1,2,3,4,5] => [1,6,2,7,3,8,4,9,5] => [2,2,2,2,1] => ? = 20
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [1,2,3,4,5,6,8,7,9] => [7,2] => ? = 7
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,8,9,1,2,3,4,10] => [5,6,1,7,2,8,3,9,4,10] => [2,2,2,2,2] => ? = 20
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,6,7,8,9,10,2,3,4,5] => [6,7,1,8,2,9,3,10,4,5] => [2,2,2,2,2] => ? = 20
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,9,10,1,2,3,4] => [5,6,7,1,8,2,9,3,10,4] => [3,2,2,2,1] => ? = 24
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,9,10,1,2,3,4,5] => [6,1,7,2,8,3,9,4,10,5] => [1,2,2,2,2,1] => ? = 25
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,7,8,9,10,1,2,3,4,6] => [7,1,8,2,9,3,10,5,4,6] => [1,2,2,2,1,2] => ? = 24
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000728
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000728: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 89%●distinct values known / distinct values provided: 62%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000728: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 89%●distinct values known / distinct values provided: 62%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => {{1,2,3}}
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,2,3,4}}
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,4,5},{2},{3}}
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,3,5},{2},{4}}
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,3,4},{2},{5}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,3,4,5},{2}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,2,5},{3},{4}}
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,2,4},{3},{5}}
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,2,4,5},{3}}
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,4,5},{2,3}}
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4,7,8] => [1,6,3,5,4,2,7,8] => {{1},{2,6},{3},{4,5},{7},{8}}
=> ? = 5
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,7,1,8,2,4] => [8,7,3,5,4,6,2,1] => {{1,8},{2,7},{3},{4,5},{6}}
=> ? = 13
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,7,1,2,4,8] => [7,6,3,5,4,2,1,8] => {{1,7},{2,6},{3},{4,5},{8}}
=> ? = 11
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,6,7,8,1,2,4,5] => [8,7,3,6,4,5,2,1] => {{1,8},{2,7},{3},{4,5,6}}
=> ? = 14
[1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3,7,8] => [1,6,5,4,3,2,7,8] => {{1},{2,6},{3,5},{4},{7},{8}}
=> ? = 6
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,6,7,8,2,3,5] => [1,8,7,4,6,5,3,2] => {{1},{2,8},{3,7},{4},{5,6}}
=> ? = 11
[1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,6,1,7,2,3,8] => [7,6,4,3,5,2,1,8] => {{1,7},{2,6},{3,4},{5},{8}}
=> ? = 11
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,8,1,2,3] => [8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? = 15
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3,8] => [7,6,5,4,3,2,1,8] => {{1,7},{2,6},{3,5},{4},{8}}
=> ? = 12
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,6,8,1,2,3,7] => [8,6,5,4,3,2,1,7] => {{1,7,8},{2,6},{3,5},{4}}
=> ? = 13
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,6,7,8,1,2,3,5] => [8,7,6,4,3,5,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> ? = 15
[1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,2,5,6,7,3,4,8] => ? => ?
=> ? = 6
[1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [1,5,6,7,2,8,3,4] => [1,8,7,5,4,6,3,2] => {{1},{2,8},{3,7},{4,5},{6}}
=> ? = 11
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,2,3,4] => [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
=> ? = 12
[1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,5,6,7,2,3,4,8] => [1,7,6,5,4,3,2,8] => {{1},{2,7},{3,6},{4,5},{8}}
=> ? = 9
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,5,6,2,3,4,7,8] => [1,6,5,3,4,2,7,8] => ?
=> ? = 6
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [5,1,6,7,8,2,3,4] => [8,1,7,6,5,4,3,2] => {{1,2,8},{3,7},{4,6},{5}}
=> ? = 13
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [5,6,1,7,2,3,4,8] => [7,6,2,5,4,3,1,8] => {{1,7},{2,3,6},{4,5},{8}}
=> ? = 11
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,7,1,8,2,3,4] => [8,7,6,3,5,4,2,1] => {{1,8},{2,7},{3,4,6},{5}}
=> ? = 15
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 16
[1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,1,2,3,4,8] => [7,6,5,3,4,2,1,8] => {{1,7},{2,6},{3,4,5},{8}}
=> ? = 12
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [1,2,6,7,8,3,4,5] => [1,2,8,7,6,5,4,3] => {{1},{2},{3,8},{4,7},{5,6}}
=> ? = 9
[1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,2,6,7,3,4,5,8] => [1,2,7,6,4,5,3,8] => ?
=> ? = 6
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,6,7,8,2,3,4,5] => [1,8,7,6,4,5,3,2] => {{1},{2,8},{3,7},{4,5,6}}
=> ? = 12
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [6,7,1,8,2,3,4,5] => [8,7,2,6,4,5,3,1] => {{1,8},{2,3,7},{4,5,6}}
=> ? = 14
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7,8] => {{1,2,3,4,5,6},{7},{8}}
=> ? = 5
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 1
[]
=> [] => [] => {}
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,7,8,1,2,3,9] => [8,7,6,4,5,3,2,1,9] => {{1,8},{2,7},{3,6},{4},{5},{9}}
=> ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,9,2,3,4] => [1,9,8,7,5,6,4,3,2] => {{1},{2,9},{3,8},{4,7},{5},{6}}
=> ? = 15
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,5,6,7,8,2,3,4,9] => ? => ?
=> ? = 12
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,7,8,1,9,2,3,4] => [9,8,7,5,4,6,3,2,1] => {{1,9},{2,8},{3,7},{4,5},{6}}
=> ? = 19
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,9,1,2,3,4] => [9,8,7,6,5,4,3,2,1] => {{1,9},{2,8},{3,7},{4,6},{5}}
=> ? = 20
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,8,1,2,3,4,9] => [8,7,6,5,4,3,2,1,9] => {{1,8},{2,7},{3,6},{4,5},{9}}
=> ? = 16
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,6,7,8,9,3,4,5] => ? => ?
=> ? = 12
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,6,7,8,9,2,3,4,5] => [1,9,8,7,6,5,4,3,2] => {{1},{2,9},{3,8},{4,7},{5,6}}
=> ? = 16
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,9,1,2,3,4,5] => [9,8,7,6,4,5,3,2,1] => {{1,9},{2,8},{3,7},{4,5,6}}
=> ? = 20
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7,9] => {{1,2,3,4,5,6,7,8},{9}}
=> ? = 7
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 8
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,8,9,1,2,3,4,10] => [9,8,7,6,5,4,3,2,1,10] => {{1,9},{2,8},{3,7},{4,6},{5},{10}}
=> ? = 20
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,6,7,8,9,10,2,3,4,5] => [1,10,9,8,7,6,5,4,3,2] => {{1},{2,10},{3,9},{4,8},{5,7},{6}}
=> ? = 20
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 9
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,9,10,1,2,3,4] => [10,9,8,7,5,6,4,3,2,1] => {{1,10},{2,9},{3,8},{4,7},{5},{6}}
=> ? = 24
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,9,10,1,2,3,4,5] => [10,9,8,7,6,5,4,3,2,1] => {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> ? = 25
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,7,8,9,10,1,2,3,4,6] => [10,9,8,7,5,4,6,3,2,1] => ?
=> ? = 24
Description
The dimension of a set partition.
This is the sum of the lengths of the arcs of a set partition. Equivalently, one obtains that this is the sum of the maximal entries of the blocks minus the sum of the minimal entries of the blocks.
A slightly shifted definition of the dimension is [[St000572]].
Matching statistic: St001579
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001579: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
St001579: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ? = 8
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 7
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 8
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ? = 8
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ? = 9
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 7
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 8
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,1,3,7,4] => ? = 8
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => ? = 9
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ? = 10
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? = 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 8
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,1,6,2,7] => ? = 7
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 10
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 8
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => ? = 9
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,6,7,1,2,5] => ? = 10
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,6] => ? = 6
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [3,5,1,2,6,4,7] => ? = 6
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => ? = 9
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,5,1,6,2,4,7] => ? = 7
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => ? = 9
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => ? = 10
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,5,6,7,1,2,4] => ? = 11
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,1,2,4,7] => ? = 8
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,7,4,6] => ? = 7
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,7,2,4,6] => ? = 8
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,2,4,6,7,5] => ? = 4
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,2,6,4,5,7] => ? = 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,6,2,7,4,5] => ? = 7
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,6,1,2,7,4,5] => ? = 8
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? = 10
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 4
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,4,6,7,2,3,5] => ? = 8
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? = 6
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ? = 7
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => ? = 8
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 9
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,1,5,6,2,3,7] => ? = 7
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => ? = 8
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 9
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 10
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,1,6,2,3,7] => ? = 8
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ? = 10
Description
The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation.
This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by
$$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$
Matching statistic: St000809
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000809: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
St000809: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
Values
[1,0]
=> [1] => ? = 0
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ? = 8
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 7
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 8
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ? = 8
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ? = 9
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 7
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 8
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,1,3,7,4] => ? = 8
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => ? = 9
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ? = 10
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? = 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 8
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,1,6,2,7] => ? = 7
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 10
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 8
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => ? = 9
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,6,7,1,2,5] => ? = 10
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,6] => ? = 6
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [3,5,1,2,6,4,7] => ? = 6
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => ? = 9
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,5,1,6,2,4,7] => ? = 7
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => ? = 9
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => ? = 10
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,5,6,7,1,2,4] => ? = 11
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,1,2,4,7] => ? = 8
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,7,4,6] => ? = 7
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,7,2,4,6] => ? = 8
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,2,4,6,7,5] => ? = 4
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,2,6,4,5,7] => ? = 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,6,2,7,4,5] => ? = 7
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,6,1,2,7,4,5] => ? = 8
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? = 10
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 4
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,4,6,7,2,3,5] => ? = 8
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? = 6
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ? = 7
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => ? = 8
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 9
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,1,5,6,2,3,7] => ? = 7
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => ? = 8
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 9
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 10
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,1,6,2,3,7] => ? = 8
Description
The reduced reflection length of the permutation.
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].
Matching statistic: St000957
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
St000957: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
Values
[1,0]
=> [1] => ? = 0
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ? = 8
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 7
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 8
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ? = 8
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ? = 9
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 7
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 8
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,1,3,7,4] => ? = 8
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => ? = 9
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ? = 10
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? = 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 8
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,1,6,2,7] => ? = 7
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 10
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 8
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => ? = 9
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,6,7,1,2,5] => ? = 10
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,6] => ? = 6
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [3,5,1,2,6,4,7] => ? = 6
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => ? = 9
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,5,1,6,2,4,7] => ? = 7
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => ? = 9
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => ? = 10
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,5,6,7,1,2,4] => ? = 11
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,1,2,4,7] => ? = 8
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,7,4,6] => ? = 7
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,7,2,4,6] => ? = 8
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,2,4,6,7,5] => ? = 4
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,2,6,4,5,7] => ? = 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,6,2,7,4,5] => ? = 7
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,6,1,2,7,4,5] => ? = 8
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? = 10
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 4
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,4,6,7,2,3,5] => ? = 8
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? = 6
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ? = 7
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => ? = 8
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 9
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,1,5,6,2,3,7] => ? = 7
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => ? = 8
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 9
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 10
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,1,6,2,3,7] => ? = 8
Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Matching statistic: St001076
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001076: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
St001076: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 69%●distinct values known / distinct values provided: 48%
Values
[1,0]
=> [1] => ? = 0
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 4
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ? = 8
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 7
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 8
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ? = 8
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ? = 9
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 7
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 8
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,1,3,7,4] => ? = 8
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => ? = 9
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ? = 10
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? = 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 8
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,1,6,2,7] => ? = 7
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 10
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 8
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => ? = 9
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,6,7,1,2,5] => ? = 10
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,6] => ? = 6
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [3,5,1,2,6,4,7] => ? = 6
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => ? = 9
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,5,1,6,2,4,7] => ? = 7
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => ? = 9
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => ? = 10
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,5,6,7,1,2,4] => ? = 11
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,1,2,4,7] => ? = 8
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,7,4,6] => ? = 7
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,7,2,4,6] => ? = 8
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,2,4,6,7,5] => ? = 4
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,2,6,4,5,7] => ? = 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,6,2,7,4,5] => ? = 7
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,6,1,2,7,4,5] => ? = 8
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? = 10
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 4
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,4,6,7,2,3,5] => ? = 8
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? = 6
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ? = 7
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => ? = 8
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 9
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,1,5,6,2,3,7] => ? = 7
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => ? = 8
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 9
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 10
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,1,6,2,3,7] => ? = 8
Description
The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12).
In symbols, for a permutation $\pi$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 1 \leq i_1,\ldots,i_k \leq n\},$$
where $\tau_a = (a,a+1)$ for $1 \leq a \leq n$ and $n+1$ is identified with $1$.
Put differently, this is the number of cyclically simple transpositions needed to sort a permutation.
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001726The number of visible inversions of a permutation. St000833The comajor index of a permutation. St000795The mad of a permutation. St000018The number of inversions of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St000496The rcs statistic of a set partition. St000246The number of non-inversions of a permutation. St000491The number of inversions of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000572The dimension exponent of a set partition. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000065The number of entries equal to -1 in an alternating sign matrix. St000004The major index of a permutation. St001428The number of B-inversions of a signed permutation. St001843The Z-index of a set partition. St001311The cyclomatic number of a graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000224The sorting index of a permutation. St001869The maximum cut size of a graph. St000305The inverse major index of a permutation. St000223The number of nestings in the permutation. St001645The pebbling number of a connected graph. St000359The number of occurrences of the pattern 23-1. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001083The number of boxed occurrences of 132 in a permutation. St000355The number of occurrences of the pattern 21-3. St001727The number of invisible inversions of a permutation. St000358The number of occurrences of the pattern 31-2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000039The number of crossings of a permutation. St000095The number of triangles of a graph. St001511The minimal number of transpositions needed to sort a permutation in either direction. St000450The number of edges minus the number of vertices plus 2 of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St000327The number of cover relations in a poset. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001862The number of crossings of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000516The number of stretching pairs of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001433The flag major index of a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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