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Your data matches 81 different statistics following compositions of up to 3 maps.
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Matching statistic: St001086
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [1,2] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,2,3,4] => 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,6,1,2,3,4] => 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,1,2,5,6,7] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [5,1,2,3,4,6] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,2,6,7,3,4,5] => 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5,7] => 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [3,6,1,7,8,2,4,5] => 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,1,2,7,3,4,5] => 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,4,7,8,1,2,5,6] => 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,1,2,3,4,7] => 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => 0
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => 0
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,6,7,8,1,2,4,5] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,1,2,5,6,7,8] => 0
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7] => 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => 0
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St000731
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,3,2,6] => 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [4,3,5,2,1,7,6] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 0
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => 0
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => 0
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 0
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => ? = 2
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => 0
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,2,1,5,6,7] => ? = 0
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,6,5,4,3,2,7,8] => ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ? = 0
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> [5,4,3,2,1,6,7,9,8] => ? = 0
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,1,7,6,5,4,3,8,9] => ? = 0
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,4,3,2,1,7,6,8,9] => ? = 0
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,7,6,5,4,3,8,9] => ? = 0
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St001483
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 1
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[]
=> []
=> []
=> []
=> ? = 0 + 1
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[3,3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[7,7,7]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000594
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000594: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000594: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7}}
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> {{1},{2,3,5},{4},{6}}
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5,6}}
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> {{1,2,5},{3,4},{6,7}}
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> {{1,2},{3,5},{4},{6}}
=> 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5,6,7}}
=> 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 0
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> {{1},{2},{3,5},{4},{6}}
=> 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 0
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4},{5},{6,7}}
=> 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,7}}
=> 0
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6}}
=> 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6}}
=> 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7}}
=> 0
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5},{6},{7}}
=> 0
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 0
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7},{8,9}}
=> ? = 0
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1,2},{3,4,5,6,7},{8},{9}}
=> ? = 0
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> {{1,2,3,4,5},{6,7},{8},{9}}
=> ? = 0
[2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8},{9}}
=> ? = 0
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2},{3,4,5,6,7},{8},{9}}
=> ? = 0
[]
=> []
=> []
=> {}
=> ? = 0
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
[3,3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8},{9}}
=> ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
[6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 0
[7,7,7]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block.
Matching statistic: St000648
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 75%●distinct values known / distinct values provided: 67%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 75%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,5,1,3,6,4] => 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,5,2,3,6,4,7] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,5,2,6,4] => 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 0
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => 0
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => 0
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => 0
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 0
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,4,6,2,7,5] => ? = 2
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,3] => ? = 0
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,1,3,4,5,7,8,6] => ? = 0
[4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 0
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,5] => ? = 0
[4,4,4,3]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 0
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,7,8,5,9] => ? = 0
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,3,2,4,5,6,8,9,7] => ? = 0
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,6,5,8,9,7] => ? = 0
[2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,5,7,8,9,6] => ? = 0
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,1,4,5,6,8,9,7] => ? = 0
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,3] => ? = 0
[3,3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,9,4] => ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,9,2] => ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => ? = 0
[6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,9,3] => ? = 0
[6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,9,10,2] => ? = 0
[7,7,7]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,9,5] => ? = 0
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St001568
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> []
=> []
=> ? = 0 + 1
[2]
=> [1,0,1,0]
=> [1]
=> [1]
=> ? = 0 + 1
[1,1]
=> [1,1,0,0]
=> []
=> []
=> ? = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> ? = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,2,1]
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [3,2]
=> 1 = 0 + 1
[2,2]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [4,3,2,1]
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [4,3,2]
=> 1 = 0 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [4,3,1]
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [4,2,1]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,2,1]
=> 1 = 0 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 0 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,3]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> ? = 0 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 0 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [3,3,2,1]
=> 1 = 0 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [5,4]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> ? = 0 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [2,2,2,1]
=> 1 = 0 + 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [5,2]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1,1]
=> [6,3,2,2,1]
=> ? = 0 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [2,1,1,1]
=> 2 = 1 + 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [6,5,2,1]
=> ? = 0 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [3,2,1]
=> 1 = 0 + 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [3,2]
=> 1 = 0 + 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [5]
=> 1 = 0 + 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,1,1,1]
=> 2 = 1 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> ? = 0 + 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> [6,5]
=> ? = 0 + 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1]
=> 2 = 1 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> [4,3,2,1]
=> 1 = 0 + 1
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> [4,3,2]
=> 1 = 0 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1]
=> ? = 0 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,2]
=> [2,2,1,1,1]
=> 3 = 2 + 1
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [3,2,1]
=> 1 = 0 + 1
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,1]
=> [5,4,3,2]
=> ? = 0 + 1
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [7,6]
=> ? = 0 + 1
[4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1]
=> [4,3,2,1]
=> 1 = 0 + 1
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[4,4,4,3]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [1]
=> ? = 0 + 1
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,1,1,1,1]
=> [8,4,3,2,1]
=> ? = 0 + 1
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? = 0 + 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,1,1]
=> [6,4,3,2,1]
=> ? = 0 + 1
[2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 0 + 1
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 0 + 1
[]
=> []
=> []
=> []
=> ? = 0 + 1
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[3,3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [3,2,1]
=> 1 = 0 + 1
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1]
=> [1]
=> ? = 0 + 1
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1]
=> [1]
=> ? = 0 + 1
[6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [3,2,1]
=> 1 = 0 + 1
[6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1]
=> [1]
=> ? = 0 + 1
[7,7,7]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1]
=> [4,3,2,1]
=> 1 = 0 + 1
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000128
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000128: Binary trees ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000128: Binary trees ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [.,[[.,.],[[[.,[[.,.],.]],.],.]]]
=> ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 0
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> ? = 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> 0
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> 0
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> ? = 0
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ? = 0
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> ? = 0
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> 0
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> ? = 2
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ? = 0
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> ? = 0
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],[.,.]]]]]
=> ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ? = 0
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> ? = 0
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ? = 0
[4,4,4,3]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 0
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> ? = 0
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],[[.,.],.]]]]]
=> ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 0
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[[.,.],[[[[[.,.],.],.],.],.]]]]]
=> ? = 0
[2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> ? = 0
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],[.,[.,.]]]]]]
=> ? = 0
[]
=> []
=> [1,0]
=> [.,.]
=> 0
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ? = 0
[3,3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 0
[6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> ? = 0
[6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ? = 0
[6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]]
=> ? = 0
[7,7,7]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[[.,[.,.]],.]]]}}} in a binary tree.
[[oeis:A159769]] counts binary trees avoiding this pattern.
Matching statistic: St000663
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1] => [1] => 0
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [6,5,4,1,2,3] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5,3,4,2,1] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => [4,3,2,5,6,1] => 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [6,4,5,3,1,2] => 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,3,6,7,2] => [4,5,3,6,7,1,2] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,3,2,6,1] => [4,5,3,2,6,1] => 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [5,6,7,4,1,2,3] => ? = 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => [3,4,5,6,2,1] => 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,5,2,1] => [6,3,4,5,2,1] => 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [6,5,4,3,1,2] => 0
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [5,4,3,2,6,1] => 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,3,2,1] => [4,5,6,3,2,1] => 0
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,3,2] => [4,5,6,7,3,1,2] => ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [7,6,5,4,1,2,3] => ? = 0
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => [6,5,3,4,2,1] => 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [5,4,6,3,2,1] => 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => [3,4,5,6,7,2,1] => ? = 0
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => [7,3,4,5,6,2,1] => ? = 0
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,4,3,2,1] => [5,6,4,3,2,1] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => [5,4,6,3,2,7,1] => ? = 2
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,5,6,7,3,2,1] => ? = 0
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => [5,4,6,7,3,2,1] => ? = 0
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,6,8,7,2,1] => [8,3,4,5,6,7,2,1] => ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,7,6,5,4,3] => [8,7,6,5,4,1,2,3] => ? = 0
[4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => [7,6,5,3,4,2,1] => ? = 0
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,8,3,2,1] => ? = 0
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 0
[4,4,4,3]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => [7,5,6,4,3,2,1] => ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 0
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,8,9,4,3,2] => [5,6,7,8,9,4,3,1,2] => ? = 0
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,5,6,8,9,7,2,1] => ? => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,8,9,4,2,1] => [5,6,7,8,9,3,4,2,1] => ? = 0
[2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,8,9,3,2,1] => [4,5,6,7,8,9,3,2,1] => ? = 0
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,6,9,8,7,2,1] => ? => ? = 0
[]
=> []
=> [] => [] => ? = 0
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,5,4,3,2,1] => [6,7,8,5,4,3,2,1] => ? = 0
[3,3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,9,5,4,3,2,1] => [6,7,8,9,5,4,3,2,1] => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [8,9,7,6,5,4,3,2,1] => ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,6,5,4,3,2,1] => [7,8,6,5,4,3,2,1] => ? = 0
[6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [7,8,9,6,5,4,3,2,1] => [7,8,9,6,5,4,3,2,1] => ? = 0
[6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,8,4,3,2,1] => [5,6,7,8,4,3,2,1] => ? = 0
[6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [9,10,8,7,6,5,4,3,2,1] => [9,10,8,7,6,5,4,3,2,1] => ? = 0
[7,7,7]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,8,9,4,3,2,1] => [5,6,7,8,9,4,3,2,1] => ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ? = 0
Description
The number of right floats of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St001221
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001221: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001221: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 0
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 0
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[4,4,4,3]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[]
=> []
=> []
=> ? = 0
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
[6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,7,7]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
Description
The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module.
Matching statistic: St000366
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => ? = 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [2,3,4,8,1,7,5,6] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ? = 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 0
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? = 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 0
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ? = 0
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ? = 0
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => 0
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 0
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [8,4,1,2,3,7,5,6] => ? = 2
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ? = 0
[3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,8,1,5,3,4,6,7] => ? = 0
[6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,4,6,2,8,5,9,7] => ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,1,2,3,4,5,8,9,6] => ? = 0
[4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ? = 0
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => ? = 0
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ? = 0
[4,4,4,3]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,8,1,3,4,5,6,7] => 0
[3,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,8,5,9,10,7] => ? = 0
[6,6,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0
[3,3,2,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? => ? = 0
[2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,9,6,10,8] => ? = 0
[7,7,3]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [4,1,2,5,7,3,9,6,10,8] => ? = 0
[]
=> []
=> []
=> [1] => 0
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,4,1,9,3,5,6,7,8] => ? = 0
[3,3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,6,3,10,5,7,8,9] => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,10,1,3,4,5,6,7,8,9] => 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,9,1,3,4,5,6,7,8] => 0
[6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,4,1,10,3,5,6,7,8,9] => ? = 0
[6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,6,3,9,5,7,8] => ? = 0
[6,6,6,6,6]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,11,1,3,4,5,6,7,8,9,10] => ? = 0
[7,7,7]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,10,7,9] => ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
The following 71 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001728The number of invisible descents 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. St000671The maximin edge-connectivity for choosing a subgraph. St000842The breadth of a permutation. St000534The number of 2-rises of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St001162The minimum jump of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001868The number of alignments of type NE of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St001344The neighbouring number of a permutation. St001549The number of restricted non-inversions between exceedances. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001487The number of inner corners of a skew partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St000455The second largest eigenvalue of a graph if it is integral. St001960The number of descents of a permutation minus one if its first entry is not one. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001621The number of atoms of a lattice. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001867The number of alignments of type EN of a signed permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001896The number of right descents of a signed permutations. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000264The girth of a graph, which is not a tree. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000068The number of minimal elements in a poset. St000022The number of fixed points of a permutation. St001410The minimal entry of a semistandard tableau. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000295The length of the border of a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001490The number of connected components of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers.
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