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Your data matches 20 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000438
(load all 166 compositions to match this statistic)
(load all 166 compositions to match this statistic)
St000438: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 3
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0]
=> 5
[1,1,0,1,0,0]
=> 4
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 7
[1,0,1,0,1,1,0,0]
=> 6
[1,0,1,1,0,0,1,0]
=> 7
[1,0,1,1,0,1,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 7
[1,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,0,1,0]
=> 7
[1,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> 7
[1,1,1,0,0,1,0,0]
=> 6
[1,1,1,0,1,0,0,0]
=> 5
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 9
[1,0,1,0,1,0,1,1,0,0]
=> 8
[1,0,1,0,1,1,0,0,1,0]
=> 9
[1,0,1,0,1,1,0,1,0,0]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> 9
[1,0,1,1,0,0,1,1,0,0]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> 9
[1,0,1,1,1,0,0,1,0,0]
=> 8
[1,0,1,1,1,0,1,0,0,0]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> 9
[1,1,0,0,1,0,1,1,0,0]
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> 7
[1,1,0,1,0,0,1,0,1,0]
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> 9
Description
The position of the last up step in a Dyck path.
Matching statistic: St000874
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000874: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000874: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 7 - 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 7 - 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 7 - 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 7 - 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 7 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 7 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 7 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 7 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 7 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 7 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 7 = 9 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 7 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 7 = 9 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 13 - 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 13 - 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 13 - 2
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 14 - 2
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 12 - 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 15 - 2
Description
The position of the last double rise in a Dyck path.
If the Dyck path has no double rises, this statistic is $0$.
Matching statistic: St000734
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 3
[1,1,0,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 5
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 5
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 7
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 7
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 7
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 6
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 7
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 6
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 6
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 9
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 8
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 9
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 9
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 9
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 8
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 9
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 7
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> 9
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,9,11,12,15],[2,4,6,8,10,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,7,9,10,13,15],[2,4,6,8,11,12,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,9,10,13,14],[2,4,6,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,9,11,12,15],[2,4,6,8,10,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,9,10,13,14],[2,4,6,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,7,9,10,11,15],[2,4,6,8,12,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,9,10,13,14],[2,4,6,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,3,5,7,8,11,13,15],[2,4,6,9,10,12,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,5,7,8,11,13,14],[2,4,6,9,10,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11,12,15],[2,4,6,9,10,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,5,7,8,11,13,14],[2,4,6,9,10,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,11,12,13],[2,4,6,9,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,7,9,10,13,15],[2,4,6,8,11,12,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,9,10,13,14],[2,4,6,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11,12,15],[2,4,6,9,10,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,9,10,13,14],[2,4,6,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,11,12,13],[2,4,6,9,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,7,9,10,11,15],[2,4,6,8,12,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,9,10,13,14],[2,4,6,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,11,12,13],[2,4,6,9,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,3,5,7,8,9,13,15],[2,4,6,10,11,12,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,5,7,8,9,13,14],[2,4,6,10,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11,12,15],[2,4,6,9,10,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,5,7,8,9,13,14],[2,4,6,10,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,11,12,13],[2,4,6,9,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,7,9,10,11,15],[2,4,6,8,12,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,5,7,8,9,13,14],[2,4,6,10,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,11,12,13],[2,4,6,9,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,5,7,8,9,10,15],[2,4,6,11,12,13,14,16]]
=> ? = 15
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,5,7,8,9,13,14],[2,4,6,10,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,11,12,13],[2,4,6,9,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,7,8,9,10,11],[2,4,6,12,13,14,15,16]]
=> ? = 11
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,6,9,11,13,15],[2,4,7,8,10,12,14,16]]
=> ? = 15
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,6,9,11,13,14],[2,4,7,8,10,12,15,16]]
=> ? = 14
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9,11,12,15],[2,4,7,8,10,13,14,16]]
=> ? = 15
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,6,9,11,13,14],[2,4,7,8,10,12,15,16]]
=> ? = 14
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,11,12,13],[2,4,7,8,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,6,9,10,13,15],[2,4,7,8,11,12,14,16]]
=> ? = 15
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,6,9,10,13,14],[2,4,7,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9,11,12,15],[2,4,7,8,10,13,14,16]]
=> ? = 15
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,6,9,10,13,14],[2,4,7,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,11,12,13],[2,4,7,8,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,6,9,10,11,15],[2,4,7,8,12,13,14,16]]
=> ? = 15
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,6,9,10,13,14],[2,4,7,8,11,12,15,16]]
=> ? = 14
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,11,12,13],[2,4,7,8,10,14,15,16]]
=> ? = 13
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,6,9,10,11,12],[2,4,7,8,13,14,15,16]]
=> ? = 12
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,3,5,7,8,11,13,15],[2,4,6,9,10,12,14,16]]
=> ? = 15
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,5,7,8,11,13,14],[2,4,6,9,10,12,15,16]]
=> ? = 14
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11,12,15],[2,4,6,9,10,13,14,16]]
=> ? = 15
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,5,7,8,11,13,14],[2,4,6,9,10,12,15,16]]
=> ? = 14
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000641
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000641: Posets ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 86%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000641: Posets ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 86%
Values
[1,0,1,0]
=> [[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 3
[1,1,0,0]
=> [.,[.,.]]
=> [2,1] => ([],2)
=> 2
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 5
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> 4
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 5
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 4
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 7
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 6
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 7
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 5
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 7
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 6
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 7
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 6
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> 5
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 6
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 5
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 9
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 8
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 9
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 8
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 9
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 7
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 13
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 12
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> [6,1,2,3,4,5,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 13
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 12
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[.,[.,.]]]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 11
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 13
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 12
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[[[.,.],.],.],[[.,.],[.,.]]]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 11
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[[[.,.],.],.],[.,[[.,.],.]]]
=> [6,7,5,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 11
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],[.,.]],.],.],.]
=> [4,1,2,3,5,6,7] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 13
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 12
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 13
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],.]]
=> [6,7,4,1,2,3,5] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ? = 12
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 11
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> [4,5,1,2,3,6,7] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 13
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[[[.,.],.],[[.,.],.]],[.,.]]
=> [7,4,5,1,2,3,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 12
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 13
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 12
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[[.,.],.],[[[.,.],.],[.,.]]]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 11
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[[.,.],.],[[.,.],[.,.]]],.]
=> [6,4,5,1,2,3,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 13
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],.],[[[.,.],[.,.]],.]]
=> [6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 12
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[[.,.],.],[[.,.],[[.,.],.]]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 11
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[[.,.],.],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 10
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[[[.,.],.],[.,[[.,.],.]]],.]
=> [5,6,4,1,2,3,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[[.,.],.],[[.,[[.,.],.]],.]]
=> [5,6,4,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 12
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[[.,.],.],[.,[[[.,.],.],.]]]
=> [5,6,7,4,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 11
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 10
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[[.,.],.],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 10
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 13
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 12
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],[.,.]],.],[[.,.],.]]
=> [6,7,3,1,2,4,5] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 12
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],[.,[.,.]]]
=> [7,6,3,1,2,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [[[[.,.],[.,.]],[[.,.],.]],.]
=> [5,6,3,1,2,4,7] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 13
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [[[.,.],[.,.]],[[[.,.],.],.]]
=> [5,6,7,3,1,2,4] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 12
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => ([(1,6),(2,4),(3,5),(5,6)],7)
=> ? = 11
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [[[.,.],[.,.]],[.,[[.,.],.]]]
=> [6,7,5,3,1,2,4] => ([(1,6),(2,4),(3,5),(5,6)],7)
=> ? = 11
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ([(3,6),(4,5),(5,6)],7)
=> ? = 10
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 13
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 12
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [[[[.,.],[[.,.],.]],[.,.]],.]
=> [6,3,4,1,2,5,7] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 13
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [[[.,.],[[.,.],.]],[[.,.],.]]
=> [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 12
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],[.,[.,.]]]
=> [7,6,3,4,1,2,5] => ([(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 11
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [[[[.,.],[[[.,.],.],.]],.],.]
=> [3,4,5,1,2,6,7] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 13
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 12
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [[[.,.],[[[[.,.],.],.],.]],.]
=> [3,4,5,6,1,2,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 13
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 12
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [[.,.],[[[[.,.],.],.],[.,.]]]
=> [7,3,4,5,6,1,2] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 11
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[[.,.],.],[.,.]]],.]
=> [6,3,4,5,1,2,7] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 13
Description
The number of non-empty boolean intervals in a poset.
Matching statistic: St000507
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 79%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 79%
Values
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> 3
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 5
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 4
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 5
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 4
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 7
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 6
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 7
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 6
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 5
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 7
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 6
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 7
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 6
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 5
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 7
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 6
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> 8
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> 8
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> 7
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,6,9],[7,8,10,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,7,9],[6,8,10,11,12,13,14]]
=> ? = 11
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[1,2,3,4,5,6,10],[7,8,9,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[1,2,3,4,5,7,10],[6,8,9,11,12,13,14]]
=> ? = 11
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[1,2,3,4,5,6,10],[7,8,9,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [[1,2,3,4,5,8,10],[6,7,9,11,12,13,14]]
=> ? = 11
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[1,2,3,4,6,8,10],[5,7,9,11,12,13,14]]
=> ? = 10
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,6,9],[7,8,10,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,7,9],[6,8,10,11,12,13,14]]
=> ? = 11
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,6,11],[7,8,9,10,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,7,11],[6,8,9,10,12,13,14]]
=> ? = 11
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,6,11],[7,8,9,10,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [[1,2,3,4,5,8,11],[6,7,9,10,12,13,14]]
=> ? = 11
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [[1,2,3,4,6,8,11],[5,7,9,10,12,13,14]]
=> ? = 10
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,6,11],[7,8,9,10,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,7,11],[6,8,9,10,12,13,14]]
=> ? = 11
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,6,11],[7,8,9,10,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [[1,2,3,4,5,9,11],[6,7,8,10,12,13,14]]
=> ? = 11
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [[1,2,3,4,6,9,11],[5,7,8,10,12,13,14]]
=> ? = 10
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,6,11],[7,8,9,10,12,13,14]]
=> ? = 12
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [[1,2,3,4,5,9,11],[6,7,8,10,12,13,14]]
=> ? = 11
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [[1,2,3,4,7,9,11],[5,6,8,10,12,13,14]]
=> ? = 10
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7,9,11],[4,6,8,10,12,13,14]]
=> ? = 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 12
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,6,9],[7,8,10,11,12,13,14]]
=> ? = 12
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,7,9],[6,8,10,11,12,13,14]]
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,8],[7,9,10,11,12,13,14]]
=> ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000725
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 3
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 5
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 4
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => 5
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 4
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => 7
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => 6
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => 7
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => 6
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => 5
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => 7
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => 6
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => 7
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => 6
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => 5
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => 7
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => 6
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => 5
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => 9
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [2,4,6,9,10,1,3,5,7,8] => 8
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [2,4,7,8,10,1,3,5,6,9] => 9
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [2,4,7,9,10,1,3,5,6,8] => 8
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [2,4,8,9,10,1,3,5,6,7] => 7
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [2,5,6,8,10,1,3,4,7,9] => 9
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [2,5,6,9,10,1,3,4,7,8] => 8
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [2,5,7,8,10,1,3,4,6,9] => 9
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,5,7,9,10,1,3,4,6,8] => 8
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [2,5,8,9,10,1,3,4,6,7] => 7
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [2,6,7,8,10,1,3,4,5,9] => 9
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,6,7,9,10,1,3,4,5,8] => 8
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [2,6,8,9,10,1,3,4,5,7] => 7
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [2,7,8,9,10,1,3,4,5,6] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [3,4,6,8,10,1,2,5,7,9] => 9
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [3,4,6,9,10,1,2,5,7,8] => 8
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [3,4,7,8,10,1,2,5,6,9] => 9
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [3,4,7,9,10,1,2,5,6,8] => 8
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [3,4,8,9,10,1,2,5,6,7] => 7
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [3,5,6,8,10,1,2,4,7,9] => 9
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [3,5,6,9,10,1,2,4,7,8] => 8
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [3,5,7,8,10,1,2,4,6,9] => 9
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [3,5,7,9,10,1,2,4,6,8] => 8
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [3,5,8,9,10,1,2,4,6,7] => 7
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [3,6,7,8,10,1,2,4,5,9] => 9
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [3,6,7,9,10,1,2,4,5,8] => 8
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [3,6,8,9,10,1,2,4,5,7] => 7
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [3,7,8,9,10,1,2,4,5,6] => 6
[1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [4,5,6,8,10,1,2,3,7,9] => 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> [2,4,6,8,10,12,14,1,3,5,7,9,11,13] => ? = 13
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> [2,4,6,8,10,13,14,1,3,5,7,9,11,12] => ? = 12
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,9,10,13],[2,4,6,8,11,12,14]]
=> [2,4,6,8,11,12,14,1,3,5,7,9,10,13] => ? = 13
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,9,10,12],[2,4,6,8,11,13,14]]
=> [2,4,6,8,11,13,14,1,3,5,7,9,10,12] => ? = 12
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> [2,4,6,8,12,13,14,1,3,5,7,9,10,11] => ? = 11
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,7,8,11,13],[2,4,6,9,10,12,14]]
=> [2,4,6,9,10,12,14,1,3,5,7,8,11,13] => ? = 13
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,8,11,12],[2,4,6,9,10,13,14]]
=> [2,4,6,9,10,13,14,1,3,5,7,8,11,12] => ? = 12
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,3,5,7,8,10,13],[2,4,6,9,11,12,14]]
=> [2,4,6,9,11,12,14,1,3,5,7,8,10,13] => ? = 13
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12],[2,4,6,9,11,13,14]]
=> [2,4,6,9,11,13,14,1,3,5,7,8,10,12] => ? = 12
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,3,5,7,8,10,11],[2,4,6,9,12,13,14]]
=> [2,4,6,9,12,13,14,1,3,5,7,8,10,11] => ? = 11
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,7,8,9,13],[2,4,6,10,11,12,14]]
=> [2,4,6,10,11,12,14,1,3,5,7,8,9,13] => ? = 13
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,3,5,7,8,9,12],[2,4,6,10,11,13,14]]
=> [2,4,6,10,11,13,14,1,3,5,7,8,9,12] => ? = 12
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,3,5,7,8,9,11],[2,4,6,10,12,13,14]]
=> [2,4,6,10,12,13,14,1,3,5,7,8,9,11] => ? = 11
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> [2,4,6,11,12,13,14,1,3,5,7,8,9,10] => ? = 10
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,3,5,6,9,11,13],[2,4,7,8,10,12,14]]
=> [2,4,7,8,10,12,14,1,3,5,6,9,11,13] => ? = 13
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,5,6,9,11,12],[2,4,7,8,10,13,14]]
=> [2,4,7,8,10,13,14,1,3,5,6,9,11,12] => ? = 12
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,3,5,6,9,10,13],[2,4,7,8,11,12,14]]
=> [2,4,7,8,11,12,14,1,3,5,6,9,10,13] => ? = 13
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,3,5,6,9,10,12],[2,4,7,8,11,13,14]]
=> [2,4,7,8,11,13,14,1,3,5,6,9,10,12] => ? = 12
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,10,11],[2,4,7,8,12,13,14]]
=> [2,4,7,8,12,13,14,1,3,5,6,9,10,11] => ? = 11
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,3,5,6,8,11,13],[2,4,7,9,10,12,14]]
=> [2,4,7,9,10,12,14,1,3,5,6,8,11,13] => ? = 13
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,3,5,6,8,11,12],[2,4,7,9,10,13,14]]
=> [2,4,7,9,10,13,14,1,3,5,6,8,11,12] => ? = 12
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,3,5,6,8,10,13],[2,4,7,9,11,12,14]]
=> [2,4,7,9,11,12,14,1,3,5,6,8,10,13] => ? = 13
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10,12],[2,4,7,9,11,13,14]]
=> [2,4,7,9,11,13,14,1,3,5,6,8,10,12] => ? = 12
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[1,3,5,6,8,10,11],[2,4,7,9,12,13,14]]
=> [2,4,7,9,12,13,14,1,3,5,6,8,10,11] => ? = 11
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,3,5,6,8,9,13],[2,4,7,10,11,12,14]]
=> [2,4,7,10,11,12,14,1,3,5,6,8,9,13] => ? = 13
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[1,3,5,6,8,9,12],[2,4,7,10,11,13,14]]
=> [2,4,7,10,11,13,14,1,3,5,6,8,9,12] => ? = 12
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,3,5,6,8,9,11],[2,4,7,10,12,13,14]]
=> [2,4,7,10,12,13,14,1,3,5,6,8,9,11] => ? = 11
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,5,6,8,9,10],[2,4,7,11,12,13,14]]
=> [2,4,7,11,12,13,14,1,3,5,6,8,9,10] => ? = 10
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,3,5,6,7,11,13],[2,4,8,9,10,12,14]]
=> [2,4,8,9,10,12,14,1,3,5,6,7,11,13] => ? = 13
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,5,6,7,11,12],[2,4,8,9,10,13,14]]
=> [2,4,8,9,10,13,14,1,3,5,6,7,11,12] => ? = 12
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,3,5,6,7,10,13],[2,4,8,9,11,12,14]]
=> [2,4,8,9,11,12,14,1,3,5,6,7,10,13] => ? = 13
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,5,6,7,10,12],[2,4,8,9,11,13,14]]
=> [2,4,8,9,11,13,14,1,3,5,6,7,10,12] => ? = 12
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,3,5,6,7,10,11],[2,4,8,9,12,13,14]]
=> [2,4,8,9,12,13,14,1,3,5,6,7,10,11] => ? = 11
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,3,5,6,7,9,13],[2,4,8,10,11,12,14]]
=> [2,4,8,10,11,12,14,1,3,5,6,7,9,13] => ? = 13
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,3,5,6,7,9,12],[2,4,8,10,11,13,14]]
=> [2,4,8,10,11,13,14,1,3,5,6,7,9,12] => ? = 12
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,3,5,6,7,9,11],[2,4,8,10,12,13,14]]
=> [2,4,8,10,12,13,14,1,3,5,6,7,9,11] => ? = 11
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,5,6,7,9,10],[2,4,8,11,12,13,14]]
=> [2,4,8,11,12,13,14,1,3,5,6,7,9,10] => ? = 10
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,5,6,7,8,13],[2,4,9,10,11,12,14]]
=> [2,4,9,10,11,12,14,1,3,5,6,7,8,13] => ? = 13
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,5,6,7,8,12],[2,4,9,10,11,13,14]]
=> [2,4,9,10,11,13,14,1,3,5,6,7,8,12] => ? = 12
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,5,6,7,8,11],[2,4,9,10,12,13,14]]
=> [2,4,9,10,12,13,14,1,3,5,6,7,8,11] => ? = 11
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,5,6,7,8,10],[2,4,9,11,12,13,14]]
=> [2,4,9,11,12,13,14,1,3,5,6,7,8,10] => ? = 10
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> [2,4,10,11,12,13,14,1,3,5,6,7,8,9] => ? = 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,3,4,7,9,11,13],[2,5,6,8,10,12,14]]
=> [2,5,6,8,10,12,14,1,3,4,7,9,11,13] => ? = 13
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,3,4,7,9,11,12],[2,5,6,8,10,13,14]]
=> [2,5,6,8,10,13,14,1,3,4,7,9,11,12] => ? = 12
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[1,3,4,7,9,10,13],[2,5,6,8,11,12,14]]
=> [2,5,6,8,11,12,14,1,3,4,7,9,10,13] => ? = 13
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[1,3,4,7,9,10,12],[2,5,6,8,11,13,14]]
=> [2,5,6,8,11,13,14,1,3,4,7,9,10,12] => ? = 12
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[1,3,4,7,9,10,11],[2,5,6,8,12,13,14]]
=> [2,5,6,8,12,13,14,1,3,4,7,9,10,11] => ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,8,11,13],[2,5,6,9,10,12,14]]
=> [2,5,6,9,10,12,14,1,3,4,7,8,11,13] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8,11,12],[2,5,6,9,10,13,14]]
=> [2,5,6,9,10,13,14,1,3,4,7,8,11,12] => ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,7,8,10,13],[2,5,6,9,11,12,14]]
=> [2,5,6,9,11,12,14,1,3,4,7,8,10,13] => ? = 13
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000839
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> {{1,2},{3,4}}
=> 3
[1,1,0,0]
=> [(1,4),(2,3)]
=> {{1,4},{2,3}}
=> 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> 5
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> 4
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> 5
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> 4
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> 3
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 7
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> {{1,2},{3,4},{5,8},{6,7}}
=> 6
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> 7
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> 6
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> {{1,2},{3,8},{4,7},{5,6}}
=> 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> 7
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> {{1,4},{2,3},{5,8},{6,7}}
=> 6
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> 7
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> 6
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> {{1,8},{2,3},{4,7},{5,6}}
=> 5
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> 7
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> 6
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> {{1,8},{2,7},{3,4},{5,6}}
=> 5
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> 9
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9}}
=> 8
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10}}
=> 9
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9}}
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> {{1,2},{3,4},{5,10},{6,9},{7,8}}
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> {{1,2},{3,6},{4,5},{7,8},{9,10}}
=> 9
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> {{1,2},{3,6},{4,5},{7,10},{8,9}}
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> {{1,2},{3,8},{4,5},{6,7},{9,10}}
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> {{1,2},{3,10},{4,5},{6,9},{7,8}}
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> {{1,2},{3,8},{4,7},{5,6},{9,10}}
=> 9
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> 8
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> {{1,2},{3,10},{4,9},{5,6},{7,8}}
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> {{1,2},{3,10},{4,9},{5,8},{6,7}}
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> {{1,4},{2,3},{5,6},{7,8},{9,10}}
=> 9
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> {{1,4},{2,3},{5,6},{7,10},{8,9}}
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> {{1,4},{2,3},{5,8},{6,7},{9,10}}
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> {{1,4},{2,3},{5,10},{6,7},{8,9}}
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> {{1,4},{2,3},{5,10},{6,9},{7,8}}
=> 7
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> {{1,6},{2,3},{4,5},{7,8},{9,10}}
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> {{1,6},{2,3},{4,5},{7,10},{8,9}}
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> {{1,8},{2,3},{4,5},{6,7},{9,10}}
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> {{1,10},{2,3},{4,5},{6,9},{7,8}}
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> {{1,8},{2,3},{4,7},{5,6},{9,10}}
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> {{1,10},{2,3},{4,7},{5,6},{8,9}}
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> {{1,10},{2,3},{4,9},{5,6},{7,8}}
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> {{1,10},{2,3},{4,9},{5,8},{6,7}}
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> {{1,6},{2,5},{3,4},{7,8},{9,10}}
=> 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12},{13,14}}
=> ? = 13
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,14},{12,13}}
=> ? = 12
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11),(13,14)]
=> {{1,2},{3,4},{5,6},{7,8},{9,12},{10,11},{13,14}}
=> ? = 13
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> {{1,2},{3,4},{5,6},{7,8},{9,14},{10,11},{12,13}}
=> ? = 12
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> {{1,2},{3,4},{5,6},{7,8},{9,14},{10,13},{11,12}}
=> ? = 11
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9},{11,12},{13,14}}
=> ? = 13
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9},{11,14},{12,13}}
=> ? = 12
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11),(13,14)]
=> {{1,2},{3,4},{5,6},{7,12},{8,9},{10,11},{13,14}}
=> ? = 13
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
=> {{1,2},{3,4},{5,6},{7,14},{8,9},{10,11},{12,13}}
=> ? = 12
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,9),(10,13),(11,12)]
=> {{1,2},{3,4},{5,6},{7,14},{8,9},{10,13},{11,12}}
=> ? = 11
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> {{1,2},{3,4},{5,6},{7,12},{8,11},{9,10},{13,14}}
=> ? = 13
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
=> {{1,2},{3,4},{5,6},{7,14},{8,11},{9,10},{12,13}}
=> ? = 12
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,10),(11,12)]
=> {{1,2},{3,4},{5,6},{7,14},{8,13},{9,10},{11,12}}
=> ? = 11
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> {{1,2},{3,4},{5,6},{7,14},{8,13},{9,12},{10,11}}
=> ? = 10
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12),(13,14)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10},{11,12},{13,14}}
=> ? = 13
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10},{11,14},{12,13}}
=> ? = 12
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11),(13,14)]
=> {{1,2},{3,4},{5,8},{6,7},{9,12},{10,11},{13,14}}
=> ? = 13
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,11),(12,13)]
=> {{1,2},{3,4},{5,8},{6,7},{9,14},{10,11},{12,13}}
=> ? = 12
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> {{1,2},{3,4},{5,8},{6,7},{9,14},{10,13},{11,12}}
=> ? = 11
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12),(13,14)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9},{11,12},{13,14}}
=> ? = 13
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,14),(12,13)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9},{11,14},{12,13}}
=> ? = 12
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11),(13,14)]
=> {{1,2},{3,4},{5,12},{6,7},{8,9},{10,11},{13,14}}
=> ? = 13
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
=> {{1,2},{3,4},{5,14},{6,7},{8,9},{10,11},{12,13}}
=> ? = 12
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,9),(10,13),(11,12)]
=> {{1,2},{3,4},{5,14},{6,7},{8,9},{10,13},{11,12}}
=> ? = 11
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10),(13,14)]
=> {{1,2},{3,4},{5,12},{6,7},{8,11},{9,10},{13,14}}
=> ? = 13
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,11),(9,10),(12,13)]
=> {{1,2},{3,4},{5,14},{6,7},{8,11},{9,10},{12,13}}
=> ? = 12
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,13),(9,10),(11,12)]
=> {{1,2},{3,4},{5,14},{6,7},{8,13},{9,10},{11,12}}
=> ? = 11
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> {{1,2},{3,4},{5,14},{6,7},{8,13},{9,12},{10,11}}
=> ? = 10
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> {{1,2},{3,4},{5,10},{6,9},{7,8},{11,12},{13,14}}
=> ? = 13
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> {{1,2},{3,4},{5,10},{6,9},{7,8},{11,14},{12,13}}
=> ? = 12
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11),(13,14)]
=> {{1,2},{3,4},{5,12},{6,9},{7,8},{10,11},{13,14}}
=> ? = 13
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
=> {{1,2},{3,4},{5,14},{6,9},{7,8},{10,11},{12,13}}
=> ? = 12
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,9),(7,8),(10,13),(11,12)]
=> {{1,2},{3,4},{5,14},{6,9},{7,8},{10,13},{11,12}}
=> ? = 11
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10),(13,14)]
=> {{1,2},{3,4},{5,12},{6,11},{7,8},{9,10},{13,14}}
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,11),(7,8),(9,10),(12,13)]
=> {{1,2},{3,4},{5,14},{6,11},{7,8},{9,10},{12,13}}
=> ? = 12
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> {{1,2},{3,4},{5,14},{6,13},{7,8},{9,10},{11,12}}
=> ? = 11
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> {{1,2},{3,4},{5,14},{6,13},{7,8},{9,12},{10,11}}
=> ? = 10
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9),(13,14)]
=> {{1,2},{3,4},{5,12},{6,11},{7,10},{8,9},{13,14}}
=> ? = 13
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> {{1,2},{3,4},{5,14},{6,11},{7,10},{8,9},{12,13}}
=> ? = 12
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> {{1,2},{3,4},{5,14},{6,13},{7,10},{8,9},{11,12}}
=> ? = 11
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> {{1,2},{3,4},{5,14},{6,13},{7,12},{8,9},{10,11}}
=> ? = 10
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> {{1,2},{3,4},{5,14},{6,13},{7,12},{8,11},{9,10}}
=> ? = 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12),(13,14)]
=> {{1,2},{3,6},{4,5},{7,8},{9,10},{11,12},{13,14}}
=> ? = 13
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,14),(12,13)]
=> {{1,2},{3,6},{4,5},{7,8},{9,10},{11,14},{12,13}}
=> ? = 12
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11),(13,14)]
=> {{1,2},{3,6},{4,5},{7,8},{9,12},{10,11},{13,14}}
=> ? = 13
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,14),(10,11),(12,13)]
=> {{1,2},{3,6},{4,5},{7,8},{9,14},{10,11},{12,13}}
=> ? = 12
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,14),(10,13),(11,12)]
=> {{1,2},{3,6},{4,5},{7,8},{9,14},{10,13},{11,12}}
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12),(13,14)]
=> {{1,2},{3,6},{4,5},{7,10},{8,9},{11,12},{13,14}}
=> ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,14),(12,13)]
=> {{1,2},{3,6},{4,5},{7,10},{8,9},{11,14},{12,13}}
=> ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11),(13,14)]
=> {{1,2},{3,6},{4,5},{7,12},{8,9},{10,11},{13,14}}
=> ? = 13
Description
The largest opener of a set partition.
An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.
Matching statistic: St000019
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[1,1,0,0]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 5
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 7
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => 6
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => 7
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => 6
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 7
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 6
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => 7
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => 6
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => 5
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => 7
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 6
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => 5
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => 9
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => 8
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => 9
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => 9
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => 9
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => 8
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => 9
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => 8
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => 7
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => 8
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => 8
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => 7
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => 9
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => 9
[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]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [14,13,12,11,10,9,8,7,6,5,4,3,2,1] => ? = 13
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? = 12
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,14),(2,3),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [14,3,2,13,12,11,10,9,8,7,6,5,4,1] => ? = 13
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? = 12
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,4,3,14,13,12,11,10,9,8,7,6,5] => ? = 11
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [14,13,4,3,12,11,10,9,8,7,6,5,2,1] => ? = 13
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,14,5,4,13,12,11,10,9,8,7,6,3] => ? = 12
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,9),(5,8),(6,7),(12,13)]
=> [14,11,10,9,8,7,6,5,4,3,2,13,12,1] => ? = 13
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [4,3,2,1,14,13,12,11,10,9,8,7,6,5] => ? = 12
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> [2,1,12,11,10,9,8,7,6,5,4,3,14,13] => ? = 11
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> [14,3,2,5,4,13,12,11,10,9,8,7,6,1] => ? = 13
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> [12,3,2,11,10,9,8,7,6,5,4,1,14,13] => ? = 12
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12),(13,14)]
=> [10,9,8,7,6,5,4,3,2,1,12,11,14,13] => ? = 11
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> [2,1,4,3,6,5,14,13,12,11,10,9,8,7] => ? = 10
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [14,13,12,5,4,11,10,9,8,7,6,3,2,1] => ? = 13
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,14,13,6,5,12,11,10,9,8,7,4,3] => ? = 12
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> [14,3,2,13,6,5,12,11,10,9,8,7,4,1] => ? = 13
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> [12,11,4,3,10,9,8,7,6,5,2,1,14,13] => ? = 12
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> [2,1,4,3,14,7,6,13,12,11,10,9,8,5] => ? = 11
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [(1,14),(2,13),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [14,13,10,9,8,7,6,5,4,3,12,11,2,1] => ? = 13
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> [2,1,14,11,10,9,8,7,6,5,4,13,12,3] => ? = 12
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,5),(3,4),(6,13),(7,12),(8,11),(9,10)]
=> [14,5,4,3,2,13,12,11,10,9,8,7,6,1] => ? = 13
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> [10,9,8,7,6,5,4,3,2,1,14,13,12,11] => ? = 12
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> [2,1,6,5,4,3,14,13,12,11,10,9,8,7] => ? = 11
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,14),(2,3),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> [14,3,2,11,10,9,8,7,6,5,4,13,12,1] => ? = 13
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> [12,9,8,7,6,5,4,3,2,11,10,1,14,13] => ? = 12
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> [4,3,2,1,6,5,14,13,12,11,10,9,8,7] => ? = 11
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9),(13,14)]
=> [2,1,4,3,12,11,10,9,8,7,6,5,14,13] => ? = 10
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [14,13,4,3,6,5,12,11,10,9,8,7,2,1] => ? = 13
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,5),(6,7),(8,13),(9,12),(10,11)]
=> [2,1,14,5,4,7,6,13,12,11,10,9,8,3] => ? = 12
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13)]
=> [14,11,4,3,10,9,8,7,6,5,2,13,12,1] => ? = 13
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> [4,3,2,1,14,7,6,13,12,11,10,9,8,5] => ? = 12
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9),(13,14)]
=> [2,1,12,5,4,11,10,9,8,7,6,3,14,13] => ? = 11
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,8),(4,7),(5,6),(10,11),(12,13)]
=> [14,9,8,7,6,5,4,3,2,11,10,13,12,1] => ? = 13
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> [6,3,2,5,4,1,14,13,12,11,10,9,8,7] => ? = 12
[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]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9),(13,14)]
=> [4,3,2,1,12,11,10,9,8,7,6,5,14,13] => ? = 11
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12),(13,14)]
=> [2,1,10,9,8,7,6,5,4,3,12,11,14,13] => ? = 10
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
=> [14,3,2,5,4,7,6,13,12,11,10,9,8,1] => ? = 13
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
=> [12,3,2,5,4,11,10,9,8,7,6,1,14,13] => ? = 12
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12),(13,14)]
=> [10,3,2,9,8,7,6,5,4,1,12,11,14,13] => ? = 11
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12),(13,14)]
=> [8,7,6,5,4,3,2,1,10,9,12,11,14,13] => ? = 10
[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),(7,8),(9,14),(10,13),(11,12)]
=> [2,1,4,3,6,5,8,7,14,13,12,11,10,9] => ? = 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [14,13,12,11,6,5,10,9,8,7,4,3,2,1] => ? = 13
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,14,13,12,7,6,11,10,9,8,5,4,3] => ? = 12
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> [14,3,2,13,12,7,6,11,10,9,8,5,4,1] => ? = 13
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> [12,11,10,5,4,9,8,7,6,3,2,1,14,13] => ? = 12
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> [2,1,4,3,14,13,8,7,12,11,10,9,6,5] => ? = 11
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [14,13,4,3,12,7,6,11,10,9,8,5,2,1] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,5),(6,13),(7,8),(9,12),(10,11)]
=> [2,1,14,5,4,13,8,7,12,11,10,9,6,3] => ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,5),(6,9),(7,8),(12,13)]
=> [14,11,10,5,4,9,8,7,6,3,2,13,12,1] => ? = 13
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000074
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2 = 3 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 4 = 5 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 4 = 5 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 6 = 7 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 6 = 7 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 6 = 7 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 6 = 7 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 6 = 7 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 5 = 6 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 3 = 4 - 1
[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],[6,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 8 = 9 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 7 = 8 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 8 = 9 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 7 = 8 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 6 = 7 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 8 = 9 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 7 = 8 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 8 = 9 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 7 = 8 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 6 = 7 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 8 = 9 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 7 = 8 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 6 = 7 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 8 = 9 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 7 = 8 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 8 = 9 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 7 = 8 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 6 = 7 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 8 = 9 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 7 = 8 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 8 = 9 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 7 = 8 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 6 = 7 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 8 = 9 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 7 = 8 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 8 = 9 - 1
[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],[8,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,2,4,5,6,7,8],[3,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[1,2,3,5,6,7,8],[4,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,6,7,8,9],[2,5,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[1,2,3,4,5,6,12],[7,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,2,5,6,7,8,9],[3,4,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,3,4,5,6,7,13],[2,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,2,4,6,7,8,9],[3,5,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,4,5,6,7,13],[3,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[1,2,3,4,5,11,13],[6,7,8,9,10,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 11 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 10 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[1,2,3,4,6,7,8],[5,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,7,8,9],[2,6,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[1,2,4,5,7,8,9],[3,6,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,5,6,7,13],[4,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,5,6,8,9,10],[2,4,7,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,6,11],[7,8,9,10,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,3,4,5,6,7,12],[2,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,6,7,8,9],[4,5,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,2,4,5,6,7,12],[3,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[1,2,3,4,5,10,13],[6,7,8,9,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,7,8,9,10],[3,4,6,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 11 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,5,6,7,8,13],[2,4,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 10 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[1,2,3,5,7,8,9],[4,6,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,8,9,10],[2,5,7,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [[1,2,3,5,6,7,12],[4,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,2,5,6,8,9,10],[3,4,7,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [[1,3,4,6,7,8,13],[2,5,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [[1,2,3,4,5,10,12],[6,7,8,9,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,4,7,8,9,10],[3,5,6,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 - 1
[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]
=> [[1,2,5,6,7,8,13],[3,4,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 11 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[1,3,4,5,6,11,13],[2,7,8,9,10,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 10 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,6,8,9,10],[3,5,7,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[1,2,4,6,7,8,13],[3,5,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[1,2,4,5,6,11,13],[3,7,8,9,10,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 11 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[1,2,3,4,9,11,13],[5,6,7,8,10,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10 - 1
[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]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 9 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,7,8],[6,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,8,9],[2,7,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,2,4,5,6,8,9],[3,7,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,6,7,13],[5,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,5,6,7,9,10],[2,4,8,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[1,2,3,5,6,8,9],[4,7,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,6,7,9,10],[2,5,8,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [[1,2,3,4,6,7,12],[5,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 - 1
Description
The number of special entries.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000176: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000176: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 4 = 3 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 6 = 5 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 6 = 5 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 5 = 4 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 7 = 6 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 6 = 5 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 8 = 7 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 7 = 6 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 6 = 5 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 8 = 7 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 7 = 6 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 6 = 5 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 5 = 4 + 1
[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],[6,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 10 = 9 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 10 = 9 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 10 = 9 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 9 = 8 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 10 = 9 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 9 = 8 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 10 = 9 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 10 = 9 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 9 = 8 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 10 = 9 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 8 = 7 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 10 = 9 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 10 = 9 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 9 = 8 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> 10 = 9 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 9 = 8 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> 7 = 6 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 10 = 9 + 1
[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],[8,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,2,4,5,6,7,8],[3,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[1,2,3,5,6,7,8],[4,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,6,7,8,9],[2,5,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[1,2,3,4,5,6,12],[7,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,2,5,6,7,8,9],[3,4,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,3,4,5,6,7,13],[2,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,2,4,6,7,8,9],[3,5,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,4,5,6,7,13],[3,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[1,2,3,4,5,11,13],[6,7,8,9,10,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 11 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 10 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[1,2,3,4,6,7,8],[5,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,7,8,9],[2,6,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[1,2,4,5,7,8,9],[3,6,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,5,6,7,13],[4,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,5,6,8,9,10],[2,4,7,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[1,2,3,4,5,6,11],[7,8,9,10,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,3,4,5,6,7,12],[2,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,6,7,8,9],[4,5,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,2,4,5,6,7,12],[3,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[1,2,3,4,5,10,13],[6,7,8,9,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,7,8,9,10],[3,4,6,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 11 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,5,6,7,8,13],[2,4,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 10 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[1,2,3,5,7,8,9],[4,6,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,8,9,10],[2,5,7,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [[1,2,3,5,6,7,12],[4,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,2,5,6,8,9,10],[3,4,7,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [[1,3,4,6,7,8,13],[2,5,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [[1,2,3,4,5,10,12],[6,7,8,9,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,4,7,8,9,10],[3,5,6,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 + 1
[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]
=> [[1,2,5,6,7,8,13],[3,4,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 11 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[1,3,4,5,6,11,13],[2,7,8,9,10,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 10 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,6,8,9,10],[3,5,7,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[1,2,4,6,7,8,13],[3,5,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[1,2,4,5,6,11,13],[3,7,8,9,10,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 11 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[1,2,3,4,9,11,13],[5,6,7,8,10,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10 + 1
[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]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,7,8],[6,9,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,8,9],[2,7,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,2,4,5,6,8,9],[3,7,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,6,7,13],[5,8,9,10,11,12,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[6,6,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 12 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,5,6,7,9,10],[2,4,8,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? = 11 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[1,2,3,5,6,8,9],[4,7,10,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[7,2,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,6,7,9,10],[2,5,8,11,12,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[7,4,0,0,0,0,0,0,0,0,0],[7,3,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? = 12 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [[1,2,3,4,6,7,12],[5,8,9,10,11,13,14]]
=> [[7,7,0,0,0,0,0,0,0,0,0,0,0,0],[7,6,0,0,0,0,0,0,0,0,0,0,0],[7,5,0,0,0,0,0,0,0,0,0,0],[6,5,0,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 13 + 1
Description
The total number of tiles in the Gelfand-Tsetlin pattern.
The tiling of a Gelfand-Tsetlin pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called tiles, and each entry in a pattern belongs to exactly one tile.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000301The number of facets of the stable set polytope of a graph. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000459The hook length of the base cell of a partition. St000841The largest opener of a perfect matching. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000738The first entry in the last row of a standard tableau. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St000393The number of strictly increasing runs in a binary word. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.
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