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Your data matches 130 different statistics following compositions of up to 3 maps.
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Matching statistic: St001091
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 1
[3]
=> 0
[2,1]
=> 0
[1,1,1]
=> 2
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 3
[5]
=> 0
[4,1]
=> 0
[3,2]
=> 0
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 4
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 0
[4,1,1]
=> 1
[3,3]
=> 1
[3,2,1]
=> 0
[3,1,1,1]
=> 2
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 3
[1,1,1,1,1,1]
=> 5
[7]
=> 0
[6,1]
=> 0
[5,2]
=> 0
[5,1,1]
=> 1
[4,3]
=> 0
[4,2,1]
=> 0
[4,1,1,1]
=> 2
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 4
[1,1,1,1,1,1,1]
=> 6
[8]
=> 0
[7,1]
=> 0
[6,2]
=> 0
[6,1,1]
=> 1
[5,3]
=> 0
[5,2,1]
=> 0
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000052
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [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]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 0
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,4,5,5,5,6,6,7,7,8,10}
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11}
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 83%
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 83%
Values
[1]
=> 10 => 01 => 01 => 1 = 0 + 1
[2]
=> 100 => 010 => 001 => 2 = 1 + 1
[1,1]
=> 110 => 011 => 011 => 1 = 0 + 1
[3]
=> 1000 => 0100 => 0001 => 3 = 2 + 1
[2,1]
=> 1010 => 0101 => 0101 => 1 = 0 + 1
[1,1,1]
=> 1110 => 0111 => 0111 => 1 = 0 + 1
[4]
=> 10000 => 01000 => 00001 => 4 = 3 + 1
[3,1]
=> 10010 => 01001 => 00101 => 2 = 1 + 1
[2,2]
=> 1100 => 0110 => 0011 => 2 = 1 + 1
[2,1,1]
=> 10110 => 01011 => 01011 => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 01111 => 01111 => 1 = 0 + 1
[5]
=> 100000 => 010000 => 000001 => 5 = 4 + 1
[4,1]
=> 100010 => 010001 => 000101 => 3 = 2 + 1
[3,2]
=> 10100 => 01010 => 00101 => 2 = 1 + 1
[3,1,1]
=> 100110 => 010011 => 001101 => 2 = 1 + 1
[2,2,1]
=> 11010 => 01101 => 01011 => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 010111 => 010111 => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 011111 => 011111 => 1 = 0 + 1
[6]
=> 1000000 => 0100000 => 0000001 => 6 = 5 + 1
[5,1]
=> 1000010 => 0100001 => 0000101 => 4 = 3 + 1
[4,2]
=> 100100 => 010010 => 000101 => 3 = 2 + 1
[4,1,1]
=> 1000110 => 0100011 => 0001101 => 3 = 2 + 1
[3,3]
=> 11000 => 01100 => 00011 => 3 = 2 + 1
[3,2,1]
=> 101010 => 010101 => 010101 => 1 = 0 + 1
[3,1,1,1]
=> 1001110 => 0100111 => 0011101 => 2 = 1 + 1
[2,2,2]
=> 11100 => 01110 => 00111 => 2 = 1 + 1
[2,2,1,1]
=> 110110 => 011011 => 011011 => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0101111 => 0101111 => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 0111111 => 1 = 0 + 1
[7]
=> 10000000 => 01000000 => 00000001 => 7 = 6 + 1
[6,1]
=> 10000010 => 01000001 => 00000101 => 5 = 4 + 1
[5,2]
=> 1000100 => 0100010 => 0000101 => 4 = 3 + 1
[5,1,1]
=> 10000110 => 01000011 => 00001101 => 4 = 3 + 1
[4,3]
=> 101000 => 010100 => 000101 => 3 = 2 + 1
[4,2,1]
=> 1001010 => 0100101 => 0010101 => 2 = 1 + 1
[4,1,1,1]
=> 10001110 => 01000111 => 00011101 => 3 = 2 + 1
[3,3,1]
=> 110010 => 011001 => 001011 => 2 = 1 + 1
[3,2,2]
=> 101100 => 010110 => 001011 => 2 = 1 + 1
[3,2,1,1]
=> 1010110 => 0101011 => 0101011 => 1 = 0 + 1
[3,1,1,1,1]
=> 10011110 => 01001111 => 00111101 => 2 = 1 + 1
[2,2,2,1]
=> 111010 => 011101 => 010111 => 1 = 0 + 1
[2,2,1,1,1]
=> 1101110 => 0110111 => 0110111 => 1 = 0 + 1
[2,1,1,1,1,1]
=> 10111110 => 01011111 => 01011111 => 1 = 0 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 01111111 => 1 = 0 + 1
[8]
=> 100000000 => 010000000 => 000000001 => 8 = 7 + 1
[7,1]
=> 100000010 => 010000001 => 000000101 => 6 = 5 + 1
[6,2]
=> 10000100 => 01000010 => 00000101 => 5 = 4 + 1
[6,1,1]
=> 100000110 => 010000011 => 000001101 => 5 = 4 + 1
[5,3]
=> 1001000 => 0100100 => 0000101 => 4 = 3 + 1
[5,2,1]
=> 10001010 => 01000101 => 00010101 => 3 = 2 + 1
[11]
=> 100000000000 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[10,1]
=> 100000000010 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[9,2]
=> 10000000100 => 01000000010 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[9,1,1]
=> 100000000110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[8,2,1]
=> 10000001010 => 01000000101 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[8,1,1,1]
=> 100000001110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[7,3,1]
=> 1000010010 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[7,2,2]
=> 1000001100 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[7,2,1,1]
=> 10000010110 => 01000001011 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[6,3,1,1]
=> 1000100110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[6,2,2,1]
=> 1000011010 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[6,2,1,1,1]
=> 10000101110 => 01000010111 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[5,2,1,1,1,1]
=> 10001011110 => 01000101111 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[4,2,1,1,1,1,1]
=> 10010111110 => 01001011111 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01010111111 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,4,4,5,5,5,6,7,7,8,10} + 1
[12]
=> 1000000000000 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[11,1]
=> 1000000000010 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[10,2]
=> 100000000100 => 010000000010 => 000000000101 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[10,1,1]
=> 1000000000110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[9,3]
=> 10000001000 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[9,2,1]
=> 100000001010 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[9,1,1,1]
=> 1000000001110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[8,3,1]
=> 10000010010 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[8,2,2]
=> 10000001100 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[8,2,1,1]
=> 100000010110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[8,1,1,1,1]
=> 1000000011110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[7,4,1]
=> 1000100010 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[7,3,2]
=> 1000010100 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[7,3,1,1]
=> 10000100110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[7,2,2,1]
=> 10000011010 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[7,2,1,1,1]
=> 100000101110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[7,1,1,1,1,1]
=> 1000000111110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[6,4,1,1]
=> 1001000110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[6,3,2,1]
=> 1000101010 => 0100010101 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[6,3,1,1,1]
=> 10001001110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[6,2,2,1,1]
=> 10000110110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[6,1,1,1,1,1,1]
=> 1000001111110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
[5,3,1,1,1,1]
=> 10010011110 => ? => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,11} + 1
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000160
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 83%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 83%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1 = 0 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 1 = 0 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 1 = 0 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 1 = 0 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 3 = 2 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 1 = 0 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 1 = 0 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 4 = 3 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 1 = 0 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 1 = 0 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 1 = 0 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 2 = 1 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 5 = 4 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 1 = 0 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 1 = 0 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 1 = 0 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 2 = 1 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 2 = 1 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 1 = 0 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 3 = 2 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 6 = 5 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 1 = 0 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 1 = 0 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 1 = 0 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 2 = 1 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 1 = 0 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 1 = 0 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 3 = 2 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 4 = 3 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 5 = 4 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 7 = 6 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 1 = 0 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 1 = 0 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 1 = 0 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 2 = 1 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 1 = 0 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 1 = 0 + 1
[11]
=> 100000000000 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[10,1]
=> 100000000010 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[9,1,1]
=> 100000000110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[8,3]
=> 1000001000 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[8,2,1]
=> 10000001010 => [1,7,2,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[8,1,1,1]
=> 100000001110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[7,3,1]
=> 1000010010 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[7,2,2]
=> 1000001100 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[7,2,1,1]
=> 10000010110 => [1,6,2,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[7,1,1,1,1]
=> 100000011110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[6,3,1,1]
=> 1000100110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[6,2,2,1]
=> 1000011010 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[6,2,1,1,1]
=> 10000101110 => [1,5,2,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[6,1,1,1,1,1]
=> 100000111110 => [1,6,1,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[5,3,1,1,1]
=> 1001001110 => [1,3,3,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[5,2,2,1,1]
=> 1000110110 => [1,4,1,2,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[5,2,1,1,1,1]
=> 10001011110 => [1,4,2,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[4,2,1,1,1,1,1]
=> 10010111110 => [1,3,2,1,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => [1,2,2,1,1,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => [1,1,1,2,1,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[12]
=> 1000000000000 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[11,1]
=> 1000000000010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[10,2]
=> 100000000100 => [1,9,3] => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[10,1,1]
=> 1000000000110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[9,3]
=> 10000001000 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[9,2,1]
=> 100000001010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[9,1,1,1]
=> 1000000001110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[8,3,1]
=> 10000010010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[8,2,2]
=> 10000001100 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[8,2,1,1]
=> 100000010110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[8,1,1,1,1]
=> 1000000011110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,4,1]
=> 1000100010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,3,2]
=> 1000010100 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,3,1,1]
=> 10000100110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,2,2,1]
=> 10000011010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,2,1,1,1]
=> 100000101110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,1,1,1,1,1]
=> 1000000111110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[6,4,1,1]
=> 1001000110 => [1,3,4,1,2] => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[6,3,2,1]
=> 1000101010 => [1,4,2,2,2] => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[6,3,1,1,1]
=> 10001001110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[6,2,2,2]
=> 1000011100 => [1,5,1,1,3] => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000475
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 83%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 83%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1 = 0 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 1 = 0 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 1 = 0 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 1 = 0 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 3 = 2 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 1 = 0 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 1 = 0 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 4 = 3 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 1 = 0 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 1 = 0 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 1 = 0 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 2 = 1 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 5 = 4 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 1 = 0 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 1 = 0 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 1 = 0 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 2 = 1 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 2 = 1 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 1 = 0 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 3 = 2 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 6 = 5 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 1 = 0 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 1 = 0 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 1 = 0 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 2 = 1 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 1 = 0 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 1 = 0 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 3 = 2 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 4 = 3 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 5 = 4 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 7 = 6 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 1 = 0 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 1 = 0 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 1 = 0 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 2 = 1 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 1 = 0 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 1 = 0 + 1
[11]
=> 100000000000 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[10,1]
=> 100000000010 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[9,1,1]
=> 100000000110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[8,3]
=> 1000001000 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[8,2,1]
=> 10000001010 => [1,7,2,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[8,1,1,1]
=> 100000001110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[7,3,1]
=> 1000010010 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[7,2,2]
=> 1000001100 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[7,2,1,1]
=> 10000010110 => [1,6,2,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[7,1,1,1,1]
=> 100000011110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[6,3,1,1]
=> 1000100110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[6,2,2,1]
=> 1000011010 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[6,2,1,1,1]
=> 10000101110 => [1,5,2,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[6,1,1,1,1,1]
=> 100000111110 => [1,6,1,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[5,3,1,1,1]
=> 1001001110 => [1,3,3,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[5,2,2,1,1]
=> 1000110110 => [1,4,1,2,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[5,2,1,1,1,1]
=> 10001011110 => [1,4,2,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[4,2,1,1,1,1,1]
=> 10010111110 => [1,3,2,1,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => [1,2,2,1,1,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => [1,1,1,2,1,1,1,1,2] => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8,10} + 1
[12]
=> 1000000000000 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[11,1]
=> 1000000000010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[10,2]
=> 100000000100 => [1,9,3] => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[10,1,1]
=> 1000000000110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[9,3]
=> 10000001000 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[9,2,1]
=> 100000001010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[9,1,1,1]
=> 1000000001110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[8,3,1]
=> 10000010010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[8,2,2]
=> 10000001100 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[8,2,1,1]
=> 100000010110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[8,1,1,1,1]
=> 1000000011110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,4,1]
=> 1000100010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,3,2]
=> 1000010100 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,3,1,1]
=> 10000100110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,2,2,1]
=> 10000011010 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,2,1,1,1]
=> 100000101110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[7,1,1,1,1,1]
=> 1000000111110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[6,4,1,1]
=> 1001000110 => [1,3,4,1,2] => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[6,3,2,1]
=> 1000101010 => [1,4,2,2,2] => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[6,3,1,1,1]
=> 10001001110 => ? => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
[6,2,2,2]
=> 1000011100 => [1,5,1,1,3] => ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6,6,6,6,7,7,8,8,9,11} + 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000356
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 83%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 5
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => 6
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => 0
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => 0
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,10,9,8,7,6,5,4,3,2,1,12] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,8,7,6,5,4,3,2,1,11] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,7,9,6,5,4,3,2,1,10] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,8,6,5,4,3,2,1,10] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [7,6,5,8,4,3,2,1,9] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,5,4,3,2,1,9] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,7,6,4,3,2,1,9] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,5,4,3,7,2,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [5,6,4,7,3,2,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,4,7,6,3,2,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,5,3,2,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,6,5,4,2,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,3,2,6,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [4,3,5,6,2,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,5,4,6,2,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [3,4,6,5,2,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,4,3,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [4,5,3,2,1,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,7,6,5,4,8,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,6,7,5,8,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,7,6,9,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,5,8,7,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,5,4,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,6,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,8,10,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,4,8,7,6,5,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,6,9,8,7,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,8,10,9,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10,11,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,4,5,5,5,5,6,6,7,7,8,10}
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,11,9,8,7,6,5,4,3,2,1,12] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,10,7,6,5,4,3,2,1,11] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [8,10,9,7,6,5,4,3,2,1,11] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,7,6,9,5,4,3,2,1,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,9,6,5,4,3,2,1,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,9,8,7,5,4,3,2,1,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [7,6,5,4,8,3,2,1,9] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [6,7,5,8,4,3,2,1,9] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [6,5,8,7,4,3,2,1,9] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,7,8,6,4,3,2,1,9] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [4,8,7,6,5,3,2,1,9] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,5,4,3,2,7,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [5,6,4,3,7,2,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [5,4,6,7,3,2,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,11}
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000204
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[.,.],.]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.]
=> 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [[[.,.],.],[.,[.,[.,.]]]]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [[[.,.],[.,[.,[.,.]]]],.]
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> 5
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? ∊ {0,6}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [[[.,.],.],[.,[.,[.,.]]]]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [[[.,.],[.,.]],[.,[.,.]]]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [[[.,.],[.,[.,[.,.]]]],.]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [[[.,[.,.]],[.,[.,.]]],.]
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? ∊ {0,6}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? ∊ {0,0,5,7}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? ∊ {0,0,5,7}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[[.,.],.],[.,[.,[.,.]]]]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [[[[.,.],.],.],[.,[.,.]]]
=> 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [[[.,.],[.,[.,.]]],[.,.]]
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? ∊ {0,0,5,7}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ? ∊ {0,0,5,7}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? ∊ {0,0,0,1,5,5,6,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? ∊ {0,0,0,1,5,5,6,8}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? ∊ {0,0,0,1,5,5,6,8}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> ? ∊ {0,0,0,1,5,5,6,8}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? ∊ {0,0,0,1,5,5,6,8}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> ? ∊ {0,0,0,1,5,5,6,8}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? ∊ {0,0,0,1,5,5,6,8}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> ? ∊ {0,0,0,1,5,5,6,8}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,1,5,6,7] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => [[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,1,5,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [8,4,2,1,3,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,1,2,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,1,5,6,7] => ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,5,1,6,7] => [[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,5,7,8,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [5,2,3,4,6,7,8,9,1] => ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => [[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => [[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St001167
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,6}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,6}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,5,7}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,5,7}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,5,7}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,5,7}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,5,5,6,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,1,5,5,6,8}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,1,5,5,6,8}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,5,5,6,8}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,1,5,5,6,8}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,1,5,5,6,8}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0,1,5,5,6,8}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,5,5,6,8}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,2,4,5,5,6,6,7,9}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St000065
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 75%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> 5
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> 6
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> 4
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> 0
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> 7
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> 5
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> 4
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> 4
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,1}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,1}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,1,2,5}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,1,2,5}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,1,2,5}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,1,2,5}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,1,2,5}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0,0],[1,0,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0,0,0],[1,0,-1,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0,0],[1,0,-1,1,0,0,0,0,0],[0,1,0,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,4,5,5,5,5,6,6,7,7,8,10}
Description
The number of entries equal to -1 in an alternating sign matrix.
The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St001176
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 0
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,0,2}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,2}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,2}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,3}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,1,3}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,3}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,3}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,4}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,2,4}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,2,4}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,4}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,4}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,2,3,5}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,5}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,5}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,3,5}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,5}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,3,5}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,2,2,3,5}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,2,3,4,6}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3,4,6}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3,4,6}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,3,4,6}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3,4,6}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,3,4,6}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,3,4,6}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,2,3,4,6}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,7}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,2,3,3,4,5,6,8}
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 5
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
The following 120 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000011The number of touch points (or returns) of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000711The number of big exceedences of a permutation. St000359The number of occurrences of the pattern 23-1. St001777The number of weak descents in an integer composition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000028The number of stack-sorts needed to sort a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000214The number of adjacencies of a permutation. St000441The number of successions of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000358The number of occurrences of the pattern 31-2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000021The number of descents of a permutation. St000314The number of left-to-right-maxima of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001498The normalised height of a Nakayama algebra with magnitude 1. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000444The length of the maximal rise of a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000654The first descent of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000445The number of rises of length 1 of a Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000013The height of a Dyck path. St000074The number of special entries. St000931The number of occurrences of the pattern UUU in a Dyck path. St000674The number of hills of a Dyck path. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000996The number of exclusive left-to-right maxima of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000740The last entry of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000215The number of adjacencies of a permutation, zero appended. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000365The number of double ascents of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000989The number of final rises of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000051The size of the left subtree of a binary tree. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000338The number of pixed points of a permutation. St000461The rix statistic of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000710The number of big deficiencies of a permutation. St000990The first ascent of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000738The first entry in the last row of a standard tableau. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000686The finitistic dominant dimension of a Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001438The number of missing boxes of a skew partition. St001435The number of missing boxes in the first row. St001868The number of alignments of type NE of a signed permutation. St001487The number of inner corners of a skew partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000259The diameter of a connected graph. St001867The number of alignments of type EN of a signed permutation.
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