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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000003
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 3
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 4
[3,2]
=> 5
[3,1,1]
=> 6
[2,2,1]
=> 5
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 5
[4,2]
=> 9
[4,1,1]
=> 10
[3,3]
=> 5
[3,2,1]
=> 16
[3,1,1,1]
=> 10
[2,2,2]
=> 5
[2,2,1,1]
=> 9
[2,1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> 1
[7]
=> 1
[6,1]
=> 6
[5,2]
=> 14
[5,1,1]
=> 15
[4,3]
=> 14
[4,2,1]
=> 35
[4,1,1,1]
=> 20
[3,3,1]
=> 21
[3,2,2]
=> 21
[3,2,1,1]
=> 35
[3,1,1,1,1]
=> 15
[2,2,2,1]
=> 14
[2,2,1,1,1]
=> 14
[2,1,1,1,1,1]
=> 6
[1,1,1,1,1,1,1]
=> 1
[8]
=> 1
[7,1]
=> 7
[6,2]
=> 20
[6,1,1]
=> 21
[5,3]
=> 28
[5,2,1]
=> 64
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St000001
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 58%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[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] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 6
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 9
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 10
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 5
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 16
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 10
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 5
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 9
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 5
[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] => 1
[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] => ? ∊ {1,1}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 14
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 15
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 14
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 35
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 20
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 21
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 21
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 35
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 15
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 14
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 14
[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] => 6
[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] => ? ∊ {1,1}
[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] => ? ∊ {1,1,7,7}
[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] => ? ∊ {1,1,7,7}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 20
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 21
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 28
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 64
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 35
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 14
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 70
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 56
[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] => ? ∊ {1,1,7,7}
[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] => ? ∊ {1,1,7,7}
[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] => ? ∊ {1,1,8,8,27,27,28,28}
[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] => ? ∊ {1,1,8,8,27,27,28,28}
[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] => ? ∊ {1,1,8,8,27,27,28,28}
[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] => ? ∊ {1,1,8,8,27,27,28,28}
[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] => ? ∊ {1,1,8,8,27,27,28,28}
[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] => ? ∊ {1,1,8,8,27,27,28,28}
[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] => ? ∊ {1,1,8,8,27,27,28,28}
[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] => ? ∊ {1,1,8,8,27,27,28,28}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,9,9,35,35,36,36,75,75,84,84,160,160}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,2,5,6] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,3,1,5,6] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,1,6] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [6,7,2,1,3,4,5] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,7,1] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[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] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,165,165,210,210,231,231,252,330,330,385,385,550,550,594,594,693,693,924,924,990,990,1100,1100,1232,1232}
Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Matching statistic: St000100
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 47% ●values known / values provided: 51%●distinct values known / distinct values provided: 47%
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 47% ●values known / values provided: 51%●distinct values known / distinct values provided: 47%
Values
[1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 6
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 9
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 10
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 16
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 10
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 9
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 6
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 14
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 15
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 35
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 20
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 21
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 21
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 35
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 15
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 14
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 6
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> 7
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> 20
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> 21
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> 28
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> 64
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> 35
[11]
=> [[11],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[10,1]
=> [[10,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[9,2]
=> [[9,2],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[9,1,1]
=> [[9,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[8,3]
=> [[8,3],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[8,2,1]
=> [[8,2,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[8,1,1,1]
=> [[8,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[7,4]
=> [[7,4],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[7,3,1]
=> [[7,3,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[7,2,2]
=> [[7,2,2],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[7,2,1,1]
=> [[7,2,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[7,1,1,1,1]
=> [[7,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[6,5]
=> [[6,5],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(3,9),(4,3),(4,10),(5,1),(5,8),(6,4),(6,7),(7,10),(9,8),(10,9)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[6,4,1]
=> [[6,4,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[6,3,2]
=> [[6,3,2],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[6,3,1,1]
=> [[6,3,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[6,2,2,1]
=> [[6,2,2,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[6,2,1,1,1]
=> [[6,2,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[6,1,1,1,1,1]
=> [[6,1,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,5,1]
=> [[5,5,1],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,9),(4,3),(4,10),(5,4),(5,7),(6,1),(6,7),(7,10),(9,8),(10,9)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,4,2]
=> [[5,4,2],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,4,1,1]
=> [[5,4,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,5),(4,10),(5,1),(5,9),(6,4),(6,8),(7,3),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,3,3]
=> [[5,3,3],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,3,2,1]
=> [[5,3,2,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,2,2,2]
=> [[5,2,2,2],[]]
=> ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,2,1,1,1,1]
=> [[5,2,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[5,1,1,1,1,1,1]
=> [[5,1,1,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,4,3]
=> [[4,4,3],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(8,10),(9,7),(9,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,4,2,1]
=> [[4,4,2,1],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(10,8)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,3,3,1]
=> [[4,3,3,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,3,2,2]
=> [[4,3,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(10,8)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,3,2,1,1]
=> [[4,3,2,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,3,1,1,1,1]
=> [[4,3,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ([(0,6),(0,7),(3,2),(4,5),(4,10),(5,1),(5,9),(6,4),(6,8),(7,3),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,2,2,1,1,1]
=> [[4,2,2,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,2,1,1,1,1,1]
=> [[4,2,1,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[4,1,1,1,1,1,1,1]
=> [[4,1,1,1,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,3,3,2]
=> [[3,3,3,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(8,10),(9,7),(9,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,3,2,2,1]
=> [[3,3,2,2,1],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,3,2,1,1,1]
=> [[3,3,2,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,3,1,1,1,1,1]
=> [[3,3,1,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,2,2,2,2]
=> [[3,2,2,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,9),(4,3),(4,10),(5,4),(5,7),(6,1),(6,7),(7,10),(9,8),(10,9)],11)
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,2,2,2,1,1]
=> [[3,2,2,2,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,2,2,1,1,1,1]
=> [[3,2,2,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
[3,2,1,1,1,1,1,1]
=> [[3,2,1,1,1,1,1,1],[]]
=> ?
=> ? ∊ {1,1,10,10,44,44,45,45,110,110,120,120,132,132,165,165,210,210,231,231,252,330,330,385,385,462,462,550,550,594,594,660,660,693,693,825,825,924,924,990,990,990,990,1100,1100,1155,1155,1188,1232,1232,1320,1320,1540,1540,2310,2310}
Description
The number of linear extensions of a poset.
Matching statistic: St001595
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 14% ●values known / values provided: 16%●distinct values known / distinct values provided: 14%
St001595: Skew partitions ⟶ ℤResult quality: 14% ●values known / values provided: 16%●distinct values known / distinct values provided: 14%
Values
[1]
=> [[1],[]]
=> 1
[2]
=> [[2],[]]
=> 1
[1,1]
=> [[1,1],[]]
=> 1
[3]
=> [[3],[]]
=> 1
[2,1]
=> [[2,1],[]]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> 1
[4]
=> [[4],[]]
=> 1
[3,1]
=> [[3,1],[]]
=> 3
[2,2]
=> [[2,2],[]]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> 3
[1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[5]
=> [[5],[]]
=> 1
[4,1]
=> [[4,1],[]]
=> 4
[3,2]
=> [[3,2],[]]
=> 5
[3,1,1]
=> [[3,1,1],[]]
=> 6
[2,2,1]
=> [[2,2,1],[]]
=> 5
[2,1,1,1]
=> [[2,1,1,1],[]]
=> 4
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[6]
=> [[6],[]]
=> 1
[5,1]
=> [[5,1],[]]
=> 5
[4,2]
=> [[4,2],[]]
=> 9
[4,1,1]
=> [[4,1,1],[]]
=> 10
[3,3]
=> [[3,3],[]]
=> 5
[3,2,1]
=> [[3,2,1],[]]
=> 16
[3,1,1,1]
=> [[3,1,1,1],[]]
=> 10
[2,2,2]
=> [[2,2,2],[]]
=> 5
[2,2,1,1]
=> [[2,2,1,1],[]]
=> 9
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 5
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 1
[7]
=> [[7],[]]
=> 1
[6,1]
=> [[6,1],[]]
=> 6
[5,2]
=> [[5,2],[]]
=> 14
[5,1,1]
=> [[5,1,1],[]]
=> 15
[4,3]
=> [[4,3],[]]
=> 14
[4,2,1]
=> [[4,2,1],[]]
=> 35
[4,1,1,1]
=> [[4,1,1,1],[]]
=> 20
[3,3,1]
=> [[3,3,1],[]]
=> 21
[3,2,2]
=> [[3,2,2],[]]
=> 21
[3,2,1,1]
=> [[3,2,1,1],[]]
=> 35
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> 15
[2,2,2,1]
=> [[2,2,2,1],[]]
=> 14
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 14
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> 6
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> 1
[8]
=> [[8],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[7,1]
=> [[7,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[6,2]
=> [[6,2],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[6,1,1]
=> [[6,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[5,3]
=> [[5,3],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[5,2,1]
=> [[5,2,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[4,4]
=> [[4,4],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[4,3,1]
=> [[4,3,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[4,2,2]
=> [[4,2,2],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[3,3,2]
=> [[3,3,2],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90}
[9]
=> [[9],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[8,1]
=> [[8,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[7,2]
=> [[7,2],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[7,1,1]
=> [[7,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[6,3]
=> [[6,3],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[6,2,1]
=> [[6,2,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[6,1,1,1]
=> [[6,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[5,4]
=> [[5,4],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[5,3,1]
=> [[5,3,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[5,2,2]
=> [[5,2,2],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[5,2,1,1]
=> [[5,2,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[4,4,1]
=> [[4,4,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[4,3,2]
=> [[4,3,2],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[4,3,1,1]
=> [[4,3,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[4,2,2,1]
=> [[4,2,2,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[4,2,1,1,1]
=> [[4,2,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[3,3,3]
=> [[3,3,3],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[3,3,2,1]
=> [[3,3,2,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[3,2,2,2]
=> [[3,2,2,2],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[3,2,1,1,1,1]
=> [[3,2,1,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[3,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[2,2,2,1,1,1]
=> [[2,2,2,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
[2,2,1,1,1,1,1]
=> [[2,2,1,1,1,1,1],[]]
=> ? ∊ {1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84,84,105,105,120,120,162,162,168,168,189,189,216,216}
Description
The number of standard Young tableaux of the skew partition.
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