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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St000001
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 16
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 6
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 768
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 168
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 216
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 70
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 14
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 216
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 56
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 90
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 35
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 9
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 20
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 10
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 168
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 42
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 56
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 21
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 70
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 21
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 35
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 16
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 10
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 6
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
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: St000003
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> 1
[1,0,1,0]
=> [1]
=> 1
[1,1,0,0]
=> []
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 16
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 5
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 6
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 768
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 168
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 216
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 70
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 216
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 56
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 90
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 35
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 20
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 168
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 42
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 56
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 21
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 70
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 21
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 35
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 16
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 10
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St000100
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 58% ●values known / values provided: 71%●distinct values known / distinct values provided: 58%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 58% ●values known / values provided: 71%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
[1,1,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
[1,1,1,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 16
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 6
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
[1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> 768
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> 168
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> 216
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 70
[1,0,1,0,1,1,1,0,0,0]
=> [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
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> 216
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 56
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> 90
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 35
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 20
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> 168
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> 42
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 56
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 21
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 70
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 21
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 35
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 16
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 10
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 9
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,1}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[4,4,3,2,1],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(11,13),(12,10),(12,13)],14)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[4,3,3,2,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ([(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),(9,11),(10,8),(10,11)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(13,11)],14)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[4,4,2,2,1],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(12,10)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[4,3,2,2,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(11,9)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[4,4,3,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[5,3,3,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[4,3,3,1,1],[]]
=> ([(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),(9,11),(10,11)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[4,4,2,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(11,9)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[5,4,1,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[5,4,3,2],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(11,13),(12,10),(12,13)],14)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[4,4,3,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,2),(3,10),(4,1),(4,11),(5,3),(5,7),(6,4),(6,7),(7,10),(7,11),(10,8),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[5,3,3,2],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[4,3,3,2],[]]
=> ([(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),(9,11),(10,8),(10,11)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[5,4,2,2],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(12,10)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[4,4,2,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,2),(3,10),(4,1),(4,11),(5,3),(5,7),(6,4),(6,7),(7,10),(7,11),(10,8),(11,9)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[5,3,2,2],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(11,9)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[5,4,3,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[4,4,3,1],[]]
=> ([(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),(9,11),(10,8),(10,11)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[5,3,3,1],[]]
=> ([(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),(9,11),(10,11)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[5,4,2,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(11,9)],12)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [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)
=> ? ∊ {1,1,462,462,660,660,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640}
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)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 23% ●values known / values provided: 44%●distinct values known / distinct values provided: 23%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 23% ●values known / values provided: 44%●distinct values known / distinct values provided: 23%
Values
[1,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> 16
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 5
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 6
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> 5
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> 35
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> 20
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> 21
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> 21
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> 35
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> 16
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> 10
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> 14
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> 5
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> 9
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? ∊ {1,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[4,4,3,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[4,3,3,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[4,4,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[4,3,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[3,3,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[4,4,3,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[5,3,3,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[4,3,3,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[4,4,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[4,3,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[3,3,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[4,2,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[5,4,1,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[4,3,1,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? ∊ {1,14,14,28,28,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 14
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> 15
Description
The number of standard Young tableaux of the skew partition.
Matching statistic: St001686
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 5% ●values known / values provided: 16%●distinct values known / distinct values provided: 5%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 5% ●values known / values provided: 16%●distinct values known / distinct values provided: 5%
Values
[1,0]
=> []
=> []
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1}
[1,1,0,0]
=> []
=> []
=> ?
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1}
[1,1,1,0,0,0]
=> []
=> []
=> ?
=> ? ∊ {1,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,5,5,6,16}
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,5,5,6,16}
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,5,5,6,16}
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,5,5,6,16}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1,5,5,6,16}
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ?
=> ? ∊ {1,1,5,5,6,16}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [[4,3,2,1,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [[4,2,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [[4,3,1,1,0,0,0,0,0],[3,3,1,1,0,0,0,0],[3,2,1,1,0,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [[3,3,1,1,0,0,0,0],[3,2,1,1,0,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [[4,2,1,1,0,0,0,0],[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[4,1,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [[4,3,2,0,0,0,0,0,0],[3,3,2,0,0,0,0,0],[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [[3,3,2,0,0,0,0,0],[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[4,2,2,0,0,0,0,0],[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [[4,3,1,0,0,0,0,0],[3,3,1,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[3,3,1,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[4,2,1,0,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ?
=> ? ∊ {1,1,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[5,4,3,2,1,0,0,0,0,0,0,0,0,0,0],[4,4,3,2,1,0,0,0,0,0,0,0,0,0],[4,3,3,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[4,3,2,1,1,0,0,0,0,0,0],[4,3,2,1,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,1,1,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[4,4,3,2,1,0,0,0,0,0,0,0,0,0],[4,3,3,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[4,3,2,1,1,0,0,0,0,0,0],[4,3,2,1,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,1,1,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> [[5,3,3,2,1,0,0,0,0,0,0,0,0,0],[4,3,3,2,1,0,0,0,0,0,0,0,0],[3,3,3,2,1,0,0,0,0,0,0,0],[3,3,2,2,1,0,0,0,0,0,0],[3,3,2,1,1,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,1,1,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> [[4,3,3,2,1,0,0,0,0,0,0,0,0],[3,3,3,2,1,0,0,0,0,0,0,0],[3,3,2,2,1,0,0,0,0,0,0],[3,3,2,1,1,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,1,1,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[3,3,3,2,1,0,0,0,0,0,0,0],[3,3,2,2,1,0,0,0,0,0,0],[3,3,2,1,1,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,1,1,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> [[5,4,2,2,1,0,0,0,0,0,0,0,0,0],[4,4,2,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[4,2,2,2,1,0,0,0,0,0,0],[4,2,2,1,1,0,0,0,0,0],[4,2,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,1,1,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> [[4,4,2,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[4,2,2,2,1,0,0,0,0,0,0],[4,2,2,1,1,0,0,0,0,0],[4,2,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,1,1,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
Description
The order of promotion on a Gelfand-Tsetlin pattern.
Matching statistic: St001491
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 13%●distinct values known / distinct values provided: 3%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 13%●distinct values known / distinct values provided: 3%
Values
[1,0]
=> []
=> => => ? = 1
[1,0,1,0]
=> [1]
=> 10 => 10 => 1
[1,1,0,0]
=> []
=> => => ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 1100 => 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 110 => 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 010 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 10 => 1
[1,1,1,0,0,0]
=> []
=> => => ? = 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 111000 => ? ∊ {1,3,3,5,5,6,16}
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 11100 => ? ∊ {1,3,3,5,5,6,16}
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 101010 => ? ∊ {1,3,3,5,5,6,16}
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 11010 => ? ∊ {1,3,3,5,5,6,16}
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1110 => 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01100 => ? ∊ {1,3,3,5,5,6,16}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0110 => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 10100 => ? ∊ {1,3,3,5,5,6,16}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 1100 => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 110 => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0010 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 010 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 10 => 1
[1,1,1,1,0,0,0,0]
=> []
=> => => ? ∊ {1,3,3,5,5,6,16}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 11110000 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 1111000 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 11010100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 1110100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 111100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 10110010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 1011010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 11010010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 1110010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 111010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 10010110 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 1010110 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 110110 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 11110 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0111000 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 011100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0101010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 011010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 01110 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 1011000 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 101100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 1101000 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 111000 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 11100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 1001010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 101010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 11010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 1110 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 001100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 00110 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 010100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 0110 => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 100100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 10100 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 1100 => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => 110 => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 10000 => 00010 => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1000 => 0010 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 100 => 010 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 10 => 10 => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? ∊ {1,1,1,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 1010101010 => 1111100000 => ? ∊ {1,1,1,1,1,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 110101010 => 111110000 => ? ∊ {1,1,1,1,1,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 1001101010 => 1110101000 => ? ∊ {1,1,1,1,1,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 101101010 => 111101000 => ? ∊ {1,1,1,1,1,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 11101010 => 11111000 => ? ∊ {1,1,1,1,1,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1110 => 1110 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1100 => 0110 => 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1010 => 1100 => 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 110 => 110 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1000 => 0010 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 100 => 010 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 10 => 10 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001722
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 13%●distinct values known / distinct values provided: 3%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 13%●distinct values known / distinct values provided: 3%
Values
[1,0]
=> []
=> []
=> => ? = 1
[1,0,1,0]
=> [1]
=> [1,0]
=> 10 => 1
[1,1,0,0]
=> []
=> []
=> => ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 10 => 1
[1,1,1,0,0,0]
=> []
=> []
=> => ? = 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {2,3,3,5,5,6,16}
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? ∊ {2,3,3,5,5,6,16}
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? ∊ {2,3,3,5,5,6,16}
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? ∊ {2,3,3,5,5,6,16}
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {2,3,3,5,5,6,16}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? ∊ {2,3,3,5,5,6,16}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 10 => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> => ? ∊ {2,3,3,5,5,6,16}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 10111011000100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 111011000100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 10101111000100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> 10111010010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 111010010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 101110100100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> 10 => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> => ? ∊ {1,1,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> 101110111001000100 => ? ∊ {1,1,1,1,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1110111001000100 => ? ∊ {1,1,1,1,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 101011111001000100 => ? ∊ {1,1,1,1,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1011111001000100 => ? ∊ {1,1,1,1,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 11111001000100 => ? ∊ {1,1,1,1,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1,0]
=> 10 => 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St001435
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 13%●distinct values known / distinct values provided: 6%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 13%●distinct values known / distinct values provided: 6%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {5,5,6,16} - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {5,5,6,16} - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {5,5,6,16} - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {5,5,6,16} - 1
[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]
=> [[1,1,1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {1,1,1,1,1,2,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} - 1
[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]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[5,5],[4]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[5,5],[4]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} - 1
Description
The number of missing boxes in the first row.
Matching statistic: St001719
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 12%●distinct values known / distinct values provided: 2%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 12%●distinct values known / distinct values provided: 2%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {1,1,1,2,2,3,3,5,5,6,16}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? ∊ {2,2,3,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,16),(2,16),(3,15),(4,14),(5,14),(5,18),(6,15),(6,19),(7,18),(7,19),(9,13),(10,13),(11,9),(12,10),(13,8),(14,11),(15,12),(16,8),(17,9),(17,10),(18,11),(18,17),(19,12),(19,17)],20)
=> ? ∊ {1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,1,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,15),(2,15),(3,14),(4,13),(5,8),(6,13),(6,16),(7,14),(7,16),(9,12),(10,9),(11,9),(12,8),(13,10),(14,11),(15,12),(16,10),(16,11)],17)
=> ? ∊ {1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,11),(4,11),(5,9),(6,8),(7,8),(7,9),(8,10),(9,10),(10,11)],12)
=> ? ∊ {1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,5,7,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,12),(3,9),(4,8),(5,10),(6,10),(7,8),(7,9),(8,11),(9,11),(10,12),(11,12)],13)
=> ? ∊ {1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,7,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 12%●distinct values known / distinct values provided: 5%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 12%●distinct values known / distinct values provided: 5%
Values
[1,0]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0]
=> [3,1,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0]
=> [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,3,5,5,6,16} + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,3,5,5,6,16} + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,3,5,5,6,16} + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,3,5,5,6,16} + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,3,5,5,6,16} + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,3,5,5,6,16} + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,3,5,5,6,16} + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,3,5,5,6,16} + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,4] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,3,4,4,5,5,5,5,6,9,9,10,10,14,14,16,20,21,21,35,35,42,56,56,70,70,90,168,168,216,216,768} + 1
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,6] => ([(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,5,5,5,5,6,9,9,10,10,14,14,14,14,14,14,15,15,16,20,21,21,28,28,35,35,35,35,42,42,42,42,56,56,64,64,70,70,70,84,84,90,120,120,162,162,168,168,189,189,210,210,216,216,252,252,288,288,300,300,448,450,450,462,462,525,525,567,567,660,660,768,825,825,990,990,1155,1155,1188,1320,1320,1540,1540,2112,2112,2310,2310,2640,2970,2970,3520,3520,4158,4158,4455,4455,5775,5775,7700,8580,11583,11583,12870,12870,15015,15015,16016,21450,21450,48048,48048,64064,64064,68640,292864} + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001720The minimal length of a chain of small intervals in a lattice. St001768The number of reduced words of a signed permutation. St000884The number of isolated descents of a permutation. St000764The number of strong records in an integer composition. St000665The number of rafts of a permutation. St000237The number of small exceedances. St000534The number of 2-rises of a permutation. St001438The number of missing boxes of a skew partition.
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