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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000806
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000806: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => 3
[1,1,0,0]
=> [2] => 3
[1,0,1,0,1,0]
=> [1,1,1] => 4
[1,0,1,1,0,0]
=> [1,2] => 4
[1,1,0,0,1,0]
=> [2,1] => 4
[1,1,0,1,0,0]
=> [3] => 4
[1,1,1,0,0,0]
=> [3] => 4
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 5
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 5
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 5
[1,0,1,1,0,1,0,0]
=> [1,3] => 5
[1,0,1,1,1,0,0,0]
=> [1,3] => 5
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 5
[1,1,0,0,1,1,0,0]
=> [2,2] => 4
[1,1,0,1,0,0,1,0]
=> [3,1] => 5
[1,1,0,1,0,1,0,0]
=> [4] => 5
[1,1,0,1,1,0,0,0]
=> [4] => 5
[1,1,1,0,0,0,1,0]
=> [3,1] => 5
[1,1,1,0,0,1,0,0]
=> [4] => 5
[1,1,1,0,1,0,0,0]
=> [4] => 5
[1,1,1,1,0,0,0,0]
=> [4] => 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 6
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 6
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 6
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 6
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 6
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 6
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St001880
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 62%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 62%
Values
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? = 3 - 1
[1,1,0,0]
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> ? = 3 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 4 - 1
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 4 - 1
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 5 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 5 - 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 5 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 5 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 1
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 1
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 5 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ? = 7 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 62%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 62%
Values
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? = 3 - 2
[1,1,0,0]
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> ? = 3 - 2
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 4 - 2
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 4 - 2
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 5 - 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 5 - 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 5 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 5 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 4 - 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? = 6 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 6 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 6 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 5 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 7 - 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001645
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 75%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 75%
Values
[1,0,1,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 3 - 1
[1,1,0,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 4 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 4 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 5 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 5 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 5 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 5 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 5 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 5 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 5 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 5 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,3,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,4,6,3,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,4,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,5,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,4,6,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,3,6,5,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,5,3,6,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,5,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,4,3,6,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,3,1] => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,6,3,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,5,7,3,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,5,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,6,5,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,5,4,6,7,3,1] => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,5,4,7,6,3,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,5,6,4,7,3,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,4,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,6,4,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,6,5,4,7,3,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,5,7,4,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,5,4,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,5,7,2,1] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 8 - 1
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 75%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 75%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ? = 4 - 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> ? = 4 - 2
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> ? = 4 - 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> ? = 4 - 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ? = 5 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? = 4 - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ? = 5 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ? = 5 - 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ? = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ? = 6 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ? = 5 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ? = 6 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ? = 6 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> ? = 6 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 6 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> ? = 6 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 7 - 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 7 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 7 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,3,2,4,5,1] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [6,3,2,5,4,1] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 7 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 7 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,2,3,4,5,1,7] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 8 - 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,5,4,1,7] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 8 - 2
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [7,2,3,5,4,6,1] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => [1,7,2,3,4,6,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [6,2,4,3,5,1,7] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [7,2,4,3,5,6,1] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [7,2,4,3,6,5,1] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,4,5,3,1,7] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [7,2,4,5,3,6,1] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => [1,7,2,3,4,6,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,2,5,4,3,1,7] => [1,6,2,3,5,4,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [7,2,5,4,3,6,1] => [1,7,2,3,5,4,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [7,2,5,4,6,3,1] => [1,7,2,3,5,6,4] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => [1,7,2,3,6,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 8 - 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000718
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 62%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 62%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 3
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 5
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 5
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 5
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 6
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 6
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 5
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 6
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 6
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 6
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 6
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 6
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 7
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 7
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 6
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 6
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 7
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 7
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 7
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St001637
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 5 - 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 5 - 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 6 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 6 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 6 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 6 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 6 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 7 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 7 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 7 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 7 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 7 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 6 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 7 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 6 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 5 - 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 5 - 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 6 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 6 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 6 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 6 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 6 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 7 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 7 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 7 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 7 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 7 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 7 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 6 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 7 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 6 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 7 - 2
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St000528
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 75%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 75%
Values
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 4 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 5 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 6 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,6,4,3,2,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,6,5,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [5,4,6,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,5,6,3,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [3,6,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [5,3,6,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [4,3,6,5,2,1] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [5,4,3,6,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,5,3,6,2,1] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000643
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000643: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 62%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000643: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 62%
Values
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 3
[1,1,0,0]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 3
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 4
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 5
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 5
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 5
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 5
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 5
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 5
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 5
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 6
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 6
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 6
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 5
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 5
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 6
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 5
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 6
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 6
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 5
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 6
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 6
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 6
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 6
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 6
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,6,4,3,2,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,6,5,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [5,4,6,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,5,6,3,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [3,6,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 7
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [5,3,6,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [4,3,6,5,2,1] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [5,4,3,6,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,5,3,6,2,1] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? = 7
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7
Description
The size of the largest orbit of antichains under Panyushev complementation.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000080The rank of the poset. St000906The length of the shortest maximal chain in a poset. St001875The number of simple modules with projective dimension at most 1. St000019The cardinality of the support of a permutation. St000209Maximum difference of elements in cycles. St000216The absolute length of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
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