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Matching statistic: St000806
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(load all 13 compositions to match this statistic)
St000806: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => 3
[2] => 3
[1,1,1] => 4
[1,2] => 4
[2,1] => 4
[3] => 4
[1,1,1,1] => 5
[1,1,2] => 5
[1,2,1] => 5
[1,3] => 5
[2,1,1] => 5
[2,2] => 4
[3,1] => 5
[4] => 5
[1,1,1,1,1] => 6
[1,1,1,2] => 6
[1,1,2,1] => 6
[1,1,3] => 6
[1,2,1,1] => 6
[1,2,2] => 5
[1,3,1] => 6
[1,4] => 6
[2,1,1,1] => 6
[2,1,2] => 6
[2,2,1] => 5
[2,3] => 5
[3,1,1] => 6
[3,2] => 5
[4,1] => 6
[5] => 6
[1,1,1,1,1,1] => 7
[1,1,1,1,2] => 7
[1,1,1,2,1] => 7
[1,1,1,3] => 7
[1,1,2,1,1] => 7
[1,1,2,2] => 6
[1,1,3,1] => 7
[1,1,4] => 7
[1,2,1,1,1] => 7
[1,2,1,2] => 7
[1,2,2,1] => 6
[1,2,3] => 6
[1,3,1,1] => 7
[1,3,2] => 6
[1,4,1] => 7
[1,5] => 7
[2,1,1,1,1] => 7
[2,1,1,2] => 7
[2,1,2,1] => 7
[2,1,3] => 7
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: St000672
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [3,1,2] => [1,3,2] => 1 = 3 - 2
[2] => [1,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 3 - 2
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 2 = 4 - 2
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 2 = 4 - 2
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => [1,3,4,2] => 2 = 4 - 2
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 2 = 4 - 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 3 = 5 - 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,4,5,1,3] => 3 = 5 - 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,4,5,2] => 3 = 5 - 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,2,4] => 3 = 5 - 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,3,2,5,4] => 2 = 4 - 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3 = 5 - 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,3,4,5,2] => 3 = 5 - 2
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 3 = 5 - 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 4 = 6 - 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,5,6,1,4] => 4 = 6 - 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,4,1,5,6,3] => 4 = 6 - 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,4,6,1,3,5] => 4 = 6 - 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,4,2,6,5] => 3 = 5 - 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,5,2,6,1,3] => 4 = 6 - 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,5,6,2] => 4 = 6 - 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,6,1,2,4,5] => 4 = 6 - 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,3,2,4,6,5] => 3 = 5 - 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,4,5,6,1,3] => 4 = 6 - 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,2,1,5,6,3] => 3 = 5 - 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,6,1,3,5] => 4 = 6 - 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,3,4,2,6,5] => 3 = 5 - 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => 4 = 6 - 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,3,4,5,6,2] => 4 = 6 - 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 4 = 6 - 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 5 = 7 - 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [2,3,4,6,7,1,5] => 5 = 7 - 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [2,3,5,1,6,7,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [2,3,5,7,1,4,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [2,4,1,5,3,7,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [3,5,6,2,7,1,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [2,4,1,5,6,7,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [2,4,7,1,3,5,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,4,2,5,7,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [4,2,5,6,7,1,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [4,5,2,1,6,7,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [4,5,2,7,1,3,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,4,5,2,7,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [4,6,2,3,7,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,4,5,6,7,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,7,1,2,4,5,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [1,3,2,4,5,7,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,4,3,6,7,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,4,5,1,6,7,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,4,5,7,1,3,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [4,2,1,5,3,7,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [5,3,6,2,7,1,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [4,2,1,5,6,7,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,7,1,3,5,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [1,3,4,2,5,7,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,4,5,6,7,1,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [5,2,3,1,6,7,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,2,3,7,1,4,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [1,3,4,5,2,7,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,3,4,5,6,7,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 5 = 7 - 2
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 6 = 8 - 2
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [2,3,4,5,7,8,1,6] => 6 = 8 - 2
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [2,3,4,6,1,7,8,5] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [2,3,4,6,8,1,5,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => [2,3,5,1,6,4,8,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => [3,4,6,7,2,8,1,5] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => [2,3,5,1,6,7,8,4] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [2,3,5,8,1,4,6,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [2,4,1,5,3,6,8,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => [3,5,2,6,7,8,1,4] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [3,5,6,2,1,7,8,4] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => [3,5,6,2,8,1,4,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => [2,4,1,5,6,3,8,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => [3,5,7,2,4,8,1,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => [2,4,1,5,6,7,8,3] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [2,4,8,1,3,5,6,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => [3,1,4,2,5,6,8,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => [4,2,5,3,7,8,1,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => [4,2,5,6,1,7,8,3] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => [4,2,5,6,8,1,3,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => [4,5,2,1,6,3,8,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => [5,6,3,7,2,8,1,4] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => [4,5,2,1,6,7,8,3] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => 6 = 8 - 2
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 7 = 9 - 2
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => 7 = 9 - 2
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 8 = 10 - 2
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [10,1,2,3,4,5,6,7,8,9] => 8 = 10 - 2
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000216
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 2 = 3 - 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 4 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 4 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3 = 4 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4 = 5 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 4 = 5 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4 = 5 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 4 = 5 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 4 = 5 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4 = 5 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 4 = 5 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 4 = 5 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 4 = 5 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 5 = 6 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 5 = 6 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 5 = 6 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 5 = 6 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 5 = 6 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 4 = 5 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 5 = 6 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 5 = 6 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 5 = 6 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 5 = 6 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 5 = 6 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5 = 6 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,4,8,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [6,4,1,2,3,7,8,5] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [8,4,1,2,6,3,5,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,4,1,2,7,3,8,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [5,3,1,2,6,7,8,4] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [8,3,1,6,2,4,5,7] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
Description
The absolute length of a permutation.
The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.
Matching statistic: St001583
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 62%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 62%
Values
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 3 - 1
[2] => [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3 = 4 - 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 4 - 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 4 - 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 4 - 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ? ∊ {4,5} - 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 5 - 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 5 - 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 5 - 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 5 - 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 5 - 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 5 - 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ? ∊ {4,5} - 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {5,5,5,5,6,6,6,6} - 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {5,5,5,5,6,6,6,6} - 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {5,5,5,5,6,6,6,6} - 1
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {5,5,5,5,6,6,6,6} - 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5 = 6 - 1
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? ∊ {5,5,5,5,6,6,6,6} - 1
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {5,5,5,5,6,6,6,6} - 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5 = 6 - 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5 = 6 - 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 6 - 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 6 - 1
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? ∊ {5,5,5,5,6,6,6,6} - 1
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ? ∊ {5,5,5,5,6,6,6,6} - 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6 = 7 - 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6 = 7 - 1
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6 = 7 - 1
[2,2,1,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,3,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6 = 7 - 1
[2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,1,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,1,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6 = 7 - 1
[3,2,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6 = 7 - 1
[3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[4,1,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[4,2] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[5,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,2,2] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,1,4] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,2,1,2] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,2,2,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,2,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,3,1,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,1,3,2] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001637
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[1,1] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 3 - 2
[2] => [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[2,1] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[3] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 5 - 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[2,2] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[3,1] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[4] => [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
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[1,4] => [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)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[4,1] => [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)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[5] => [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)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(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)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,3,1] => [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)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,4] => [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)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(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)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[1,1] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 3 - 2
[2] => [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[2,1] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[3] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 5 - 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[2,2] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[3,1] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[4] => [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
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[1,4] => [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)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[4,1] => [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)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[5] => [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)
=> ? ∊ {5,5,5,5,6,6,6,6} - 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(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)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,3,1] => [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)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[2,4] => [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)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(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)
=> ? ∊ {5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 2
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
Description
The number of points of the poset minus the width of the poset.
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