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Your data matches 30 different statistics following compositions of up to 3 maps.
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Matching statistic: St000678
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Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000052
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [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
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [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
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [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
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000025
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000053
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St001498
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001733
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
Description
The number of weak left to right maxima of a Dyck path.
A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its
left.
Matching statistic: St000439
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000674
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St001068
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 2 = 1 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001622The number of join-irreducible elements of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001596The number of two-by-two squares inside a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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