searching the database
Your data matches 20 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001184
(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
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 2
([(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
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
([(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
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> 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]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
([(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]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 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]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(4,5)],6)
=> [1]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(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]
=> 2
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 2
([(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
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St000011
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ 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
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 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
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(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
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 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
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 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]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(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
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 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
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> [1,1,0,1,0,0,1,0]
=> 2
([(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]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 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
Mp00028: Dyck paths —reverse⟶ 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
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(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]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(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]
=> 2
([(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]
=> 2
([(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
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 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]
=> 2
([(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]
=> 2
([(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]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 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]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(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]
=> 2
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(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]
=> 2
([(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]
=> 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: St001210
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001210: 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
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(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]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(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]
=> 2
([(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]
=> 2
([(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
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 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]
=> 2
([(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]
=> 2
([(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]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 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]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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]
=> 2
([(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]
=> 2
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
([(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
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(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]
=> 2
([(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]
=> 2
Description
Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001239
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
St001239: 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
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 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
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 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]
=> 2
([(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
([(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
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 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
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 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]
=> [1,1,1,0,0,0,1,0,1,0]
=> 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]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
([(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
([(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]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 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
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> 2
([(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]
=> 2
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 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]
=> 2
([(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]
=> 2
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001530
(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
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001530: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001530: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 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
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> [1,1,1,0,1,0,0,0]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
([(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
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 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
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 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]
=> [1,1,1,1,0,0,0,1,0,0]
=> 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]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
([(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
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 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
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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]
=> [1,1,1,0,1,0,0,0]
=> 2
([(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]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(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
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
Description
The depth of a Dyck path. That is the depth of the corresponding Nakayama algebra with a linear quiver.
Matching statistic: St000439
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ 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
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
([(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]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
([(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 = 2 + 1
([(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]
=> 3 = 2 + 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]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> [1,1,0,1,0,0,1,0,1,0]
=> 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]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(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]
=> 3 = 2 + 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]
=> 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]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 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]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 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]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
([(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 = 2 + 1
([(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 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St001067
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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]
=> [1,0,1,0]
=> 1 = 2 - 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
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 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]
=> [1,1,1,0,1,1,0,0,0,0]
=> 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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,0,1,1,0,0,0,0]
=> 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]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001185
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
St001185: 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
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 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]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 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]
=> 1 = 2 - 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]
=> 1 = 2 - 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]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 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]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 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]
=> [1,1,1,0,0,0,1,0,1,0]
=> 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]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 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]
=> 1 = 2 - 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]
=> 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]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 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]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 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]
=> 1 = 2 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 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]
=> 1 = 2 - 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]
=> 1 = 2 - 1
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Matching statistic: St001223
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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]
=> [1,0,1,0]
=> 1 = 2 - 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
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 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]
=> [1,1,1,0,1,1,0,0,0,0]
=> 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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,0,1,1,0,0,0,0]
=> 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]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 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]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 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
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000444The length of the maximal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000640The rank of the largest boolean interval in a poset.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!