searching the database
Your data matches 76 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: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001232: 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]
=> [1,1,0,0]
=> 0
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(5,6)],7)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(4,6),(5,6)],7)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000676
(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
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 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]
=> 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]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 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
([(0,1),(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]
=> 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]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 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]
=> 3 = 2 + 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
([(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),(2,3),(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
([(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),(1,4),(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
([(1,2),(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]
=> 3 = 2 + 1
([(1,4),(2,3),(2,5),(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
([(0,1),(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
([(0,4),(1,5),(2,3),(2,5),(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),(1,4),(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
([(0,5),(1,4),(2,3),(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),(2,3),(2,4),(3,4)],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
([(0,4),(1,2),(1,3),(2,5),(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
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],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
([(0,4),(1,2),(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
([(0,4),(1,4),(2,3),(2,5),(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
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [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
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [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
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [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
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 1
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001183
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
St001183: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001183: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 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]
=> 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 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]
=> 3 = 2 + 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]
=> 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 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]
=> 3 = 2 + 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(1,2),(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]
=> 3 = 2 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(1,2),(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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
Description
The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001258
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
St001258: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 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]
=> 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 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]
=> 3 = 2 + 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]
=> 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 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]
=> 3 = 2 + 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(1,2),(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]
=> 3 = 2 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(1,2),(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]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 11 + 1
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.
For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
Matching statistic: St000444
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 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]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 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]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 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]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(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]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
([(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]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(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]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(0,4),(1,2),(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]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000659
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 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]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 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]
=> 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 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]
=> 1 = 2 - 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 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]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,2),(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 = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [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 = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [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 = 2 - 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [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 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [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 = 2 - 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [4,1,1]
=> [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 = 2 - 1
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11 - 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11 - 1
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11 - 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001031
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 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]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 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]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 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]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 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]
=> [1,1,0,1,0,1,1,0,0,0]
=> 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]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 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]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 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]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(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]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11 - 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11 - 1
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 11 - 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St001035
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 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]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 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]
=> 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 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]
=> 1 = 2 - 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 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]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,2),(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 = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [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 = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 - 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [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 = 2 - 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [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 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [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 = 2 - 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [4,1,1]
=> [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 = 2 - 1
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11 - 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11 - 1
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 11 - 1
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St001039
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 + 1
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 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]
=> 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]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 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
([(0,1),(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]
=> 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]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 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]
=> 3 = 2 + 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
([(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),(2,3),(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
([(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),(1,4),(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
([(1,2),(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]
=> 3 = 2 + 1
([(1,4),(2,3),(2,5),(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
([(0,1),(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
([(0,4),(1,5),(2,3),(2,5),(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),(1,4),(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
([(0,5),(1,4),(2,3),(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),(2,3),(2,4),(3,4)],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
([(0,4),(1,2),(1,3),(2,5),(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
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],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
([(0,4),(1,2),(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
([(0,4),(1,4),(2,3),(2,5),(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
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [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
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [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
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [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
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [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
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 4 = 3 + 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [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
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [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
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [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
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 1
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001198
(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
St001198: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 76%●distinct values known / distinct values provided: 40%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 76%●distinct values known / distinct values provided: 40%
Values
([(0,1)],2)
=> [1]
=> [1,0]
=> ? = 0
([(1,2)],3)
=> [1]
=> [1,0]
=> ? = 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(2,3)],4)
=> [1]
=> [1,0]
=> ? = 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
([(3,4)],5)
=> [1]
=> [1,0]
=> ? = 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(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
([(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
([(0,1),(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
([(4,5)],6)
=> [1]
=> [1,0]
=> ? = 0
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
([(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
([(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),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(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),(1,4),(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
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,4),(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
([(0,5),(1,4),(2,3),(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),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(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
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(5,6)],7)
=> [1]
=> [1,0]
=> ? = 0
([(4,6),(5,6)],7)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> ? = 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 11
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 11
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 11
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 11
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 66 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. 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. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001624The breadth of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St000071The number of maximal chains in a poset. St000080The rank of the poset. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000527The width of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001779The order of promotion on the set of linear extensions of a poset. St000528The height of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001782The order of rowmotion on the set of order ideals of a poset. St001812The biclique partition number of a graph. St001645The pebbling number of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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!