Your data matches 49 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [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: St000025
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(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
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,3),(0,4),(1,2),(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]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [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]
 => 6
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
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: St000676
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(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
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,3),(0,4),(1,2),(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]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 6
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 6
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
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: St001007
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(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
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,3),(0,4),(1,2),(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]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [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]
 => 6
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001809
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(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
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,3),(0,4),(1,2),(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]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [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]
 => 6
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
Description
The index of the step at the first peak of maximal height in a Dyck path.
Matching statistic: St000024
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 4 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4 = 5 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4 = 5 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(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]
 => 5 = 6 - 1
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4 = 5 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4 = 5 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [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]
 => 5 = 6 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 - 1
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 - 1
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 - 1
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 - 1
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 - 1
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6 - 1
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6 - 1
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 - 1
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000439
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(0,1),(0,2),(0,3),(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
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 7 = 6 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,2),(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]
 => 7 = 6 + 1
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [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]
 => 7 = 6 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St001504
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(0,1),(0,2),(0,3),(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
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 7 = 6 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,2),(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]
 => 7 = 6 + 1
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [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]
 => 7 = 6 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10 + 1
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000444
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 91%●distinct values known / distinct values provided: 71%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(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
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,3),(0,4),(1,2),(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]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [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]
 => 6
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => [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]
 => 6
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 6
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001039
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 91%●distinct values known / distinct values provided: 71%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([],2)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(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
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],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]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,3),(0,4),(1,2),(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]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 5
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 6
([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 6
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000443The number of long tunnels of a Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000738The first entry in the last row of a standard tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000005The bounce statistic of a Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001268The size of the largest ordinal summand in the poset. St001779The order of promotion on the set of linear extensions of a poset. St000172The Grundy number of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000537The cutwidth of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001963The tree-depth of a graph. St000171The degree of the graph. St001349The number of different graphs obtained from the given graph by removing an edge.