Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000053
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
Description
The number of valleys of the Dyck path.
Matching statistic: St000331
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
Description
The number of upper interactions of a Dyck path. An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≥ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Matching statistic: St001169
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001185
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Matching statistic: St001215
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001215: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
Description
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module. For the first 196 values, the statistic also gives the number of indecomposable non-projective modules $X$ such that $\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.
Matching statistic: St001227
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001509
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001509: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
Description
The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$. This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly below the diagonal and relative to the trivial lower boundary.
Matching statistic: St000015
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000015: 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 = 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
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
Description
The number of peaks of a Dyck path.
Matching statistic: St001068
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001239
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 1 = 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
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 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),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 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),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 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),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001622The number of join-irreducible elements of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000640The rank of the largest boolean interval in a poset.