Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000676
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1,0,0,1,0]
 => 2
([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 4
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 4
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 4
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 4
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: St000443
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 4
Description
The number of long tunnels of a Dyck path. A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St001007
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 4
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001187
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001187: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 4
Description
The number of simple modules with grade at least one in the corresponding Nakayama algebra.
Matching statistic: St001224
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001224: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 4
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 first Ext-group between X and the regular module.
Matching statistic: St000024
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 2 = 3 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 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: St001875
Mapping
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 2
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 3
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 3
([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 3
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 3
([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 4
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ? = 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ? = 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ? = 4
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ? = 3
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ? = 4
([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ? = 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([],1)
 => ? = 4
([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([],1)
 => ? = 4
([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([],1)
 => ? = 4
([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(5,3),(6,2)],7)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ? = 4
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ? = 4
([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ? = 4
([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ? = 4
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001630
Mapping
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 2 - 2
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 2 - 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 3 - 2
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 3 - 2
([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 4 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 4 - 2
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 4 - 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ? = 3 - 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ? = 4 - 2
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ? = 4 - 2
([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ? = 3 - 2
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([],1)
 => ? = 4 - 2
([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([],1)
 => ? = 4 - 2
([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([],1)
 => ? = 4 - 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(5,3),(6,2)],7)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ? = 4 - 2
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ? = 4 - 2
([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ? = 4 - 2
([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ? = 4 - 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St000937
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1]
 => [1]
 => []
 => ? = 2 - 2
([],2)
 => [1,1]
 => [2]
 => []
 => ? = 2 - 2
([(0,1)],2)
 => [2]
 => [1,1]
 => [1]
 => ? = 3 - 2
([],3)
 => [1,1,1]
 => [2,1]
 => [1]
 => ? = 3 - 2
([(0,1),(0,2)],3)
 => [2,1]
 => [3]
 => []
 => ? = 3 - 2
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,2),(1,2)],3)
 => [2,1]
 => [3]
 => []
 => ? = 3 - 2
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [2,2]
 => [2]
 => 2 = 4 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [2,2]
 => [2]
 => 2 = 4 - 2
([(0,3),(1,2)],4)
 => [2,2]
 => [4]
 => []
 => ? = 4 - 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [4]
 => []
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [4]
 => []
 => ? = 2 - 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => [1]
 => ? = 3 - 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => [1]
 => ? = 3 - 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [5]
 => []
 => ? = 3 - 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [5]
 => []
 => ? = 4 - 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [5]
 => []
 => ? = 3 - 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [4,1]
 => [1]
 => ? = 3 - 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [4,1]
 => [1]
 => ? = 3 - 2
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [4,1]
 => [1]
 => ? = 3 - 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [5]
 => []
 => ? = 3 - 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [4,1]
 => [1]
 => ? = 3 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [5]
 => []
 => ? = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [3,2]
 => [5]
 => []
 => ? = 3 - 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [2,2,1]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [5]
 => []
 => ? = 3 - 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [5,1]
 => [1]
 => ? = 4 - 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [3,2,1]
 => [5,1]
 => [1]
 => ? = 4 - 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [3,2,1]
 => [5,1]
 => [1]
 => ? = 4 - 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [3,3]
 => [6]
 => []
 => ? = 4 - 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => [3,3]
 => [6]
 => []
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [3,3]
 => [6]
 => []
 => ? = 4 - 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => [3,3]
 => [6]
 => []
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [3,2,1]
 => [5,1]
 => [1]
 => ? = 4 - 2
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3]
 => [6]
 => []
 => ? = 3 - 2
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 4 - 2
([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(5,3),(6,2)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => [4,3]
 => [7]
 => []
 => ? = 4 - 2
Description
The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Values
([],1)
 => ([],1)
 => ([],1)
 => ? = 2 - 2
([],2)
 => ([(0,1)],2)
 => ([],1)
 => ? = 2 - 2
([(0,1)],2)
 => ([],2)
 => ([],2)
 => ? = 3 - 2
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ? = 3 - 2
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => ? = 3 - 2
([(0,2),(2,1)],3)
 => ([],3)
 => ([],3)
 => ? = 3 - 2
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => ? = 3 - 2
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 4 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([],3)
 => ? = 3 - 2
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([],3)
 => ? = 3 - 2
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([],3)
 => ? = 3 - 2
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 4 - 2
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ? = 4 - 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => ? = 2 - 2
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],4)
 => ? = 4 - 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 3 - 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([],4)
 => ? = 4 - 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 3 - 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 4 - 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 3 - 2
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 3 - 2
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 3 - 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 3 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 3 - 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([],4)
 => ? = 4 - 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 3 - 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 3 - 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([],4)
 => ? = 4 - 2
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 3 - 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([],4)
 => ? = 4 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ? = 4 - 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 4 - 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => ([],4)
 => ? = 4 - 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([],4)
 => ? = 4 - 2
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 4 - 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ? = 4 - 2
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ? = 4 - 2
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 4 - 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => ([],4)
 => ? = 4 - 2
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([],3)
 => ? = 4 - 2
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ? = 4 - 2
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4)],6)
 => ([],4)
 => ? = 4 - 2
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 4 - 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],2)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 4 - 2
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.