Identifier
-
Mp00110:
Posets
—Greene-Kleitman invariant⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
([],1) => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 2
([],2) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 4
([(0,1)],2) => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
([],3) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6
([(0,2),(2,1)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([],4) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 8
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
([(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
([],5) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 10
([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,5),(1,4),(1,5),(4,2),(5,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,4),(0,5),(1,4),(1,5),(2,3)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,3),(1,2),(1,4),(2,5),(3,4),(3,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,5),(1,4),(4,2),(5,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,3),(1,4),(3,5),(4,2),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,5),(1,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,5),(1,4),(2,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,5),(1,4),(1,5),(2,3),(2,5)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6),(6,1)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,6),(6,1)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,2),(1,3),(1,4),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(2,3),(2,4),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(2,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(2,5),(5,6),(6,3),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,5),(5,6),(6,2),(6,3),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,5),(5,6),(6,1),(6,2),(6,3),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,6),(3,6),(4,5),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,6),(2,6),(3,6),(4,5),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,6),(3,5),(4,5),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(2,6),(3,6),(4,5),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,6),(2,5),(3,5),(5,6),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,2),(1,3),(1,5),(4,6),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,3),(0,4),(0,5),(1,6),(2,6),(5,1),(5,2)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,2),(1,4),(1,5),(3,6),(4,6),(5,3)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,3),(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,3),(2,4),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,3),(2,4),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,3),(1,5),(4,6),(5,2),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,4),(0,5),(2,6),(3,6),(5,1),(5,2),(5,3)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,4),(1,5),(3,6),(4,6),(5,2),(5,3)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,3),(2,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,2),(1,5),(3,6),(4,6),(5,3),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,4),(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,6),(2,6),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,5),(4,6),(5,2),(5,3),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,5),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,3),(1,6),(2,6),(3,4),(3,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,5),(3,6),(4,6),(5,2),(5,3),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,5),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,3),(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(1,6),(2,5),(3,5),(4,6),(5,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,5),(3,5),(5,6),(6,4)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
Greene-Kleitman invariant
Description
The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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