Identifier
-
Mp00251:
Graphs
—clique sizes⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001493: Dyck paths ⟶ ℤ
Values
([],1) => [1] => [1,0] => [1,0] => 1
([],2) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(0,1)],2) => [2] => [1,0,1,0] => [1,1,0,0] => 1
([],3) => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
([(1,2)],3) => [2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(0,2),(1,2)],3) => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 1
([(0,1),(0,2),(1,2)],3) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
([],4) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 1
([(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 1
([(1,3),(2,3)],4) => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 2
([(0,3),(1,3),(2,3)],4) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 1
([(0,3),(1,2)],4) => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 1
([(0,3),(1,2),(2,3)],4) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 1
([(1,2),(1,3),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 1
([(0,2),(0,3),(1,2),(1,3)],4) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
([],5) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
([(3,4)],5) => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
([(2,4),(3,4)],5) => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 2
([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 2
([(0,4),(1,4),(2,4),(3,4)],5) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1
([(1,4),(2,3)],5) => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 2
([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 2
([(0,1),(2,4),(3,4)],5) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 1
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1
([(1,4),(2,3),(2,4),(3,4)],5) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1
([(1,3),(1,4),(2,3),(2,4)],5) => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1
([(0,4),(1,3),(2,3),(2,4)],5) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1
([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(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
([],6) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
([(4,5)],6) => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 1
([(3,5),(4,5)],6) => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
([(2,5),(3,5),(4,5)],6) => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(2,5),(3,4)],6) => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 2
([(2,5),(3,4),(4,5)],6) => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
([(1,2),(3,5),(4,5)],6) => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 2
([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1
([(1,5),(2,5),(3,4),(4,5)],6) => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1
([(2,5),(3,4),(3,5),(4,5)],6) => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 1
([(1,5),(2,4),(3,4),(3,5)],6) => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 1
>>> Load all 252 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
clique sizes
Description
The integer partition of the sizes of the maximal cliques of a graph.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!