Identifier
-
Mp00251:
Graphs
—clique sizes⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001296: Dyck paths ⟶ ℤ
Values
([],1) => [1] => [1,0] => [1,0] => 0
([],2) => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
([(0,1)],2) => [2] => [1,0,1,0] => [1,1,0,0] => 0
([],3) => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 2
([(1,2)],3) => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
([(0,2),(1,2)],3) => [2,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 1
([(0,1),(0,2),(1,2)],3) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0
([],4) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 3
([(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 2
([(1,3),(2,3)],4) => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
([(0,3),(1,3),(2,3)],4) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(0,3),(1,2)],4) => [2,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 1
([(0,3),(1,2),(2,3)],4) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(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
([(0,3),(1,2),(1,3),(2,3)],4) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 1
([(0,2),(0,3),(1,2),(1,3)],4) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 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] => 0
([],5) => [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
([(3,4)],5) => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 3
([(2,4),(3,4)],5) => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 3
([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3
([(0,4),(1,4),(2,4),(3,4)],5) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
([(1,4),(2,3)],5) => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3
([(0,1),(2,4),(3,4)],5) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(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),(1,4),(2,3),(3,4)],5) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 2
([(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] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3
([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 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] => [1,1,1,0,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,1,0,0,0] => 2
([(0,4),(1,3),(2,3),(2,4)],5) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 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] => [1,1,1,0,0,1,0,1,0,0] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,1,0,0,0] => 2
([(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] => [1,1,1,0,1,0,0,1,0,0] => 2
([(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] => [1,1,1,1,0,0,0,0,1,0] => 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] => [1,1,1,1,0,0,0,1,0,0] => 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] => [1,1,1,1,0,0,1,0,0,0] => 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] => [1,1,1,0,1,0,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,1,0,1,0,0,0] => 2
([(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] => [1,1,1,1,0,1,0,0,0,0] => 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] => 0
([],6) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
([(4,5)],6) => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
([(3,5),(4,5)],6) => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 4
([(2,5),(3,5),(4,5)],6) => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 4
([(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] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3
([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(2,5),(3,4)],6) => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 3
([(2,5),(3,4),(4,5)],6) => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 4
([(1,2),(3,5),(4,5)],6) => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3
([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
([(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] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3
([(0,1),(2,5),(3,5),(4,5)],6) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,0,1,0,0,1,0,1,0] => 3
([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(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] => [1,1,1,0,0,1,0,1,0,0,1,0] => 3
([(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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(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] => [1,1,1,0,1,0,0,0,1,0,1,0] => 3
([(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] => [1,1,1,0,0,1,0,1,0,0,1,0] => 3
([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(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] => [1,1,1,0,1,0,0,1,0,0,1,0] => 3
([(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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3
([(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] => [1,1,1,0,1,0,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,1,0,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,1,0,1,0,0,0] => 2
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(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] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3
([(0,1),(2,5),(3,4),(4,5)],6) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 2
([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(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] => [1,1,1,0,0,1,0,1,0,0,1,0] => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,0,0,1,0] => 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] => [1,1,1,0,1,0,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,0,1,0,0,1,0] => 3
([(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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3
([(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] => [1,1,1,0,1,0,1,0,0,1,0,0] => 2
([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(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] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
([(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] => [1,1,1,0,1,0,0,1,0,0] => 2
([(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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
>>> 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 maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras.
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
Delest-Viennot-inverse
Description
Return the Dyck path obtained by applying the inverse of Delest-Viennot's bijection to the corresponding parallelogram polyomino.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\beta^{(-1)}\circ\gamma)(D)$.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\beta^{(-1)}\circ\gamma)(D)$.
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!