Identifier
-
Mp00229:
Dyck paths
—Delest-Viennot⟶
Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000786: Graphs ⟶ ℤ
Values
[1,0] => [1,0] => ([],1) => ([],1) => 1
[1,0,1,0] => [1,1,0,0] => ([(0,1)],2) => ([],2) => 2
[1,1,0,0] => [1,0,1,0] => ([(0,1)],2) => ([],2) => 2
[1,0,1,0,1,0] => [1,1,0,1,0,0] => ([(0,2),(2,1)],3) => ([],3) => 3
[1,0,1,1,0,0] => [1,1,0,0,1,0] => ([(0,2),(2,1)],3) => ([],3) => 3
[1,1,0,0,1,0] => [1,0,1,1,0,0] => ([(0,2),(2,1)],3) => ([],3) => 3
[1,1,0,1,0,0] => [1,1,1,0,0,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => ([],3) => 3
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 4
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 4
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 4
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => 4
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 4
[1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 4
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 4
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => 4
[1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 4
[1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => 4
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 4
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => 4
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 4
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([],4) => 4
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => 5
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => 5
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => 5
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => 5
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => 5
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => 5
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => 5
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => ([(3,6),(4,5)],7) => 5
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => 5
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => 5
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => 5
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => 5
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(3,6),(4,5),(5,6)],7) => 5
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => 6
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(5,6)],7) => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(5,6)],7) => 6
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(5,6)],7) => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(5,6)],7) => 6
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => 6
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => 6
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => 6
[1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => 6
[1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(5,6)],7) => 6
[1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(5,6)],7) => 6
[1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 6
[1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => ([(5,6)],7) => 6
[1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => ([(5,6)],7) => 6
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Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Map
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
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 $(\gamma^{(-1)}\circ\beta)(D)$.
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 $(\gamma^{(-1)}\circ\beta)(D)$.
Map
parallelogram poset
Description
The cell poset of the parallelogram polyomino corresponding to the Dyck path.
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.
This map returns the cell poset of $\gamma(D)$. In this partial order, the cells of the polyomino are the elements and a cell covers those cells with which it shares an edge and which are closer to the origin.
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.
This map returns the cell poset of $\gamma(D)$. In this partial order, the cells of the polyomino are the elements and a cell covers those cells with which it shares an edge and which are closer to the origin.
Map
incomparability graph
Description
The incomparability graph of a poset.
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