Identifier
-
Mp00227:
Dyck paths
—Delest-Viennot-inverse⟶
Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000741: Graphs ⟶ ℤ
Values
[1,0] => [1,0] => [.,.] => ([],1) => 0
[1,0,1,0] => [1,1,0,0] => [[.,.],.] => ([(0,1)],2) => 1
[1,1,0,0] => [1,0,1,0] => [.,[.,.]] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => 2
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [[.,[.,.]],.] => ([(0,2),(1,2)],3) => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [.,[[.,.],.]] => ([(0,2),(1,2)],3) => 2
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => ([(0,2),(1,2)],3) => 2
[1,1,1,0,0,0] => [1,1,0,1,0,0] => [[[.,.],.],.] => ([(0,2),(1,2)],3) => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [[.,[[.,.],.]],.] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [[.,[.,[.,.]]],.] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(2,3)],4) => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [.,[[.,[.,.]],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [.,[[[.,.],.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => 2
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [[[.,[.,.]],.],.] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0] => [[[[.,.],.],.],.] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [[[.,.],.],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [[[.,.],.],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [[.,[[.,[.,.]],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [[.,[.,[[.,.],.]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => [[.,[[[.,.],.],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [[.,.],[[.,[.,.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [[[.,[.,.]],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [.,[[.,[[.,.],.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [.,[[.,[.,[.,.]]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [.,[.,[[.,[.,.]],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [.,[.,[[[.,.],.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [.,[[[.,[.,.]],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [[[.,[[.,.],.]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [[[.,[.,[.,.]]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [[.,[[.,.],.]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [[.,[.,.]],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [[[[.,.],.],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [[.,[.,.]],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [[[[.,[.,.]],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [[[[[.,.],.],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[[.,.],.],[.,[[.,.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[.,.],.],[.,[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[.,.],[.,[[.,[.,.]],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[.,.],[.,[.,[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 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] => [[.,.],[.,[.,[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => [[.,.],[.,[[[.,.],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [[[.,.],.],[[.,[.,.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [[[.,.],.],[[[.,.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[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),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[.,[[.,[.,[.,.]]],.]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[.,[.,[[.,[.,.]],.]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[.,[.,[.,[[.,.],.]]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 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] => [[.,[.,[.,[.,[.,.]]]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => [[.,[.,[[[.,.],.],.]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [[.,[[[.,[.,.]],.],.]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => [[.,[[[[.,.],.],.],.]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => [[.,.],[[.,[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 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] => [[.,.],[[.,[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[.,[[.,.],.]],.],[.,.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [[[.,[.,.]],.],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [[[.,[.,.]],.],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 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] => [[.,.],[[[.,[.,.]],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[.,.],[[[[.,.],.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [[[.,[.,[.,.]]],.],[.,.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[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),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => [.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 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] => [.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [.,[.,[[[.,[.,.]],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => [.,[[[.,[[.,.],.]],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [.,[[[.,[.,[.,.]]],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [.,[[[[.,[.,.]],.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => [.,[[[[[.,.],.],.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => [[[.,[[.,[.,.]],.]],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => [[[.,[.,[[.,.],.]]],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 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] => [[[.,[.,[.,[.,.]]]],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => [[[.,[[[.,.],.],.]],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => [[.,[[.,.],.]],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => [[.,[.,.]],[.,[[.,.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 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] => [[.,[.,.]],[.,[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[.,[[.,.],.]],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [[[[.,.],.],.],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 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] => [[.,[.,.]],[[.,[.,.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
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Description
The Colin de Verdière graph invariant.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.
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
logarithmic height to pruning number
Description
Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].
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