Identifier
-
Mp00032:
Dyck paths
—inverse zeta map⟶
Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001717: Posets ⟶ ℤ
Values
[1,0] => [1,0] => [[]] => ([(0,1)],2) => 2
[1,0,1,0] => [1,1,0,0] => [[[]]] => ([(0,2),(2,1)],3) => 3
[1,1,0,0] => [1,0,1,0] => [[],[]] => ([(0,2),(1,2)],3) => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [[[[]]]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [[],[[]]] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,0,0,1,0] => [1,1,0,1,0,0] => [[[],[]]] => ([(0,3),(1,3),(3,2)],4) => 3
[1,1,0,1,0,0] => [1,1,0,0,1,0] => [[[]],[]] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [[],[],[]] => ([(0,3),(1,3),(2,3)],4) => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [[[],[[]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 4
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => [[[[],[]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 4
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [[],[[],[]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 3
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[[[]],[]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 4
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [[[]],[[]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 3
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => [[],[[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => [[[],[],[]]] => ([(0,4),(1,4),(2,4),(4,3)],5) => 3
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [[[],[]],[]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 3
[1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => [[[]],[],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 5
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [[[],[[[]]]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 5
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 5
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [[],[[],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 4
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [[[[[]]],[]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 4
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [[[],[],[[]]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 4
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [[[],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 4
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 5
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => 4
[1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [[[],[[],[]]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 4
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [[[[],[]]],[]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => 4
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 5
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [[],[[[]],[]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 4
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [[[[]],[[]]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => 4
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 4
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [[[],[[]],[]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 4
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [[[[]],[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 4
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => 4
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 3
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [[[]],[[],[]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [[[[]],[],[]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [[[],[]],[[]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [[[],[],[],[]]] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => 3
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [[[],[],[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 3
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [[[],[[[[]]]]]] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => 5
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [[[[[[]]]],[]]] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[],[[[[]]]],[]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [[[],[],[[[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => 5
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[[[]]]],[],[]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => 5
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 4
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => [[],[[[],[[]]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => 5
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => [[[[],[[]]]],[]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => [[],[],[[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [[[],[[[]]],[]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[[[]]],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [[[[],[],[[]]]]] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7) => 5
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => [[],[[],[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => [[],[[],[[]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [[[[[]]],[],[]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => 5
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => [[[],[]],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[],[[[]]],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 4
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[[],[],[],[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7) => 4
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [[[],[],[[]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => 4
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [[[],[[]]],[],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => 4
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[[[]]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 4
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3
[1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [[[[[[],[]]]]]] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => 6
[1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => [[],[[[[],[]]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7) => 5
[1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [[[],[[[],[]]]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => 5
[1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [[[[[],[]]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7) => 5
[1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => [[],[],[[[],[]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => 4
[1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [[[[],[[],[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => 5
[1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => [[],[[],[[],[]]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => 4
[1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[[],[]]],[]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => 5
[1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => [[[]],[[[],[]]]] => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7) => 4
[1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => [[],[[[],[]]],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => 4
[1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[[],[],[[],[]]]] => ([(0,6),(1,6),(2,5),(3,5),(5,6),(6,4)],7) => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [[[],[[],[]]],[]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => 4
[1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [[[[],[]]],[],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => 4
[1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => [[],[],[],[[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 3
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Description
The largest size of an interval in a poset.
Map
to poset
Description
Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.
Map
to ordered tree
Description
Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path $D$ of semilength $n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths $D_1,\dots,D_k$ of respective semilengths $n_1,\dots,n_k$ (so one has $n = n_1 + \dots n_k$) each of which has no returns.
Denote by $\tilde D_i$ the path of semilength $n_i-1$ obtained from $D_i$ by removing the initial up- and the final down-step.
This map then sends $D$ to the tree $T$ having a root note with ordered children $T_1,\dots,T_k$ which are again ordered trees computed from $D_1,\dots,D_k$ respectively.
The unique path of semilength $1$ is sent to the tree consisting of a single node.
This map is recursively defined as follows: A Dyck path $D$ of semilength $n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths $D_1,\dots,D_k$ of respective semilengths $n_1,\dots,n_k$ (so one has $n = n_1 + \dots n_k$) each of which has no returns.
Denote by $\tilde D_i$ the path of semilength $n_i-1$ obtained from $D_i$ by removing the initial up- and the final down-step.
This map then sends $D$ to the tree $T$ having a root note with ordered children $T_1,\dots,T_k$ which are again ordered trees computed from $D_1,\dots,D_k$ respectively.
The unique path of semilength $1$ is sent to the tree consisting of a single node.
Map
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
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