Identifier
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Images
=>
Cc0005;cc-rep-0Cc0021;cc-rep-1Cc0014;cc-rep-2Cc0002;cc-rep-3
[1,0]=>[[]]=>([(0,1)],2)=>[2] [1,0,1,0]=>[[],[]]=>([(0,2),(1,2)],3)=>[2,1] [1,1,0,0]=>[[[]]]=>([(0,2),(2,1)],3)=>[3] [1,0,1,0,1,0]=>[[],[],[]]=>([(0,3),(1,3),(2,3)],4)=>[2,1,1] [1,0,1,1,0,0]=>[[],[[]]]=>([(0,3),(1,2),(2,3)],4)=>[3,1] [1,1,0,0,1,0]=>[[[]],[]]=>([(0,3),(1,2),(2,3)],4)=>[3,1] [1,1,0,1,0,0]=>[[[],[]]]=>([(0,3),(1,3),(3,2)],4)=>[3,1] [1,1,1,0,0,0]=>[[[[]]]]=>([(0,3),(2,1),(3,2)],4)=>[4] [1,0,1,0,1,0,1,0]=>[[],[],[],[]]=>([(0,4),(1,4),(2,4),(3,4)],5)=>[2,1,1,1] [1,0,1,0,1,1,0,0]=>[[],[],[[]]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,1,1] [1,0,1,1,0,0,1,0]=>[[],[[]],[]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,1,1] [1,0,1,1,0,1,0,0]=>[[],[[],[]]]=>([(0,4),(1,3),(2,3),(3,4)],5)=>[3,1,1] [1,0,1,1,1,0,0,0]=>[[],[[[]]]]=>([(0,4),(1,2),(2,3),(3,4)],5)=>[4,1] [1,1,0,0,1,0,1,0]=>[[[]],[],[]]=>([(0,4),(1,4),(2,3),(3,4)],5)=>[3,1,1] [1,1,0,0,1,1,0,0]=>[[[]],[[]]]=>([(0,3),(1,2),(2,4),(3,4)],5)=>[3,2] [1,1,0,1,0,0,1,0]=>[[[],[]],[]]=>([(0,4),(1,3),(2,3),(3,4)],5)=>[3,1,1] [1,1,0,1,0,1,0,0]=>[[[],[],[]]]=>([(0,4),(1,4),(2,4),(4,3)],5)=>[3,1,1] [1,1,0,1,1,0,0,0]=>[[[],[[]]]]=>([(0,4),(1,2),(2,4),(4,3)],5)=>[4,1] [1,1,1,0,0,0,1,0]=>[[[[]]],[]]=>([(0,4),(1,2),(2,3),(3,4)],5)=>[4,1] [1,1,1,0,0,1,0,0]=>[[[[]],[]]]=>([(0,4),(1,2),(2,4),(4,3)],5)=>[4,1] [1,1,1,0,1,0,0,0]=>[[[[],[]]]]=>([(0,4),(1,4),(2,3),(4,2)],5)=>[4,1] [1,1,1,1,0,0,0,0]=>[[[[[]]]]]=>([(0,4),(2,3),(3,1),(4,2)],5)=>[5] [1,0,1,0,1,0,1,0,1,0]=>[[],[],[],[],[]]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>[2,1,1,1,1] [1,0,1,0,1,0,1,1,0,0]=>[[],[],[],[[]]]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>[3,1,1,1] [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] [1,0,1,0,1,1,0,1,0,0]=>[[],[],[[],[]]]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>[3,1,1,1] [1,0,1,0,1,1,1,0,0,0]=>[[],[],[[[]]]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,1,1] [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] [1,0,1,1,0,0,1,1,0,0]=>[[],[[]],[[]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,2,1] [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] [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] [1,0,1,1,0,1,1,0,0,0]=>[[],[[],[[]]]]=>([(0,5),(1,4),(2,3),(3,5),(5,4)],6)=>[4,1,1] [1,0,1,1,1,0,0,0,1,0]=>[[],[[[]]],[]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,1,1] [1,0,1,1,1,0,0,1,0,0]=>[[],[[[]],[]]]=>([(0,5),(1,4),(2,3),(3,5),(5,4)],6)=>[4,1,1] [1,0,1,1,1,0,1,0,0,0]=>[[],[[[],[]]]]=>([(0,5),(1,4),(2,4),(3,5),(4,3)],6)=>[4,1,1] [1,0,1,1,1,1,0,0,0,0]=>[[],[[[[]]]]]=>([(0,5),(1,4),(2,5),(3,2),(4,3)],6)=>[5,1] [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] [1,1,0,0,1,0,1,1,0,0]=>[[[]],[],[[]]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,2,1] [1,1,0,0,1,1,0,0,1,0]=>[[[]],[[]],[]]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[3,2,1] [1,1,0,0,1,1,0,1,0,0]=>[[[]],[[],[]]]=>([(0,4),(1,4),(2,3),(3,5),(4,5)],6)=>[3,2,1] [1,1,0,0,1,1,1,0,0,0]=>[[[]],[[[]]]]=>([(0,3),(1,4),(2,5),(3,5),(4,2)],6)=>[4,2] [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] [1,1,0,1,0,0,1,1,0,0]=>[[[],[]],[[]]]=>([(0,4),(1,4),(2,3),(3,5),(4,5)],6)=>[3,2,1] [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] [1,1,0,1,0,1,0,1,0,0]=>[[[],[],[],[]]]=>([(0,5),(1,5),(2,5),(3,5),(5,4)],6)=>[3,1,1,1] [1,1,0,1,0,1,1,0,0,0]=>[[[],[],[[]]]]=>([(0,5),(1,5),(2,3),(3,5),(5,4)],6)=>[4,1,1] [1,1,0,1,1,0,0,0,1,0]=>[[[],[[]]],[]]=>([(0,5),(1,4),(2,3),(3,5),(5,4)],6)=>[4,1,1] [1,1,0,1,1,0,0,1,0,0]=>[[[],[[]],[]]]=>([(0,5),(1,5),(2,3),(3,5),(5,4)],6)=>[4,1,1] [1,1,0,1,1,0,1,0,0,0]=>[[[],[[],[]]]]=>([(0,5),(1,4),(2,4),(4,5),(5,3)],6)=>[4,1,1] [1,1,0,1,1,1,0,0,0,0]=>[[[],[[[]]]]]=>([(0,5),(1,4),(2,5),(4,2),(5,3)],6)=>[5,1] [1,1,1,0,0,0,1,0,1,0]=>[[[[]]],[],[]]=>([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>[4,1,1] [1,1,1,0,0,0,1,1,0,0]=>[[[[]]],[[]]]=>([(0,3),(1,4),(2,5),(3,5),(4,2)],6)=>[4,2] [1,1,1,0,0,1,0,0,1,0]=>[[[[]],[]],[]]=>([(0,5),(1,4),(2,3),(3,5),(5,4)],6)=>[4,1,1] [1,1,1,0,0,1,0,1,0,0]=>[[[[]],[],[]]]=>([(0,5),(1,5),(2,3),(3,5),(5,4)],6)=>[4,1,1] [1,1,1,0,0,1,1,0,0,0]=>[[[[]],[[]]]]=>([(0,4),(1,3),(3,5),(4,5),(5,2)],6)=>[4,2] [1,1,1,0,1,0,0,0,1,0]=>[[[[],[]]],[]]=>([(0,5),(1,4),(2,4),(3,5),(4,3)],6)=>[4,1,1] [1,1,1,0,1,0,0,1,0,0]=>[[[[],[]],[]]]=>([(0,5),(1,4),(2,4),(4,5),(5,3)],6)=>[4,1,1] [1,1,1,0,1,0,1,0,0,0]=>[[[[],[],[]]]]=>([(0,5),(1,5),(2,5),(3,4),(5,3)],6)=>[4,1,1] [1,1,1,0,1,1,0,0,0,0]=>[[[[],[[]]]]]=>([(0,5),(1,3),(3,5),(4,2),(5,4)],6)=>[5,1] [1,1,1,1,0,0,0,0,1,0]=>[[[[[]]]],[]]=>([(0,5),(1,4),(2,5),(3,2),(4,3)],6)=>[5,1] [1,1,1,1,0,0,0,1,0,0]=>[[[[[]]],[]]]=>([(0,5),(1,4),(2,5),(4,2),(5,3)],6)=>[5,1] [1,1,1,1,0,0,1,0,0,0]=>[[[[[]],[]]]]=>([(0,5),(1,3),(3,5),(4,2),(5,4)],6)=>[5,1] [1,1,1,1,0,1,0,0,0,0]=>[[[[[],[]]]]]=>([(0,5),(1,5),(3,2),(4,3),(5,4)],6)=>[5,1] [1,1,1,1,1,0,0,0,0,0]=>[[[[[[]]]]]]=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6)=>[6] [1,0,1,0,1,0,1,0,1,0,1,0]=>[[],[],[],[],[],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>[2,1,1,1,1,1] [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)=>[3,1,1,1,1] [1,0,1,0,1,0,1,1,0,0,1,0]=>[[],[],[],[[]],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[3,1,1,1,1] [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)=>[3,1,1,1,1] [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)=>[4,1,1,1] [1,0,1,0,1,1,0,0,1,0,1,0]=>[[],[],[[]],[],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[3,1,1,1,1] [1,0,1,0,1,1,0,0,1,1,0,0]=>[[],[],[[]],[[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,0,1,0,1,1,0,1,0,0,1,0]=>[[],[],[[],[]],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[3,1,1,1,1] [1,0,1,0,1,1,0,1,0,1,0,0]=>[[],[],[[],[],[]]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)=>[3,1,1,1,1] [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)=>[4,1,1,1] [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)=>[4,1,1,1] [1,0,1,0,1,1,1,0,0,1,0,0]=>[[],[],[[[]],[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)=>[4,1,1,1] [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)=>[4,1,1,1] [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)=>[5,1,1] [1,0,1,1,0,0,1,0,1,0,1,0]=>[[],[[]],[],[],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[3,1,1,1,1] [1,0,1,1,0,0,1,0,1,1,0,0]=>[[],[[]],[],[[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,0,1,1,0,0,1,1,0,0,1,0]=>[[],[[]],[[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,0,1,1,0,0,1,1,0,1,0,0]=>[[],[[]],[[],[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,0,1,1,0,0,1,1,1,0,0,0]=>[[],[[]],[[[]]]]=>([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)=>[4,2,1] [1,0,1,1,0,1,0,0,1,0,1,0]=>[[],[[],[]],[],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[3,1,1,1,1] [1,0,1,1,0,1,0,0,1,1,0,0]=>[[],[[],[]],[[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,0,1,1,0,1,0,1,0,0,1,0]=>[[],[[],[],[]],[]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)=>[3,1,1,1,1] [1,0,1,1,0,1,0,1,0,1,0,0]=>[[],[[],[],[],[]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)=>[3,1,1,1,1] [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)=>[4,1,1,1] [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)=>[4,1,1,1] [1,0,1,1,0,1,1,0,0,1,0,0]=>[[],[[],[[]],[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)=>[4,1,1,1] [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)=>[4,1,1,1] [1,0,1,1,0,1,1,1,0,0,0,0]=>[[],[[],[[[]]]]]=>([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)=>[5,1,1] [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)=>[4,1,1,1] [1,0,1,1,1,0,0,0,1,1,0,0]=>[[],[[[]]],[[]]]=>([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)=>[4,2,1] [1,0,1,1,1,0,0,1,0,0,1,0]=>[[],[[[]],[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)=>[4,1,1,1] [1,0,1,1,1,0,0,1,0,1,0,0]=>[[],[[[]],[],[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)=>[4,1,1,1] [1,0,1,1,1,0,0,1,1,0,0,0]=>[[],[[[]],[[]]]]=>([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)=>[4,2,1] [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)=>[4,1,1,1] [1,0,1,1,1,0,1,0,0,1,0,0]=>[[],[[[],[]],[]]]=>([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)=>[4,1,1,1] [1,0,1,1,1,0,1,0,1,0,0,0]=>[[],[[[],[],[]]]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(6,4)],7)=>[4,1,1,1] [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)=>[5,1,1] [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)=>[5,1,1] [1,0,1,1,1,1,0,0,0,1,0,0]=>[[],[[[[]]],[]]]=>([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)=>[5,1,1] [1,0,1,1,1,1,0,0,1,0,0,0]=>[[],[[[[]],[]]]]=>([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)=>[5,1,1] [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] [1,0,1,1,1,1,1,0,0,0,0,0]=>[[],[[[[[]]]]]]=>([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)=>[6,1] [1,1,0,0,1,0,1,0,1,0,1,0]=>[[[]],[],[],[],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>[3,1,1,1,1] [1,1,0,0,1,0,1,0,1,1,0,0]=>[[[]],[],[],[[]]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,1,0,0,1,0,1,1,0,0,1,0]=>[[[]],[],[[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,1,0,0,1,0,1,1,0,1,0,0]=>[[[]],[],[[],[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,1,0,0,1,0,1,1,1,0,0,0]=>[[[]],[],[[[]]]]=>([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)=>[4,2,1] [1,1,0,0,1,1,0,0,1,0,1,0]=>[[[]],[[]],[],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,1,0,0,1,1,0,0,1,1,0,0]=>[[[]],[[]],[[]]]=>([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)=>[3,2,2] [1,1,0,0,1,1,0,1,0,0,1,0]=>[[[]],[[],[]],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,1,0,0,1,1,0,1,0,1,0,0]=>[[[]],[[],[],[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(6,5)],7)=>[3,2,1,1] [1,1,0,0,1,1,0,1,1,0,0,0]=>[[[]],[[],[[]]]]=>([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)=>[4,2,1] [1,1,0,0,1,1,1,0,0,0,1,0]=>[[[]],[[[]]],[]]=>([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)=>[4,2,1] [1,1,0,0,1,1,1,0,0,1,0,0]=>[[[]],[[[]],[]]]=>([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)=>[4,2,1] [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)=>[4,2,1] [1,1,0,0,1,1,1,1,0,0,0,0]=>[[[]],[[[[]]]]]=>([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)=>[5,2] [1,1,0,1,0,0,1,0,1,0,1,0]=>[[[],[]],[],[],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>[3,1,1,1,1] [1,1,0,1,0,0,1,0,1,1,0,0]=>[[[],[]],[],[[]]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,1,0,1,0,0,1,1,0,0,1,0]=>[[[],[]],[[]],[]]=>([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [1,1,0,1,0,0,1,1,0,1,0,0]=>[[[],[]],[[],[]]]=>([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)=>[3,2,1,1] [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)=>[4,2,1] [1,1,0,1,0,1,0,0,1,0,1,0]=>[[[],[],[]],[],[]]=>([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)=>[3,1,1,1,1] [1,1,0,1,0,1,0,0,1,1,0,0]=>[[[],[],[]],[[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,5),(6,5)],7)=>[3,2,1,1] [1,1,0,1,0,1,0,1,0,0,1,0]=>[[[],[],[],[]],[]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)=>[3,1,1,1,1] [1,1,0,1,0,1,0,1,0,1,0,0]=>[[[],[],[],[],[]]]=>([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)=>[3,1,1,1,1] [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)=>[4,1,1,1] [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)=>[4,1,1,1] [1,1,0,1,0,1,1,0,0,1,0,0]=>[[[],[],[[]],[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)=>[4,1,1,1] [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)=>[4,1,1,1] [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)=>[5,1,1] [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)=>[4,1,1,1] [1,1,0,1,1,0,0,0,1,1,0,0]=>[[[],[[]]],[[]]]=>([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)=>[4,2,1] [1,1,0,1,1,0,0,1,0,0,1,0]=>[[[],[[]],[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)=>[4,1,1,1] [1,1,0,1,1,0,0,1,0,1,0,0]=>[[[],[[]],[],[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)=>[4,1,1,1] [1,1,0,1,1,0,0,1,1,0,0,0]=>[[[],[[]],[[]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)=>[4,2,1] [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)=>[4,1,1,1] [1,1,0,1,1,0,1,0,0,1,0,0]=>[[[],[[],[]],[]]]=>([(0,6),(1,6),(2,5),(3,5),(5,6),(6,4)],7)=>[4,1,1,1] [1,1,0,1,1,0,1,0,1,0,0,0]=>[[[],[[],[],[]]]]=>([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7)=>[4,1,1,1] [1,1,0,1,1,0,1,1,0,0,0,0]=>[[[],[[],[[]]]]]=>([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)=>[5,1,1] [1,1,0,1,1,1,0,0,0,0,1,0]=>[[[],[[[]]]],[]]=>([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)=>[5,1,1] [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)=>[5,1,1] [1,1,0,1,1,1,0,0,1,0,0,0]=>[[[],[[[]],[]]]]=>([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)=>[5,1,1] [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] [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] [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)=>[4,1,1,1] [1,1,1,0,0,0,1,0,1,1,0,0]=>[[[[]]],[],[[]]]=>([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)=>[4,2,1] [1,1,1,0,0,0,1,1,0,0,1,0]=>[[[[]]],[[]],[]]=>([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)=>[4,2,1] [1,1,1,0,0,0,1,1,0,1,0,0]=>[[[[]]],[[],[]]]=>([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)=>[4,2,1] [1,1,1,0,0,0,1,1,1,0,0,0]=>[[[[]]],[[[]]]]=>([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)=>[4,3] [1,1,1,0,0,1,0,0,1,0,1,0]=>[[[[]],[]],[],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)=>[4,1,1,1] [1,1,1,0,0,1,0,0,1,1,0,0]=>[[[[]],[]],[[]]]=>([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)=>[4,2,1] [1,1,1,0,0,1,0,1,0,0,1,0]=>[[[[]],[],[]],[]]=>([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)=>[4,1,1,1] [1,1,1,0,0,1,0,1,0,1,0,0]=>[[[[]],[],[],[]]]=>([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)=>[4,1,1,1] [1,1,1,0,0,1,0,1,1,0,0,0]=>[[[[]],[],[[]]]]=>([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)=>[4,2,1] [1,1,1,0,0,1,1,0,0,0,1,0]=>[[[[]],[[]]],[]]=>([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)=>[4,2,1] [1,1,1,0,0,1,1,0,0,1,0,0]=>[[[[]],[[]],[]]]=>([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7)=>[4,2,1] [1,1,1,0,0,1,1,0,1,0,0,0]=>[[[[]],[[],[]]]]=>([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)=>[4,2,1] [1,1,1,0,0,1,1,1,0,0,0,0]=>[[[[]],[[[]]]]]=>([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)=>[5,2] [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)=>[4,1,1,1] [1,1,1,0,1,0,0,0,1,1,0,0]=>[[[[],[]]],[[]]]=>([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)=>[4,2,1] [1,1,1,0,1,0,0,1,0,0,1,0]=>[[[[],[]],[]],[]]=>([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)=>[4,1,1,1] [1,1,1,0,1,0,0,1,0,1,0,0]=>[[[[],[]],[],[]]]=>([(0,6),(1,6),(2,5),(3,5),(5,6),(6,4)],7)=>[4,1,1,1] [1,1,1,0,1,0,0,1,1,0,0,0]=>[[[[],[]],[[]]]]=>([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)=>[4,2,1] [1,1,1,0,1,0,1,0,0,0,1,0]=>[[[[],[],[]]],[]]=>([(0,6),(1,6),(2,6),(3,5),(4,5),(6,4)],7)=>[4,1,1,1] [1,1,1,0,1,0,1,0,0,1,0,0]=>[[[[],[],[]],[]]]=>([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7)=>[4,1,1,1] [1,1,1,0,1,0,1,0,1,0,0,0]=>[[[[],[],[],[]]]]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(6,4)],7)=>[4,1,1,1] [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)=>[5,1,1] [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)=>[5,1,1] [1,1,1,0,1,1,0,0,0,1,0,0]=>[[[[],[[]]],[]]]=>([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)=>[5,1,1] [1,1,1,0,1,1,0,0,1,0,0,0]=>[[[[],[[]],[]]]]=>([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)=>[5,1,1] [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] [1,1,1,0,1,1,1,0,0,0,0,0]=>[[[[],[[[]]]]]]=>([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)=>[6,1] [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)=>[5,1,1] [1,1,1,1,0,0,0,0,1,1,0,0]=>[[[[[]]]],[[]]]=>([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)=>[5,2] [1,1,1,1,0,0,0,1,0,0,1,0]=>[[[[[]]],[]],[]]=>([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)=>[5,1,1] [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)=>[5,1,1] [1,1,1,1,0,0,0,1,1,0,0,0]=>[[[[[]]],[[]]]]=>([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)=>[5,2] [1,1,1,1,0,0,1,0,0,0,1,0]=>[[[[[]],[]]],[]]=>([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)=>[5,1,1] [1,1,1,1,0,0,1,0,0,1,0,0]=>[[[[[]],[]],[]]]=>([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)=>[5,1,1] [1,1,1,1,0,0,1,0,1,0,0,0]=>[[[[[]],[],[]]]]=>([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)=>[5,1,1] [1,1,1,1,0,0,1,1,0,0,0,0]=>[[[[[]],[[]]]]]=>([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)=>[5,2] [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] [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] [1,1,1,1,0,1,0,0,1,0,0,0]=>[[[[[],[]],[]]]]=>([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)=>[5,1,1] [1,1,1,1,0,1,0,1,0,0,0,0]=>[[[[[],[],[]]]]]=>([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)=>[5,1,1] [1,1,1,1,0,1,1,0,0,0,0,0]=>[[[[[],[[]]]]]]=>([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)=>[6,1] [1,1,1,1,1,0,0,0,0,0,1,0]=>[[[[[[]]]]],[]]=>([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)=>[6,1] [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] [1,1,1,1,1,0,0,0,1,0,0,0]=>[[[[[[]]],[]]]]=>([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)=>[6,1] [1,1,1,1,1,0,0,1,0,0,0,0]=>[[[[[[]],[]]]]]=>([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)=>[6,1] [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] [1,1,1,1,1,1,0,0,0,0,0,0]=>[[[[[[[]]]]]]]=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)=>[7]
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.
Map
to poset
Description
Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.
Map
Greene-Kleitman invariant
Description
The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.