- FindStat

Identifier

Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤ

Values

[] => ([],1) => [1] => [[1]] => 1

[[]] => ([(0,1)],2) => [2] => [[1,2]] => 2

[[],[]] => ([(0,2),(1,2)],3) => [2,1] => [[1,2],[3]] => 2

[[[]]] => ([(0,2),(2,1)],3) => [3] => [[1,2,3]] => 3

[[],[],[]] => ([(0,3),(1,3),(2,3)],4) => [2,1,1] => [[1,2],[3],[4]] => 2

[[],[[]]] => ([(0,3),(1,2),(2,3)],4) => [3,1] => [[1,2,3],[4]] => 3

[[[]],[]] => ([(0,3),(1,2),(2,3)],4) => [3,1] => [[1,2,3],[4]] => 3

[[[],[]]] => ([(0,3),(1,3),(3,2)],4) => [3,1] => [[1,2,3],[4]] => 3

[[[[]]]] => ([(0,3),(2,1),(3,2)],4) => [4] => [[1,2,3,4]] => 4

[[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => [2,1,1,1] => [[1,2],[3],[4],[5]] => 2

[[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1] => [[1,2,3],[4],[5]] => 3

[[],[[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1] => [[1,2,3],[4],[5]] => 3

[[],[[],[]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [[1,2,3],[4],[5]] => 3

[[],[[[]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [[1,2,3,4],[5]] => 4

[[[]],[],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1] => [[1,2,3],[4],[5]] => 3

[[[]],[[]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => [[1,2,3],[4,5]] => 3

[[[],[]],[]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [[1,2,3],[4],[5]] => 3

[[[[]]],[]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [[1,2,3,4],[5]] => 4

[[[],[],[]]] => ([(0,4),(1,4),(2,4),(4,3)],5) => [3,1,1] => [[1,2,3],[4],[5]] => 3

[[[],[[]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [[1,2,3,4],[5]] => 4

[[[[]],[]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [[1,2,3,4],[5]] => 4

[[[[],[]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => [4,1] => [[1,2,3,4],[5]] => 4

[[[[[]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [[1,2,3,4,5]] => 5

[[],[],[],[],[]] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => 2

[[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[],[],[[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[],[[]],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [[1,2,3],[4,5],[6]] => 3

[[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[],[[],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[],[[],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[],[[[]],[]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[],[[[],[]]]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[],[[[[]]]]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5,1] => [[1,2,3,4,5],[6]] => 5

[[[]],[],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[[]],[],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [[1,2,3],[4,5],[6]] => 3

[[[]],[[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [[1,2,3],[4,5],[6]] => 3

[[[]],[[],[]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [[1,2,3],[4,5],[6]] => 3

[[[]],[[[]]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [[1,2,3,4],[5,6]] => 4

[[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[],[]],[[]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [[1,2,3],[4,5],[6]] => 3

[[[[]]],[[]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [[1,2,3,4],[5,6]] => 4

[[[],[],[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[[],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[[]],[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[[],[]]],[]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[[[]]]],[]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5,1] => [[1,2,3,4,5],[6]] => 5

[[[],[],[],[]]] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3

[[[],[],[[]]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[],[[]],[]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[],[[],[]]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[],[[[]]]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [5,1] => [[1,2,3,4,5],[6]] => 5

[[[[]],[],[]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[[]],[[]]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => [[1,2,3,4],[5,6]] => 4

[[[[],[]],[]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[[[]]],[]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [5,1] => [[1,2,3,4,5],[6]] => 5

[[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => 4

[[[[],[[]]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [5,1] => [[1,2,3,4,5],[6]] => 5

[[[[[]],[]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [5,1] => [[1,2,3,4,5],[6]] => 5

[[[[[],[]]]]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [5,1] => [[1,2,3,4,5],[6]] => 5

[[[[[[]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [[1,2,3,4,5,6]] => 6

[[],[],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => 2

[[],[],[],[],[[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[],[],[[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[],[],[[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[],[],[[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[],[[]],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[],[[]],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[],[],[[],[]],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[],[[[]]],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[],[[],[],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[],[[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[],[[[]],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[],[[[],[]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[],[[[[]]]]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[],[[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[[]],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[],[[]],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[],[[]],[[],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[],[[]],[[[]]]] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[],[[],[]],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[[[]]],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[],[]],[[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[],[[[]]],[[]]] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[],[[],[],[]],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[[],[[]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[[]],[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[[],[]]],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[[[]]]],[]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[],[[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[],[[],[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[],[[]],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[],[[],[]]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[],[[[]]]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[],[[[]],[],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[],[[[],[]],[]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

>>> Load all 214 entries. <<<

[[],[[[[]]],[]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[],[[[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(6,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[],[[[],[[]]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[],[[[[]],[]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[],[[[[],[]]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[],[[[[[]]]]]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[]],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[[]],[],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[]],[],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[]],[],[[],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[]],[],[[[]]]] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[]],[[]],[],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => [3,2,2] => [[1,2,3],[4,5],[6,7]] => 3

[[[]],[[],[]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[]],[[[]]],[]] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[]],[[],[],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(6,5)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[]],[[],[[]]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[]],[[[]],[]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[]],[[[],[]]]] => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[]],[[[[]]]]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => [5,2] => [[1,2,3,4,5],[6,7]] => 5

[[[],[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[[[]]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[]],[],[[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[[]]],[],[[]]] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[],[]],[[]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[[]]],[[]],[]] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[],[]],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[],[]],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[]]],[[],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[]]],[[[]]]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => [4,3] => [[1,2,3,4],[5,6,7]] => 4

[[[],[],[]],[],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[[],[[]]],[],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[]],[]],[],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[],[]]],[],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[[]]]],[],[]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[],[],[]],[[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(6,5)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 3

[[[],[[]]],[[]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[]],[]],[[]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[],[]]],[[]]] => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[[]]]],[[]]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => [5,2] => [[1,2,3,4,5],[6,7]] => 5

[[[],[],[],[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[[],[],[[]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[[]],[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[[],[]]],[]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[[[]]]],[]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[]],[],[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[],[]],[]],[]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[[]]],[]],[]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[],[],[]]],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(6,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[],[[]]]],[]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[]],[]]],[]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[],[]]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[[]]]]],[]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[],[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3

[[[],[],[],[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[],[[]],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[],[[],[]]]] => ([(0,6),(1,6),(2,5),(3,5),(5,6),(6,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[],[[[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[],[[]],[],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[[]],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[],[[],[]],[]]] => ([(0,6),(1,6),(2,5),(3,5),(5,6),(6,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[[[]]],[]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[],[[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[],[[],[[]]]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[],[[[]],[]]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[],[[[],[]]]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[],[[[[]]]]]] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[[]],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[]],[],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[]],[[]],[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[]],[[],[]]]] => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[]],[[[]]]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => [5,2] => [[1,2,3,4,5],[6,7]] => 5

[[[[],[]],[],[]]] => ([(0,6),(1,6),(2,5),(3,5),(5,6),(6,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[[]]],[],[]]] => ([(0,6),(1,6),(2,3),(3,5),(5,6),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[],[]],[[]]]] => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => 4

[[[[[]]],[[]]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => [5,2] => [[1,2,3,4,5],[6,7]] => 5

[[[[],[],[]],[]]] => ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[],[[]]],[]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[]],[]],[]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[],[]]],[]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[[]]]],[]]] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[[],[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,4)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 4

[[[[],[],[[]]]]] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[],[[]],[]]]] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[],[[],[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[],[[[]]]]]] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[[[]],[],[]]]] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[]],[[]]]]] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7) => [5,2] => [[1,2,3,4,5],[6,7]] => 5

[[[[[],[]],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[[]]],[]]]] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[[[],[],[]]]]] => ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => 5

[[[[[],[[]]]]]] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[[[[]],[]]]]] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[[[[],[]]]]]] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => 6

[[[[[[[]]]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [7] => [[1,2,3,4,5,6,7]] => 7

[[[[[[[[]]]]]]]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => [8] => [[1,2,3,4,5,6,7,8]] => 8

[[],[[],[[],[[],[]]]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 5

[[],[[],[[[],[]],[]]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 5

[[],[[[],[]],[[],[]]]] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9) => [4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 4

[[],[[[],[[],[]]],[]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 5

[[],[[[[],[]],[]],[]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 5

[[[],[]],[[],[[],[]]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 4

[[[],[]],[[[],[]],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 4

[[[],[[],[]]],[[],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 4

[[[[],[]],[]],[[],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 4

[[[],[[],[[],[]]]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 5

[[[],[[[],[]],[]]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 5

[[[[],[]],[[],[]]],[]] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9) => [4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 4

[[[[],[[],[]]],[]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 5

[[[[[],[]],[]],[]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 5

[[[[[[[[[]]]]]]]]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => [9] => [[1,2,3,4,5,6,7,8,9]] => 9

[[[[[[[[[[]]]]]]]]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => [10] => [[1,2,3,4,5,6,7,8,9,10]] => 10

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$F_{1} = q$

$F_{2} = q^{2}$

$F_{3} = q^{2} + q^{3}$

$F_{4} = q^{2} + 3\ q^{3} + q^{4}$

$F_{5} = q^{2} + 7\ q^{3} + 5\ q^{4} + q^{5}$

$F_{6} = q^{2} + 15\ q^{3} + 18\ q^{4} + 7\ q^{5} + q^{6}$

$F_{7} = q^{2} + 31\ q^{3} + 57\ q^{4} + 33\ q^{5} + 9\ q^{6} + q^{7}$

Description

The last entry in the first row of a standard tableau.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers

$1$ through

$n$ row by row.

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.