Identifier
Values
[] => ([],1) => [2] => [1,1] => 0
[[]] => ([(0,1)],2) => [3] => [1,1,1] => 0
[[],[]] => ([(0,2),(1,2)],3) => [3,2] => [2,2,1] => 1
[[[]]] => ([(0,2),(2,1)],3) => [4] => [1,1,1,1] => 0
[[],[],[]] => ([(0,3),(1,3),(2,3)],4) => [3,2,2,2] => [4,4,1] => 1
[[],[[]]] => ([(0,3),(1,2),(2,3)],4) => [7] => [1,1,1,1,1,1,1] => 0
[[[]],[]] => ([(0,3),(1,2),(2,3)],4) => [7] => [1,1,1,1,1,1,1] => 0
[[[],[]]] => ([(0,3),(1,3),(3,2)],4) => [4,2] => [2,2,1,1] => 1
[[[[]]]] => ([(0,3),(2,1),(3,2)],4) => [5] => [1,1,1,1,1] => 0
[[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => [3,2,2,2,2,2,2,2] => [8,8,1] => 1
[[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => [7,6] => [2,2,2,2,2,2,1] => 1
[[],[[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => [7,6] => [2,2,2,2,2,2,1] => 1
[[],[[],[]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [7,2,2] => [3,3,1,1,1,1,1] => 0
[[],[[[]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => [2,2,2,2,1] => 1
[[[]],[],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => [7,6] => [2,2,2,2,2,2,1] => 1
[[[]],[[]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [4,3,3] => [3,3,3,1] => 0
[[[],[]],[]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [7,2,2] => [3,3,1,1,1,1,1] => 0
[[[[]]],[]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => [2,2,2,2,1] => 1
[[[],[],[]]] => ([(0,4),(1,4),(2,4),(4,3)],5) => [4,2,2,2] => [4,4,1,1] => 1
[[[],[[]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [8] => [1,1,1,1,1,1,1,1] => 0
[[[[]],[]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [8] => [1,1,1,1,1,1,1,1] => 0
[[[[],[]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => [5,2] => [2,2,1,1,1] => 1
[[[[[]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [6] => [1,1,1,1,1,1] => 0
[[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5,4,4,4] => [4,4,4,4,1] => 1
[[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5,4,4,4] => [4,4,4,4,1] => 1
[[],[[],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [15] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[],[[[]],[]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [15] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[],[[[],[]]]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [5,4,2,2] => [4,4,2,2,1] => 1
[[],[[[[]]]]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [11] => [1,1,1,1,1,1,1,1,1,1,1] => 0
[[[]],[[],[]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [6,4,3,3] => [4,4,4,2,1,1] => 1
[[[]],[[[]]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [13] => [1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5,4,4,4] => [4,4,4,4,1] => 1
[[[],[]],[[]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [6,4,3,3] => [4,4,4,2,1,1] => 1
[[[[]]],[[]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [13] => [1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [15] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[]],[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [15] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[],[]]],[]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [5,4,2,2] => [4,4,2,2,1] => 1
[[[[[]]]],[]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [11] => [1,1,1,1,1,1,1,1,1,1,1] => 0
[[[],[],[[]]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [8,6] => [2,2,2,2,2,2,1,1] => 1
[[[],[[]],[]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [8,6] => [2,2,2,2,2,2,1,1] => 1
[[[],[[],[]]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [8,2,2] => [3,3,1,1,1,1,1,1] => 0
[[[],[[[]]]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [6,4] => [2,2,2,2,1,1] => 1
[[[[]],[],[]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [8,6] => [2,2,2,2,2,2,1,1] => 1
[[[[]],[[]]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [5,3,3] => [3,3,3,1,1] => 0
[[[[],[]],[]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [8,2,2] => [3,3,1,1,1,1,1,1] => 0
[[[[[]]],[]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [6,4] => [2,2,2,2,1,1] => 1
[[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => [5,2,2,2] => [4,4,1,1,1] => 1
[[[[],[[]]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [9] => [1,1,1,1,1,1,1,1,1] => 0
[[[[[]],[]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [9] => [1,1,1,1,1,1,1,1,1] => 0
[[[[[],[]]]]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => [2,2,1,1,1,1] => 1
[[[[[[]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [7] => [1,1,1,1,1,1,1] => 0
[[],[[[[],[]]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7) => [11,2,2] => [3,3,1,1,1,1,1,1,1,1,1] => 0
[[],[[[[[]]]]]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => [7,6] => [2,2,2,2,2,2,1] => 1
[[[]],[[[[]]]]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => [16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[]]],[[[]]]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => [5,4,4,4] => [4,4,4,4,1] => 1
[[[[[]]]],[[]]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => [16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[[],[]]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7) => [11,2,2] => [3,3,1,1,1,1,1,1,1,1,1] => 0
[[[[[[]]]]],[]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => [7,6] => [2,2,2,2,2,2,1] => 1
[[[],[[],[[]]]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[],[[[]],[]]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[],[[[],[]]]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => [6,4,2,2] => [4,4,2,2,1,1] => 1
[[[],[[[[]]]]]] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[]],[[],[]]]] => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7) => [6,5,3,3] => [4,4,4,2,2,1] => 1
[[[[]],[[[]]]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => [14] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[],[]],[[]]]] => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7) => [6,5,3,3] => [4,4,4,2,2,1] => 1
[[[[[]]],[[]]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => [14] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[],[[]]],[]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[[]],[]],[]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[[],[]]],[]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => [6,4,2,2] => [4,4,2,2,1,1] => 1
[[[[[[]]]],[]]] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 0
[[[[],[],[[]]]]] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7) => [9,6] => [2,2,2,2,2,2,1,1,1] => 1
[[[[],[[]],[]]]] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7) => [9,6] => [2,2,2,2,2,2,1,1,1] => 1
[[[[],[[],[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [9,2,2] => [3,3,1,1,1,1,1,1,1] => 0
[[[[],[[[]]]]]] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => [7,4] => [2,2,2,2,1,1,1] => 1
[[[[[]],[],[]]]] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7) => [9,6] => [2,2,2,2,2,2,1,1,1] => 1
[[[[[]],[[]]]]] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7) => [6,3,3] => [3,3,3,1,1,1] => 0
[[[[[],[]],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [9,2,2] => [3,3,1,1,1,1,1,1,1] => 0
[[[[[[]]],[]]]] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => [7,4] => [2,2,2,2,1,1,1] => 1
[[[[[],[],[]]]]] => ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7) => [6,2,2,2] => [4,4,1,1,1,1] => 1
[[[[[],[[]]]]]] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 0
[[[[[[]],[]]]]] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 0
[[[[[[],[]]]]]] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => [7,2] => [2,2,1,1,1,1,1] => 1
[[[[[[[]]]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [8] => [1,1,1,1,1,1,1,1] => 0
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Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
to poset
Description
Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.