Identifier
Values
[] => ([],1) => [1] => 1
[[]] => ([(0,1)],2) => [1] => 1
[[],[]] => ([(0,2),(1,2)],3) => [2] => 2
[[[]]] => ([(0,2),(2,1)],3) => [1] => 1
[[],[],[]] => ([(0,3),(1,3),(2,3)],4) => [3,3] => 6
[[],[[]]] => ([(0,3),(1,2),(2,3)],4) => [3] => 3
[[[]],[]] => ([(0,3),(1,2),(2,3)],4) => [3] => 3
[[[],[]]] => ([(0,3),(1,3),(3,2)],4) => [2] => 2
[[[[]]]] => ([(0,3),(2,1),(3,2)],4) => [1] => 1
[[],[[],[]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [8] => 22
[[],[[[]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4] => 5
[[[]],[[]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [4,2] => 8
[[[],[]],[]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [8] => 22
[[[[]]],[]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4] => 5
[[[],[],[]]] => ([(0,4),(1,4),(2,4),(4,3)],5) => [3,3] => 6
[[[],[[]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [3] => 3
[[[[]],[]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [3] => 3
[[[[],[]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => [2] => 2
[[[[[]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [1] => 1
[[],[[[[]]]]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5] => 7
[[[[[]]]],[]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5] => 7
[[[],[[],[]]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [8] => 22
[[[],[[[]]]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [4] => 5
[[[[]],[[]]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => 8
[[[[],[]],[]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [8] => 22
[[[[[]]],[]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [4] => 5
[[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => [3,3] => 6
[[[[],[[]]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [3] => 3
[[[[[]],[]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [3] => 3
[[[[[],[]]]]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [2] => 2
[[[[[[]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [1] => 1
[[],[[[[[]]]]]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => [6] => 11
[[[[[[]]]]],[]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => [6] => 11
[[[],[[[[]]]]]] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [5] => 7
[[[[[[]]]],[]]] => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [5] => 7
[[[[],[[],[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [8] => 22
[[[[],[[[]]]]]] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => [4] => 5
[[[[[]],[[]]]]] => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7) => [4,2] => 8
[[[[[],[]],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [8] => 22
[[[[[[]]],[]]]] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => [4] => 5
[[[[[],[],[]]]]] => ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7) => [3,3] => 6
[[[[[],[[]]]]]] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [3] => 3
[[[[[[]],[]]]]] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [3] => 3
[[[[[[],[]]]]]] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => [2] => 2
[[[[[[[]]]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [1] => 1
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Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many integer partitions $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight., St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. and St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight..
Map
to poset
Description
Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.
Map
promotion cycle type
Description
The cycle type of promotion on the linear extensions of a poset.