Identifier
Values
0 => ([(0,1)],2) => [2] => [1,1] => 1
1 => ([(0,1)],2) => [2] => [1,1] => 1
00 => ([(0,2),(2,1)],3) => [3] => [1,1,1] => 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 2
11 => ([(0,2),(2,1)],3) => [3] => [1,1,1] => 1
000 => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1] => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [4,2] => [2,2,1,1] => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [4,2] => [2,2,1,1] => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 2
111 => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1] => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1] => 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8) => [5,3] => [2,2,2,1,1] => 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 2
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 2
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8) => [5,3] => [2,2,2,1,1] => 2
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1] => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1] => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1] => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [7] => [1,1,1,1,1,1,1] => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [7] => [1,1,1,1,1,1,1] => 1
0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => [8] => [1,1,1,1,1,1,1,1] => 1
1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => [8] => [1,1,1,1,1,1,1,1] => 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
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
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.
Map
poset of factors
Description
The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that $u < v$ if and only if $u$ is a factor of $v$.