Identifier
-
Mp00040:
Integer compositions
—to partition⟶
Integer partitions
St000205: Integer partitions ⟶ ℤ
Values
[1] => [1] => 0
[1,1] => [1,1] => 0
[2] => [2] => 0
[1,1,1] => [1,1,1] => 0
[1,2] => [2,1] => 0
[2,1] => [2,1] => 0
[3] => [3] => 0
[1,1,1,1] => [1,1,1,1] => 0
[1,1,2] => [2,1,1] => 0
[1,2,1] => [2,1,1] => 0
[1,3] => [3,1] => 0
[2,1,1] => [2,1,1] => 0
[2,2] => [2,2] => 0
[3,1] => [3,1] => 0
[4] => [4] => 0
[1,1,1,1,1] => [1,1,1,1,1] => 0
[1,1,1,2] => [2,1,1,1] => 0
[1,1,2,1] => [2,1,1,1] => 0
[1,1,3] => [3,1,1] => 0
[1,2,1,1] => [2,1,1,1] => 0
[1,2,2] => [2,2,1] => 1
[1,3,1] => [3,1,1] => 0
[1,4] => [4,1] => 0
[2,1,1,1] => [2,1,1,1] => 0
[2,1,2] => [2,2,1] => 1
[2,2,1] => [2,2,1] => 1
[2,3] => [3,2] => 1
[3,1,1] => [3,1,1] => 0
[3,2] => [3,2] => 1
[4,1] => [4,1] => 0
[5] => [5] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[1,1,1,1,2] => [2,1,1,1,1] => 0
[1,1,1,2,1] => [2,1,1,1,1] => 0
[1,1,1,3] => [3,1,1,1] => 0
[1,1,2,1,1] => [2,1,1,1,1] => 0
[1,1,2,2] => [2,2,1,1] => 1
[1,1,3,1] => [3,1,1,1] => 0
[1,1,4] => [4,1,1] => 0
[1,2,1,1,1] => [2,1,1,1,1] => 0
[1,2,1,2] => [2,2,1,1] => 1
[1,2,2,1] => [2,2,1,1] => 1
[1,2,3] => [3,2,1] => 2
[1,3,1,1] => [3,1,1,1] => 0
[1,3,2] => [3,2,1] => 2
[1,4,1] => [4,1,1] => 0
[1,5] => [5,1] => 0
[2,1,1,1,1] => [2,1,1,1,1] => 0
[2,1,1,2] => [2,2,1,1] => 1
[2,1,2,1] => [2,2,1,1] => 1
[2,1,3] => [3,2,1] => 2
[2,2,1,1] => [2,2,1,1] => 1
[2,2,2] => [2,2,2] => 1
[2,3,1] => [3,2,1] => 2
[2,4] => [4,2] => 1
[3,1,1,1] => [3,1,1,1] => 0
[3,1,2] => [3,2,1] => 2
[3,2,1] => [3,2,1] => 2
[3,3] => [3,3] => 1
[4,1,1] => [4,1,1] => 0
[4,2] => [4,2] => 1
[5,1] => [5,1] => 0
[6] => [6] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 0
[1,1,1,1,2,1] => [2,1,1,1,1,1] => 0
[1,1,1,1,3] => [3,1,1,1,1] => 0
[1,1,1,2,1,1] => [2,1,1,1,1,1] => 0
[1,1,1,2,2] => [2,2,1,1,1] => 1
[1,1,1,3,1] => [3,1,1,1,1] => 0
[1,1,1,4] => [4,1,1,1] => 0
[1,1,2,1,1,1] => [2,1,1,1,1,1] => 0
[1,1,2,1,2] => [2,2,1,1,1] => 1
[1,1,2,2,1] => [2,2,1,1,1] => 1
[1,1,2,3] => [3,2,1,1] => 2
[1,1,3,1,1] => [3,1,1,1,1] => 0
[1,1,3,2] => [3,2,1,1] => 2
[1,1,4,1] => [4,1,1,1] => 0
[1,1,5] => [5,1,1] => 0
[1,2,1,1,1,1] => [2,1,1,1,1,1] => 0
[1,2,1,1,2] => [2,2,1,1,1] => 1
[1,2,1,2,1] => [2,2,1,1,1] => 1
[1,2,1,3] => [3,2,1,1] => 2
[1,2,2,1,1] => [2,2,1,1,1] => 1
[1,2,2,2] => [2,2,2,1] => 2
[1,2,3,1] => [3,2,1,1] => 2
[1,2,4] => [4,2,1] => 4
[1,3,1,1,1] => [3,1,1,1,1] => 0
[1,3,1,2] => [3,2,1,1] => 2
[1,3,2,1] => [3,2,1,1] => 2
[1,3,3] => [3,3,1] => 3
[1,4,1,1] => [4,1,1,1] => 0
[1,4,2] => [4,2,1] => 4
[1,5,1] => [5,1,1] => 0
[1,6] => [6,1] => 0
[2,1,1,1,1,1] => [2,1,1,1,1,1] => 0
[2,1,1,1,2] => [2,2,1,1,1] => 1
[2,1,1,2,1] => [2,2,1,1,1] => 1
[2,1,1,3] => [3,2,1,1] => 2
[2,1,2,1,1] => [2,2,1,1,1] => 1
[2,1,2,2] => [2,2,2,1] => 2
>>> Load all 255 entries. <<<
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Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many integer partitions $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Given $\lambda$ count how many integer partitions $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Map
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Description
Sends a composition to the partition obtained by sorting the entries.
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