Identifier
Values
[(1,2)] => [2,1] => [2] => 0
[(1,2),(3,4)] => [2,1,4,3] => [2,2] => 0
[(1,3),(2,4)] => [3,4,1,2] => [2,2] => 0
[(1,4),(2,3)] => [3,4,2,1] => [4] => 0
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,2,2] => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [2,2,2] => 1
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,2] => 1
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [4,2] => 1
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6] => 0
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6] => 0
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [4,2] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [2,2,2] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [2,2,2] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,2,2] => 1
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [4,2] => 1
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [4,2] => 1
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [4,2] => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6] => 0
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [4,2] => 1
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,2,2,2] => 2
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [2,2,2,2] => 2
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [4,2,2] => 4
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [4,2,2] => 4
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [6,2] => 1
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [6,2] => 1
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [8] => 0
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [8] => 0
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [6,2] => 1
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [6,2] => 1
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [4,2,2] => 4
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [2,2,2,2] => 2
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [2,2,2,2] => 2
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,2,2,2] => 2
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [4,2,2] => 4
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [4,2,2] => 4
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [4,2,2] => 4
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [6,2] => 1
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [4,2,2] => 4
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [4,2,2] => 4
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [6,2] => 1
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [6,2] => 1
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [4,2,2] => 4
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [6,2] => 1
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [6,2] => 1
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [4,2,2] => 4
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [4,2,2] => 4
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [4,2,2] => 4
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [6,2] => 1
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [6,2] => 1
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [6,2] => 1
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [8] => 0
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [8] => 0
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [4,4] => 3
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [8] => 0
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [8] => 0
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [4,4] => 3
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [8] => 0
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [6,2] => 1
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [6,2] => 1
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [6,2] => 1
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [6,2] => 1
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [4,2,2] => 4
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [4,2,2] => 4
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [4,2,2] => 4
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [4,2,2] => 4
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [6,2] => 1
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [4,2,2] => 4
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [6,2] => 1
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [8] => 0
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [6,2] => 1
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [4,2,2] => 4
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [2,2,2,2] => 2
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [2,2,2,2] => 2
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [2,2,2,2] => 2
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,2,2,2] => 2
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,2,2,2] => 2
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [2,2,2,2] => 2
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [2,2,2,2] => 2
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [4,2,2] => 4
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [4,2,2] => 4
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [6,2] => 1
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [8] => 0
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [8] => 0
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [6,2] => 1
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [4,2,2] => 4
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [4,2,2] => 4
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [4,2,2] => 4
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [2,2,2,2] => 2
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,2,2,2] => 2
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [4,2,2] => 4
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [4,2,2] => 4
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [4,4] => 3
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [4,4] => 3
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [4,4] => 3
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [8] => 0
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [6,2] => 1
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [6,2] => 1
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [8] => 0
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [4,4] => 3
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [4,4] => 3
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [4,2,2] => 4
>>> Load all 124 entries. <<<
[(1,3),(2,5),(4,8),(6,7)] => [3,5,1,7,2,8,6,4] => [4,2,2] => 4
[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,7,3,8,6,4] => [4,2,2] => 4
[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,7,8,3,5,4] => [4,2,2] => 4
[(1,3),(2,6),(4,8),(5,7)] => [3,6,1,7,8,2,5,4] => [4,2,2] => 4
[(1,4),(2,6),(3,8),(5,7)] => [4,6,7,1,8,2,5,3] => [4,2,2] => 4
[(1,5),(2,6),(3,8),(4,7)] => [5,6,7,8,1,2,4,3] => [4,2,2] => 4
[(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => [4,4] => 3
[(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => [8] => 0
[(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => [6,2] => 1
[(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => [4,2,2] => 4
[(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => [6,2] => 1
[(1,6),(2,7),(3,8),(4,5)] => [5,6,7,8,4,1,2,3] => [8] => 0
[(1,5),(2,7),(3,8),(4,6)] => [5,6,7,8,1,4,2,3] => [6,2] => 1
[(1,4),(2,7),(3,8),(5,6)] => [4,6,7,1,8,5,2,3] => [6,2] => 1
[(1,3),(2,7),(4,8),(5,6)] => [3,6,1,7,8,5,2,4] => [6,2] => 1
[(1,2),(3,7),(4,8),(5,6)] => [2,1,6,7,8,5,3,4] => [6,2] => 1
[(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [4,2,2] => 4
[(1,3),(2,8),(4,7),(5,6)] => [3,6,1,7,8,5,4,2] => [4,2,2] => 4
[(1,4),(2,8),(3,7),(5,6)] => [4,6,7,1,8,5,3,2] => [4,2,2] => 4
[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => [4,2,2] => 4
[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => [6,2] => 1
[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => [8] => 0
[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [4,4] => 3
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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.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
cycle type
Description
The cycle type of a permutation as a partition.