- FindStat

Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000205

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

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

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.

Matching statistic: St001114

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 40%

Values

[(1,2)]
 => [2,1] => [2,1] => [2,1] => 1 = 0 + 1
[(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 1 = 0 + 1
[(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => [4,2,1,3] => 1 = 0 + 1
[(1,4),(2,3)]
 => [3,4,2,1] => [4,1,3,2] => [2,4,3,1] => 1 = 0 + 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [3,2,5,4,1,6] => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,2,6,5] => [4,2,5,3,1,6] => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,1,3,2,6,5] => [5,2,4,3,1,6] => 2 = 1 + 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [5,1,3,2,6,4] => [6,2,4,3,1,5] => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,1,3,2,5,4] => [2,4,3,6,5,1] => 1 = 0 + 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [6,1,4,2,5,3] => [2,5,3,6,4,1] => 1 = 0 + 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [5,1,4,2,6,3] => [6,2,5,3,1,4] => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [5,2,6,3,1,4] => 2 = 1 + 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [4,2,6,3,1,5] => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,5,3,6,4] => [3,2,6,4,1,5] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,3,5,4] => [3,2,1,4,6,5] => 2 = 1 + 1
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [3,1,6,2,5,4] => [4,2,1,3,6,5] => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [4,1,6,2,5,3] => [5,2,1,3,6,4] => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [6,2,5,1,4,3] => [3,6,2,5,4,1] => 1 = 0 + 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [5,2,6,1,4,3] => [6,3,1,2,5,4] => 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => ? => ? = 2 + 1
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => ? => ? = 2 + 1
[(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,1,3,2,6,5,8,7] => ? => ? = 4 + 1
[(1,5),(2,3),(4,6),(7,8)]
 => [3,5,2,6,1,4,8,7] => [5,1,3,2,6,4,8,7] => ? => ? = 4 + 1
[(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [6,1,3,2,5,4,8,7] => ? => ? = 1 + 1
[(1,7),(2,3),(4,5),(6,8)]
 => [3,5,2,7,4,8,1,6] => [7,1,3,2,5,4,8,6] => ? => ? = 1 + 1
[(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,1,3,2,5,4,7,6] => ? => ? = 0 + 1
[(1,8),(2,4),(3,5),(6,7)]
 => [4,5,7,2,3,8,6,1] => [8,1,4,2,5,3,7,6] => ? => ? = 0 + 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [7,1,4,2,5,3,8,6] => ? => ? = 1 + 1
[(1,6),(2,4),(3,5),(7,8)]
 => [4,5,6,2,3,1,8,7] => [6,1,4,2,5,3,8,7] => ? => ? = 1 + 1
[(1,5),(2,4),(3,6),(7,8)]
 => [4,5,6,2,1,3,8,7] => [5,1,4,2,6,3,8,7] => ? => ? = 4 + 1
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => ? => ? = 2 + 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => ? => ? = 2 + 1
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => ? => ? = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,6,3,5,4,8,7] => ? => ? = 4 + 1
[(1,3),(2,6),(4,5),(7,8)]
 => [3,5,1,6,4,2,8,7] => [3,1,6,2,5,4,8,7] => ? => ? = 4 + 1
[(1,4),(2,6),(3,5),(7,8)]
 => [4,5,6,1,3,2,8,7] => [4,1,6,2,5,3,8,7] => ? => ? = 4 + 1
[(1,5),(2,6),(3,4),(7,8)]
 => [4,5,6,3,1,2,8,7] => [6,2,5,1,4,3,8,7] => ? => ? = 1 + 1
[(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [5,2,6,1,4,3,8,7] => ? => ? = 4 + 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [5,2,7,1,4,3,8,6] => ? => ? = 4 + 1
[(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [5,2,8,1,4,3,7,6] => ? => ? = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [6,2,8,1,4,3,7,5] => ? => ? = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [6,2,7,1,4,3,8,5] => ? => ? = 4 + 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [7,2,6,1,4,3,8,5] => ? => ? = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
 => [4,5,7,3,1,8,2,6] => [7,2,5,1,4,3,8,6] => ? => ? = 1 + 1
[(1,4),(2,7),(3,5),(6,8)]
 => [4,5,7,1,3,8,2,6] => [4,1,7,2,5,3,8,6] => ? => ? = 4 + 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,5,1,7,4,8,2,6] => [3,1,7,2,5,4,8,6] => ? => ? = 4 + 1
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,5,7,4,8,3,6] => [2,1,7,3,5,4,8,6] => ? => ? = 4 + 1
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,3,5,4,7,6] => ? => ? = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
 => [3,5,1,7,4,8,6,2] => [3,1,8,2,5,4,7,6] => ? => ? = 1 + 1
[(1,4),(2,8),(3,5),(6,7)]
 => [4,5,7,1,3,8,6,2] => [4,1,8,2,5,3,7,6] => ? => ? = 1 + 1
[(1,5),(2,8),(3,4),(6,7)]
 => [4,5,7,3,1,8,6,2] => [8,2,5,1,4,3,7,6] => ? => ? = 0 + 1
[(1,6),(2,8),(3,4),(5,7)]
 => [4,6,7,3,8,1,5,2] => [8,2,6,1,4,3,7,5] => ? => ? = 0 + 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [7,1,4,3,8,2,6,5] => ? => ? = 3 + 1
[(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,1,4,3,7,2,6,5] => ? => ? = 0 + 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [8,1,5,3,7,2,6,4] => ? => ? = 0 + 1
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [7,1,5,3,8,2,6,4] => ? => ? = 3 + 1
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [8,2,6,1,5,3,7,4] => ? => ? = 0 + 1
[(1,5),(2,8),(3,6),(4,7)]
 => [5,6,7,8,1,3,4,2] => [5,1,8,2,6,3,7,4] => ? => ? = 1 + 1
[(1,4),(2,8),(3,6),(5,7)]
 => [4,6,7,1,8,3,5,2] => [4,1,8,2,6,3,7,5] => ? => ? = 1 + 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,6,1,7,8,4,5,2] => [3,1,8,2,6,4,7,5] => ? => ? = 1 + 1
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,6,7,8,4,5,3] => [2,1,8,3,6,4,7,5] => ? => ? = 1 + 1
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,6,7,8,4,3,5] => [2,1,7,3,6,4,8,5] => ? => ? = 4 + 1
[(1,3),(2,7),(4,6),(5,8)]
 => [3,6,1,7,8,4,2,5] => [3,1,7,2,6,4,8,5] => ? => ? = 4 + 1
[(1,4),(2,7),(3,6),(5,8)]
 => [4,6,7,1,8,3,2,5] => [4,1,7,2,6,3,8,5] => ? => ? = 4 + 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,6,7,8,1,3,2,4] => [5,1,7,2,6,3,8,4] => ? => ? = 4 + 1
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [7,2,6,1,5,3,8,4] => ? => ? = 1 + 1
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [6,2,7,1,5,3,8,4] => ? => ? = 4 + 1
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [6,2,8,1,5,3,7,4] => ? => ? = 1 + 1
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [8,1,5,2,6,3,7,4] => ? => ? = 0 + 1

Description

The number of odd descents of a permutation.