- FindStat

Identifier

Mp00193: Lattices —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 1$

$F_{3} = 1$

$F_{4} = 1 + q$

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given

$\lambda$ count how many integer compositions

$w$ (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.
See also St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight..
Each value in this statistic is greater than or equal to corresponding value in St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition 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

rowmotion cycle type

Description

The cycle type of rowmotion on the order ideals of a poset.

Map

to poset

Description

Return the poset corresponding to the lattice.