- FindStat

Identifier

Mp00185: Skew partitions —cell poset⟶ Posets
St000181: Posets ⟶ ℤ

Values

[[1],[]] => ([],1) => 1

[[2],[]] => ([(0,1)],2) => 1

[[1,1],[]] => ([(0,1)],2) => 1

[[2,1],[1]] => ([],2) => 2

[[3],[]] => ([(0,2),(2,1)],3) => 1

[[2,1],[]] => ([(0,1),(0,2)],3) => 1

[[3,1],[1]] => ([(1,2)],3) => 2

[[2,2],[1]] => ([(0,2),(1,2)],3) => 1

[[3,2],[2]] => ([(1,2)],3) => 2

[[1,1,1],[]] => ([(0,2),(2,1)],3) => 1

[[2,2,1],[1,1]] => ([(1,2)],3) => 2

[[2,1,1],[1]] => ([(1,2)],3) => 2

[[3,2,1],[2,1]] => ([],3) => 3

[[4],[]] => ([(0,3),(2,1),(3,2)],4) => 1

[[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1

[[4,1],[1]] => ([(1,2),(2,3)],4) => 2

[[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1

[[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => 1

[[4,2],[2]] => ([(0,3),(1,2)],4) => 2

[[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 1

[[3,2,1],[1,1]] => ([(1,2),(1,3)],4) => 2

[[3,1,1],[1]] => ([(0,3),(1,2)],4) => 2

[[4,2,1],[2,1]] => ([(2,3)],4) => 3

[[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => 1

[[4,3],[3]] => ([(1,2),(2,3)],4) => 2

[[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => 1

[[3,3,1],[2,1]] => ([(1,3),(2,3)],4) => 2

[[3,2,1],[2]] => ([(1,2),(1,3)],4) => 2

[[4,3,1],[3,1]] => ([(2,3)],4) => 3

[[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => 1

[[3,3,2],[2,2]] => ([(0,3),(1,2)],4) => 2

[[3,2,2],[2,1]] => ([(1,3),(2,3)],4) => 2

[[4,3,2],[3,2]] => ([(2,3)],4) => 3

[[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 1

[[2,2,2,1],[1,1,1]] => ([(1,2),(2,3)],4) => 2

[[2,2,1,1],[1,1]] => ([(0,3),(1,2)],4) => 2

[[3,3,2,1],[2,2,1]] => ([(2,3)],4) => 3

[[2,1,1,1],[1]] => ([(1,2),(2,3)],4) => 2

[[3,2,2,1],[2,1,1]] => ([(2,3)],4) => 3

[[3,2,1,1],[2,1]] => ([(2,3)],4) => 3

[[4,3,2,1],[3,2,1]] => ([],4) => 4

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

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

[[5,1],[1]] => ([(1,4),(3,2),(4,3)],5) => 2

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

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

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

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

[[4,2,1],[1,1]] => ([(1,3),(1,4),(4,2)],5) => 2

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

[[5,2,1],[2,1]] => ([(2,3),(3,4)],5) => 3

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

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

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

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

[[3,3,1],[1,1]] => ([(1,2),(1,3),(2,4),(3,4)],5) => 2

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

[[4,3,1],[2,1]] => ([(1,4),(2,3),(2,4)],5) => 2

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

[[5,3,1],[3,1]] => ([(1,4),(2,3)],5) => 3

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

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

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

[[5,3,2],[3,2]] => ([(1,4),(2,3)],5) => 3

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

[[3,2,2,1],[1,1,1]] => ([(1,3),(1,4),(4,2)],5) => 2

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

[[4,3,2,1],[2,2,1]] => ([(2,3),(2,4)],5) => 3

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

[[4,2,2,1],[2,1,1]] => ([(1,4),(2,3)],5) => 3

[[4,2,1,1],[2,1]] => ([(1,4),(2,3)],5) => 3

[[5,3,2,1],[3,2,1]] => ([(3,4)],5) => 4

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

[[5,4],[4]] => ([(1,4),(3,2),(4,3)],5) => 2

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

[[4,4,1],[3,1]] => ([(1,4),(2,3),(3,4)],5) => 2

[[4,3,1],[3]] => ([(1,3),(1,4),(4,2)],5) => 2

[[5,4,1],[4,1]] => ([(2,3),(3,4)],5) => 3

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

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

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

[[3,2,2],[2]] => ([(1,2),(1,3),(2,4),(3,4)],5) => 2

[[4,3,2],[3,1]] => ([(1,4),(2,3),(2,4)],5) => 2

[[5,4,2],[4,2]] => ([(1,4),(2,3)],5) => 3

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

[[3,3,2,1],[2,1,1]] => ([(1,4),(2,3),(2,4)],5) => 2

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

[[4,4,2,1],[3,2,1]] => ([(2,4),(3,4)],5) => 3

[[3,2,1,1],[2]] => ([(1,3),(1,4),(4,2)],5) => 2

[[4,3,2,1],[3,1,1]] => ([(2,3),(2,4)],5) => 3

[[4,3,1,1],[3,1]] => ([(1,4),(2,3)],5) => 3

[[5,4,2,1],[4,2,1]] => ([(3,4)],5) => 4

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

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

[[4,3,3],[3,2]] => ([(1,4),(2,3),(3,4)],5) => 2

[[5,4,3],[4,3]] => ([(2,3),(3,4)],5) => 3

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

[[3,3,3,1],[2,2,1]] => ([(1,4),(2,3),(3,4)],5) => 2

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

[[4,4,3,1],[3,3,1]] => ([(1,4),(2,3)],5) => 3

[[3,2,2,1],[2,1]] => ([(1,4),(2,3),(2,4)],5) => 2

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Description

The number of connected components of the Hasse diagram for the poset.

Map

cell poset

Description

The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.