Identifier
Values
[[1],[]] => ([],1) => [1] => 1
[[2],[]] => ([(0,1)],2) => [1] => 1
[[1,1],[]] => ([(0,1)],2) => [1] => 1
[[2,1],[1]] => ([],2) => [2] => 2
[[3],[]] => ([(0,2),(2,1)],3) => [1] => 1
[[2,1],[]] => ([(0,1),(0,2)],3) => [2] => 2
[[3,1],[1]] => ([(1,2)],3) => [3] => 3
[[2,2],[1]] => ([(0,2),(1,2)],3) => [2] => 2
[[3,2],[2]] => ([(1,2)],3) => [3] => 3
[[1,1,1],[]] => ([(0,2),(2,1)],3) => [1] => 1
[[2,2,1],[1,1]] => ([(1,2)],3) => [3] => 3
[[2,1,1],[1]] => ([(1,2)],3) => [3] => 3
[[3,2,1],[2,1]] => ([],3) => [3,3] => 6
[[4],[]] => ([(0,3),(2,1),(3,2)],4) => [1] => 1
[[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => [3] => 3
[[4,1],[1]] => ([(1,2),(2,3)],4) => [4] => 4
[[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => [2] => 2
[[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => [3,2] => 5
[[4,2],[2]] => ([(0,3),(1,2)],4) => [4,2] => 6
[[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => [3] => 3
[[3,2,1],[1,1]] => ([(1,2),(1,3)],4) => [8] => 8
[[3,1,1],[1]] => ([(0,3),(1,2)],4) => [4,2] => 6
[[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => [3] => 3
[[4,3],[3]] => ([(1,2),(2,3)],4) => [4] => 4
[[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => [3,2] => 5
[[3,3,1],[2,1]] => ([(1,3),(2,3)],4) => [8] => 8
[[3,2,1],[2]] => ([(1,2),(1,3)],4) => [8] => 8
[[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => [3] => 3
[[3,3,2],[2,2]] => ([(0,3),(1,2)],4) => [4,2] => 6
[[3,2,2],[2,1]] => ([(1,3),(2,3)],4) => [8] => 8
[[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => [1] => 1
[[2,2,2,1],[1,1,1]] => ([(1,2),(2,3)],4) => [4] => 4
[[2,2,1,1],[1,1]] => ([(0,3),(1,2)],4) => [4,2] => 6
[[2,1,1,1],[1]] => ([(1,2),(2,3)],4) => [4] => 4
[[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [1] => 1
[[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => [4] => 4
[[5,1],[1]] => ([(1,4),(3,2),(4,3)],5) => [5] => 5
[[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => 5
[[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => [5,4] => 9
[[5,2],[2]] => ([(0,3),(1,4),(4,2)],5) => [5,5] => 10
[[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => [4,2] => 6
[[4,1,1],[1]] => ([(0,3),(1,4),(4,2)],5) => [5,5] => 10
[[3,3],[1]] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => 5
[[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => [5,4] => 9
[[5,3],[3]] => ([(0,3),(1,4),(4,2)],5) => [5,5] => 10
[[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => 5
[[3,3,1],[1,1]] => ([(1,2),(1,3),(2,4),(3,4)],5) => [5,5] => 10
[[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => [4] => 4
[[3,1,1,1],[1]] => ([(0,3),(1,4),(4,2)],5) => [5,5] => 10
[[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4] => 4
[[5,4],[4]] => ([(1,4),(3,2),(4,3)],5) => [5] => 5
[[2,2,2],[1]] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => 5
[[3,2,2],[2]] => ([(1,2),(1,3),(2,4),(3,4)],5) => [5,5] => 10
[[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => [5,4] => 9
[[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [4,2] => 6
[[4,4,3],[3,3]] => ([(0,3),(1,4),(4,2)],5) => [5,5] => 10
[[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => [5,4] => 9
[[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4] => 4
[[3,3,3,2],[2,2,2]] => ([(0,3),(1,4),(4,2)],5) => [5,5] => 10
[[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [1] => 1
[[2,2,2,2,1],[1,1,1,1]] => ([(1,4),(3,2),(4,3)],5) => [5] => 5
[[2,2,2,1,1],[1,1,1]] => ([(0,3),(1,4),(4,2)],5) => [5,5] => 10
[[2,2,1,1,1],[1,1]] => ([(0,3),(1,4),(4,2)],5) => [5,5] => 10
[[2,1,1,1,1],[1]] => ([(1,4),(3,2),(4,3)],5) => [5] => 5
[[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [1] => 1
[[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [5] => 5
[[6,1],[1]] => ([(1,5),(3,4),(4,2),(5,3)],6) => [6] => 6
[[4,2],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => [5,4] => 9
[[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => [5,5] => 10
[[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [3,2] => 5
[[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => [5,5] => 10
[[4,4],[2]] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6) => [5,4] => 9
[[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [3,2] => 5
[[2,2,1,1],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => [5,4] => 9
[[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [5] => 5
[[5,5],[4]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5] => 5
[[6,5],[5]] => ([(1,5),(3,4),(4,2),(5,3)],6) => [6] => 6
[[4,4,4],[3,3]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [5,5] => 10
[[2,2,2,2],[1,1]] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6) => [5,4] => 9
[[3,3,3,3],[2,2,2]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [5,5] => 10
[[2,2,2,2,2],[1,1,1,1]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5] => 5
[[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [1] => 1
[[2,2,2,2,2,1],[1,1,1,1,1]] => ([(1,5),(3,4),(4,2),(5,3)],6) => [6] => 6
[[2,1,1,1,1,1],[1]] => ([(1,5),(3,4),(4,2),(5,3)],6) => [6] => 6
[[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [1] => 1
[[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => [6] => 6
[[7,1],[1]] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7) => [7] => 7
[[2,1,1,1,1,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => [6] => 6
[[6,6],[5]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => [6] => 6
[[7,6],[6]] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7) => [7] => 7
[[2,2,2,2,2,2],[1,1,1,1,1]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => [6] => 6
[[1,1,1,1,1,1,1],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [1] => 1
[[2,2,2,2,2,2,1],[1,1,1,1,1,1]] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7) => [7] => 7
[[2,1,1,1,1,1,1],[1]] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7) => [7] => 7
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Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Map
promotion cycle type
Description
The cycle type of promotion on the linear extensions of a 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.