Processing math: 100%

Identifier
Values
[1] => [[1]] => [1] => ([],1) => 1
[2] => [[1,2]] => [1,2] => ([(0,1)],2) => 2
[1,1] => [[1],[2]] => [2,1] => ([],2) => 1
[3] => [[1,2,3]] => [1,2,3] => ([(0,2),(2,1)],3) => 6
[2,1] => [[1,2],[3]] => [3,1,2] => ([(1,2)],3) => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => ([],3) => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 24
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => ([(1,2),(2,3)],4) => 6
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => ([(0,3),(1,2)],4) => 4
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => ([(2,3)],4) => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => ([],4) => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 120
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 24
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5) => 12
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 6
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => ([(1,4),(2,3)],5) => 4
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => ([(3,4)],5) => 2
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => ([],5) => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 720
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6) => 120
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6) => 48
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6) => 24
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6) => 36
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6) => 12
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 6
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6) => 8
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => ([(2,5),(3,4)],6) => 4
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => ([(4,5)],6) => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => ([],6) => 1
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7) => 24
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7) => 6
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7) => 4
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => ([(5,6)],7) => 2
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => ([],7) => 1
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Description
The product of the sizes of the principal order filters in a poset.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers 1 through n row by row.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1in}
and the cover relation is given by (w,x)(y,z) if wy and xz.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)(5,2), (2,1)(4,4), (2,1)(3,5), (1,3)(4,4), (1,3)(3,5)}.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.