Identifier
Values
[2] => [[1,2]] => [1,2] => ([(0,1)],2) => 1
[1,1] => [[1],[2]] => [2,1] => ([],2) => 2
[3] => [[1,2,3]] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[2,1] => [[1,2],[3]] => [3,1,2] => ([(1,2)],3) => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => ([],3) => 3
[4] => [[1,2,3,4]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => ([(1,2),(2,3)],4) => 1
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => ([(0,3),(1,2)],4) => 2
[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) => 4
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 1
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5) => 1
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 2
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => ([(1,4),(2,3)],5) => 1
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => ([(3,4)],5) => 3
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => ([],5) => 5
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6) => 1
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6) => 1
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6) => 2
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6) => 2
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6) => 1
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 3
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6) => 3
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => ([(2,5),(3,4)],6) => 2
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => ([(4,5)],6) => 4
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => ([],6) => 6
[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) => 3
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7) => 4
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7) => 3
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => ([(5,6)],7) => 5
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => ([],7) => 7
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of maximal chains of minimal length in a poset.
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.
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 $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$