- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000907: Posets ⟶ ℤ

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) => 3
[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) => 4
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => ([(1,2),(2,3)],4) => 3
[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) => 5
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 4
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5) => 6
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 3
[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) => 6
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6) => 5
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6) => 8
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6) => 4
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6) => 9
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6) => 6
[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) => 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
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[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) => 4
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7) => 3
[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
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8) => 16
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8) => 9
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8) => 16
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8) => 4
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => ([],8) => 1

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + q^{2} + q^{3}$

$F_{4} = q + q^{2} + q^{3} + 2\ q^{4}$

$F_{5} = q + q^{2} + q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{6} = q + q^{2} + q^{3} + 2\ q^{4} + q^{5} + 2\ q^{6} + 2\ q^{8} + q^{9}$

Description

The number of maximal antichains 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

$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$