- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000884: Permutations ⟶ ℤ

Values

[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [2,1] => 1
[1,1] => [[1],[2]] => [2,1] => [1,2] => 0
[3] => [[1,2,3]] => [1,2,3] => [3,2,1] => 0
[2,1] => [[1,2],[3]] => [3,1,2] => [2,1,3] => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 0
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [3,2,1,4] => 0
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => 2
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [2,1,3,4] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [4,3,2,1,5] => 0
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,2,1,5,4] => 1
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,2,1,4,5] => 0
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,1,4,3,5] => 2
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,1,3,4,5] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 0
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 1
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 0
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 0
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 1
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 0
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 3
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 2
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [2,1,3,4,5,6] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 0
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 0
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 1
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 0
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 0
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => 0
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => 1
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 0
[7,1] => [[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => 0
[6,2] => [[1,2,3,4,5,6],[7,8]] => [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 1
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 0
[5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => 0
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => 0
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 0
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => 2
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => 0
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => 0
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 4
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => 1
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => 0
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => 0
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [9,8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8,9] => 0
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [9,8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8,9] => 0
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [9,8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8,9] => 0
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8,9] => 0
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8,9] => 0
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8,9] => 1
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => 0
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => 0
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [10,1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1,10] => 0
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [9,10,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,10,9] => 1
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [10,9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9,10] => 0
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [10,9,8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8,9,10] => 0
[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [7,8,9,10,1,2,3,4,5,6] => [6,5,4,3,2,1,10,9,8,7] => 0
[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [9,10,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,10,9] => 2
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [10,9,8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8,9,10] => 0
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8,9,10] => 0
[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [9,10,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,10,9] => 1
[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,10,9] => 3
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8,9,10] => 0
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8,9,10] => 0
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,10,9] => 5
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8,9,10] => 1
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 0
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [7,8,9,10,11,12,1,2,3,4,5,6] => [6,5,4,3,2,1,12,11,10,9,8,7] => 0
[6,4,2] => [[1,2,3,4,5,6],[7,8,9,10],[11,12]] => [11,12,7,8,9,10,1,2,3,4,5,6] => [6,5,4,3,2,1,10,9,8,7,12,11] => 1
[4,4,2,2] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]] => [11,12,9,10,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,10,9,12,11] => 2
[4,2,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]] => [11,12,9,10,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,10,9,12,11] => 4
[] => [] => [] => [] => 0

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + q$

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

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

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

Description

The number of isolated descents of a permutation.
A descent

$i$ is isolated if neither

$i+1$ nor

$i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

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.