- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000873: Permutations ⟶ ℤ

Values

[2] => [1,0,1,0] => [1,2] => [1,2] => 2
[1,1] => [1,1,0,0] => [2,1] => [2,1] => 0
[3] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 3
[2,1] => [1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 1
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => [3,1,2] => 1
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 4
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 2
[2,2] => [1,1,1,0,0,0] => [3,1,2] => [2,3,1] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,2,3] => 2
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,1,2,3] => 1
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 3
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,3,4,2] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,3,4] => 3
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => [4,2,3,1] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,2,3,4] => 2
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 4
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => 2
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 4
[3,3] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => [2,4,1,3] => 2
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,5,3,4,2] => 1
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => [2,3,4,1] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [5,2,3,1,4] => 1
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 1
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 5
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 3
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => 5
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,3,5,2,4] => 3
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => 2
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => 4
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [5,2,4,1,3] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,6,3,4,2,5] => 2
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => 3
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [2,5,3,4,1] => 0
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => [6,2,3,1,4,5] => 1
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => 4
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,4,6,3,5] => 4
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => 3
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [2,5,1,3,4] => 2
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,6,3,5,2,4] => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 2
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,3,6,7,4] => [1,2,7,4,5,3,6] => 3
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [2,4,5,1,3] => 3
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => [6,2,4,1,3,5] => 1
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,6,4] => [1,3,6,4,5,2] => 1
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [2,3,5,1,4] => 3
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => [2,6,3,4,1,5] => 2
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => 5
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,3,6,2,4,5] => 3
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,5,6,3,7,4] => [1,2,7,4,6,3,5] => 3
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => 3
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => [6,2,5,1,3,4] => 1
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,3,5,6,2,4] => 4
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,3,4,7,5] => [1,2,4,7,5,6,3] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => [2,6,4,5,1,3] => 2
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,2,3,4] => [1,3,4,6,2,5] => 4
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,2,3,6,7,4] => [1,3,7,4,5,2,6] => 3
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => [2,6,3,5,1,4] => 2
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,5,6,7,3,4] => [1,2,4,7,3,5,6] => 4
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => [2,6,1,3,4,5] => 2
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,5,7,3,4,6] => [1,2,4,6,7,3,5] => 5
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => [2,5,6,1,3,4] => 3
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => [1,3,4,5,6,2] => 1
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,6,2,3,7,5] => [1,3,7,5,6,2,4] => 3
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,6,7,3,4,5] => [1,2,4,5,7,3,6] => 5
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => [2,3,6,4,5,1] => 0
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => [2,4,6,1,3,5] => 3
[3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,6,2,3,7,4] => [1,3,7,4,6,2,5] => 3
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,1,2,3] => [2,3,6,1,4,5] => 3
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,2,3] => [1,3,7,2,4,5,6] => 3
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,4,5,7,2,3,6] => [1,3,6,7,2,4,5] => 4
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => 2
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,1,2,4,5] => [2,4,5,6,1,3] => 4
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,2,3,4,7,5] => [1,3,4,7,5,6,2] => 1
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [1,4,6,7,2,3,5] => [1,3,5,7,2,4,6] => 4
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [5,1,6,2,3,4] => [3,4,6,2,5,1] => 0
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,5,6,7,2,3,4] => [1,3,4,7,2,5,6] => 4
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,4,7,2,3,5,6] => [1,3,5,6,7,2,4] => 5
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,1,2,3,4] => [2,3,4,6,1,5] => 4
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,7,3,4,5] => [1,4,5,7,3,6,2] => 1
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,6,7,2,3,4,5] => [1,3,4,5,7,2,6] => 5
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,2,3,4,5,6] => [1,3,4,5,6,7,2] => 1

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$F_{2} = 1 + q^{2}$

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

$F_{4} = 1 + q + 2\ q^{2} + q^{4}$

$F_{5} = 1 + 2\ q + q^{2} + 2\ q^{3} + q^{5}$

$F_{6} = 1 + 3\ q + 3\ q^{2} + q^{3} + 2\ q^{4} + q^{6}$

Description

The aix statistic of a permutation.
According to [1], this statistic on finite strings

$\pi$ of integers is given as follows: let

$m$ be the leftmost occurrence of the minimal entry and let

$\pi = \alpha\ m\ \beta$ . Then

$\operatorname{aix}\pi = \begin{cases} \operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\ 1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\ 0 & \text{ if } \beta = \emptyset \end{cases}\ .$

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let

$\pi$ be a permutation. Following Viennot [1], we associate to

$\pi$ a heap of pieces, by considering each decreasing run

$(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of

$\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.

Map

to 321-avoiding permutation (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength

$n$ in an

$n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.