- FindStat

Identifier

Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words

Images

[1] => [1] => =>

[1,2] => [1,2] => 0 => 0

[2,1] => [2,1] => 1 => 1

[1,2,3] => [1,2,3] => 00 => 00

[1,3,2] => [1,3,2] => 01 => 10

[2,1,3] => [2,1,3] => 10 => 01

[2,3,1] => [3,1,2] => 10 => 01

[3,1,2] => [2,3,1] => 01 => 10

[3,2,1] => [3,2,1] => 11 => 11

[1,2,3,4] => [1,2,3,4] => 000 => 000

[1,2,4,3] => [1,2,4,3] => 001 => 100

[1,3,2,4] => [1,3,2,4] => 010 => 010

[1,3,4,2] => [1,4,2,3] => 010 => 010

[1,4,2,3] => [1,3,4,2] => 001 => 100

[1,4,3,2] => [1,4,3,2] => 011 => 110

[2,1,3,4] => [2,1,3,4] => 100 => 001

[2,1,4,3] => [2,1,4,3] => 101 => 101

[2,3,1,4] => [3,1,2,4] => 100 => 001

[2,3,4,1] => [4,1,2,3] => 100 => 001

[2,4,1,3] => [3,4,1,2] => 010 => 010

[2,4,3,1] => [4,3,1,2] => 110 => 011

[3,1,2,4] => [2,3,1,4] => 010 => 010

[3,1,4,2] => [4,2,3,1] => 101 => 101

[3,2,1,4] => [3,2,1,4] => 110 => 011

[3,2,4,1] => [4,1,3,2] => 101 => 101

[3,4,1,2] => [2,4,1,3] => 010 => 010

[3,4,2,1] => [4,2,1,3] => 110 => 011

[4,1,2,3] => [2,3,4,1] => 001 => 100

[4,1,3,2] => [3,2,4,1] => 101 => 101

[4,2,1,3] => [3,4,2,1] => 011 => 110

[4,2,3,1] => [3,1,4,2] => 101 => 101

[4,3,1,2] => [2,4,3,1] => 011 => 110

[4,3,2,1] => [4,3,2,1] => 111 => 111

[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000

[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1000

[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0100

[1,2,4,5,3] => [1,2,5,3,4] => 0010 => 0100

[1,2,5,3,4] => [1,2,4,5,3] => 0001 => 1000

[1,2,5,4,3] => [1,2,5,4,3] => 0011 => 1100

[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0010

[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010

[1,3,4,2,5] => [1,4,2,3,5] => 0100 => 0010

[1,3,4,5,2] => [1,5,2,3,4] => 0100 => 0010

[1,3,5,2,4] => [1,4,5,2,3] => 0010 => 0100

[1,3,5,4,2] => [1,5,4,2,3] => 0110 => 0110

[1,4,2,3,5] => [1,3,4,2,5] => 0010 => 0100

[1,4,2,5,3] => [1,5,3,4,2] => 0101 => 1010

[1,4,3,2,5] => [1,4,3,2,5] => 0110 => 0110

[1,4,3,5,2] => [1,5,2,4,3] => 0101 => 1010

[1,4,5,2,3] => [1,3,5,2,4] => 0010 => 0100

[1,4,5,3,2] => [1,5,3,2,4] => 0110 => 0110

[1,5,2,3,4] => [1,3,4,5,2] => 0001 => 1000

[1,5,2,4,3] => [1,4,3,5,2] => 0101 => 1010

[1,5,3,2,4] => [1,4,5,3,2] => 0011 => 1100

[1,5,3,4,2] => [1,4,2,5,3] => 0101 => 1010

[1,5,4,2,3] => [1,3,5,4,2] => 0011 => 1100

[1,5,4,3,2] => [1,5,4,3,2] => 0111 => 1110

[2,1,3,4,5] => [2,1,3,4,5] => 1000 => 0001

[2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1001

[2,1,4,3,5] => [2,1,4,3,5] => 1010 => 0101

[2,1,4,5,3] => [2,1,5,3,4] => 1010 => 0101

[2,1,5,3,4] => [2,1,4,5,3] => 1001 => 1001

[2,1,5,4,3] => [2,1,5,4,3] => 1011 => 1101

[2,3,1,4,5] => [3,1,2,4,5] => 1000 => 0001

[2,3,1,5,4] => [3,1,2,5,4] => 1001 => 1001

[2,3,4,1,5] => [4,1,2,3,5] => 1000 => 0001

[2,3,4,5,1] => [5,1,2,3,4] => 1000 => 0001

[2,3,5,1,4] => [4,5,1,2,3] => 0100 => 0010

[2,3,5,4,1] => [5,4,1,2,3] => 1100 => 0011

[2,4,1,3,5] => [3,4,1,2,5] => 0100 => 0010

[2,4,1,5,3] => [5,3,4,1,2] => 1010 => 0101

[2,4,3,1,5] => [4,3,1,2,5] => 1100 => 0011

[2,4,3,5,1] => [5,1,2,4,3] => 1001 => 1001

[2,4,5,1,3] => [3,5,1,2,4] => 0100 => 0010

[2,4,5,3,1] => [5,3,1,2,4] => 1100 => 0011

[2,5,1,3,4] => [3,4,5,1,2] => 0010 => 0100

[2,5,1,4,3] => [4,3,5,1,2] => 1010 => 0101

[2,5,3,1,4] => [4,5,3,1,2] => 0110 => 0110

[2,5,3,4,1] => [4,1,2,5,3] => 1001 => 1001

[2,5,4,1,3] => [3,5,4,1,2] => 0110 => 0110

[2,5,4,3,1] => [5,4,3,1,2] => 1110 => 0111

[3,1,2,4,5] => [2,3,1,4,5] => 0100 => 0010

[3,1,2,5,4] => [2,3,1,5,4] => 0101 => 1010

[3,1,4,2,5] => [4,2,3,1,5] => 1010 => 0101

[3,1,4,5,2] => [5,2,3,1,4] => 1010 => 0101

[3,1,5,2,4] => [4,5,2,3,1] => 0101 => 1010

[3,1,5,4,2] => [5,4,2,3,1] => 1101 => 1011

[3,2,1,4,5] => [3,2,1,4,5] => 1100 => 0011

[3,2,1,5,4] => [3,2,1,5,4] => 1101 => 1011

[3,2,4,1,5] => [4,1,3,2,5] => 1010 => 0101

[3,2,4,5,1] => [5,1,3,2,4] => 1010 => 0101

[3,2,5,1,4] => [4,5,1,3,2] => 0101 => 1010

[3,2,5,4,1] => [5,4,1,3,2] => 1101 => 1011

[3,4,1,2,5] => [2,4,1,3,5] => 0100 => 0010

[3,4,1,5,2] => [5,2,4,1,3] => 1010 => 0101

[3,4,2,1,5] => [4,2,1,3,5] => 1100 => 0011

[3,4,2,5,1] => [5,1,4,2,3] => 1010 => 0101

[3,4,5,1,2] => [2,5,1,3,4] => 0100 => 0010

[3,4,5,2,1] => [5,2,1,3,4] => 1100 => 0011

[3,5,1,2,4] => [2,4,5,1,3] => 0010 => 0100

[3,5,1,4,2] => [4,2,5,1,3] => 1010 => 0101

>>> Load all 11325 entries. <<<

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

descent word

Description

The descent positions of a permutation as a binary word.
For a permutation

$\pi$ of

$n$ letters and each

$1\leq i\leq n-1$ such that

$\pi(i) > \pi(i+1)$ we set

$w_i=1$ , otherwise

$w_i=0$ .
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.

Map

reverse

Description

Return the reversal of a binary word.