- FindStat

Identifier

Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1111

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 11305 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

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

Map

descent bottoms

Description

The descent bottoms of a permutation as a binary word.