- FindStat

Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words

Images

[1] => [1] =>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 7320 entries. <<<

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..

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.