- FindStat

Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤ

Values

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

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

[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0

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

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

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

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

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

[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0

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

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

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

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

[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 011 => 5

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

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

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

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

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

[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 011 => 5

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

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

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

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

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

[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 011 => 5

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

[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 011 => 5

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

[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => 011 => 5

[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 011 => 5

[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 111 => 6

[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0

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

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

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

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

[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7

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

[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6

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

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

[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 6

[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7

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

[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6

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

[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6

[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 6

[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7

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

[1,5,2,4,3] => [1,5,2,4,3] => [1,3,5,4,2] => 0011 => 7

[1,5,3,2,4] => [1,5,3,2,4] => [1,4,3,5,2] => 0101 => 6

[1,5,3,4,2] => [1,5,2,4,3] => [1,3,5,4,2] => 0011 => 7

[1,5,4,2,3] => [1,5,4,2,3] => [1,4,5,3,2] => 0011 => 7

[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0111 => 9

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

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

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

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

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

[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1011 => 8

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

[2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6

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

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

[2,3,5,1,4] => [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 6

[2,3,5,4,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7

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

[2,4,1,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6

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

[2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6

[2,4,5,1,3] => [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 6

[2,4,5,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7

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

[2,5,1,4,3] => [2,5,1,4,3] => [3,1,5,4,2] => 1011 => 8

[2,5,3,1,4] => [1,5,3,2,4] => [1,4,3,5,2] => 0101 => 6

[2,5,3,4,1] => [1,5,2,4,3] => [1,3,5,4,2] => 0011 => 7

[2,5,4,1,3] => [2,5,4,1,3] => [4,1,5,3,2] => 1011 => 8

[2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0111 => 9

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

[3,1,2,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 0101 => 6

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

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

[3,1,5,2,4] => [3,1,5,2,4] => [2,4,1,5,3] => 0101 => 6

[3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => 1011 => 8

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

[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1101 => 7

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

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

[3,2,5,1,4] => [3,2,5,1,4] => [4,2,1,5,3] => 1101 => 7

[3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => 1011 => 8

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

[3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6

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

[3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6

[3,4,5,1,2] => [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 6

[3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7

[3,5,1,2,4] => [3,5,1,2,4] => [3,4,1,5,2] => 0101 => 6

[3,5,1,4,2] => [2,5,1,4,3] => [3,1,5,4,2] => 1011 => 8

[3,5,2,1,4] => [3,5,2,1,4] => [4,3,1,5,2] => 1101 => 7

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Description

The sum of the positions of the ones in a binary word.

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

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

inverse

Description

Sends a permutation to its inverse.