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Identifier

Mp00310: Permutations —toric promotion⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ 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] => [3,2,1] => [3,2,1] => 11 => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,5,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => 1111 => 10

[2,5,4,3,1] => [1,5,4,3,2] => [5,4,3,1,2] => 1110 => 6

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

[3,1,2,5,4] => [2,5,4,1,3] => [5,4,2,1,3] => 1110 => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.

Map

toric promotion

Description

Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.