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Identifier

Mp00109: Permutations —descent word⟶ Binary words

Images

[1] =>

[1,2] => 0

[2,1] => 1

[1,2,3] => 00

[1,3,2] => 01

[2,1,3] => 10

[2,3,1] => 01

[3,1,2] => 10

[3,2,1] => 11

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 42558 entries. <<<

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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.

Properties

surjective, graded

Sage code

def mapping(X):
    if X:
        D = X.descents(from_zero=True)
        w = [1 if i in D else 0 for i in range(len(X)-1)]
        return Words([0,1])(w)
    raise ValueError("The descent word is not defined for the empty permutation")

Weight

Created

Sep 22, 2020 at 11:07 by FindStatCrew

Updated

Sep 22, 2020 at 11:07 by Martin Rubey