- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words

Images

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

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

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

[1,0,1,0,1,0] => [3,2,1] => 11 => 11

[1,0,1,1,0,0] => [2,3,1] => 01 => 10

[1,1,0,0,1,0] => [3,1,2] => 10 => 01

[1,1,0,1,0,0] => [2,1,3] => 10 => 01

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

[1,0,1,0,1,0,1,0] => [4,3,2,1] => 111 => 111

[1,0,1,0,1,1,0,0] => [3,4,2,1] => 011 => 110

[1,0,1,1,0,0,1,0] => [4,2,3,1] => 101 => 101

[1,0,1,1,0,1,0,0] => [3,2,4,1] => 101 => 101

[1,0,1,1,1,0,0,0] => [2,3,4,1] => 001 => 100

[1,1,0,0,1,0,1,0] => [4,3,1,2] => 110 => 011

[1,1,0,0,1,1,0,0] => [3,4,1,2] => 010 => 010

[1,1,0,1,0,0,1,0] => [4,2,1,3] => 110 => 011

[1,1,0,1,0,1,0,0] => [3,2,1,4] => 110 => 011

[1,1,0,1,1,0,0,0] => [2,3,1,4] => 010 => 010

[1,1,1,0,0,0,1,0] => [4,1,2,3] => 100 => 001

[1,1,1,0,0,1,0,0] => [3,1,2,4] => 100 => 001

[1,1,1,0,1,0,0,0] => [2,1,3,4] => 100 => 001

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

[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 1111 => 1111

[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 0111 => 1110

[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 1011 => 1101

[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 1011 => 1101

[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 0011 => 1100

[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 1101 => 1011

[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 0101 => 1010

[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 1101 => 1011

[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 1101 => 1011

[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 0101 => 1010

[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1001 => 1001

[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 1001 => 1001

[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 1001 => 1001

[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0001 => 1000

[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 1110 => 0111

[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 0110 => 0110

[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 1010 => 0101

[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 1010 => 0101

[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 0010 => 0100

[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 1110 => 0111

[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 0110 => 0110

[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1110 => 0111

[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 1110 => 0111

[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 0110 => 0110

[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1010 => 0101

[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 1010 => 0101

[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 1010 => 0101

[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 0010 => 0100

[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1100 => 0011

[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 0100 => 0010

[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1100 => 0011

[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 1100 => 0011

[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 0100 => 0010

[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1100 => 0011

[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 1100 => 0011

[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 1100 => 0011

[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 0100 => 0010

[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1000 => 0001

[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 1000 => 0001

[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 1000 => 0001

[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1000 => 0001

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 11111 => 11111

[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => 01111 => 11110

[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 10111 => 11101

[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => 10111 => 11101

[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 00111 => 11100

[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => 11011 => 11011

[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => 01011 => 11010

[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => 11011 => 11011

[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => 11011 => 11011

[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => 01011 => 11010

[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 10011 => 11001

[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => 10011 => 11001

[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => 10011 => 11001

[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 00011 => 11000

[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 11101 => 10111

[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 01101 => 10110

[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 10101 => 10101

[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => 10101 => 10101

[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 00101 => 10100

[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => 11101 => 10111

[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => 01101 => 10110

[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => 11101 => 10111

[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => 11101 => 10111

[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => 01101 => 10110

[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => 10101 => 10101

[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => 10101 => 10101

[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => 10101 => 10101

[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => 00101 => 10100

[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 11001 => 10011

[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 01001 => 10010

[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => 11001 => 10011

[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => 11001 => 10011

[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => 01001 => 10010

[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => 11001 => 10011

[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => 11001 => 10011

[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => 11001 => 10011

[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => 01001 => 10010

>>> Load all 2215 entries. <<<

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

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

reverse

Description

Return the reversal of a binary word.