- FindStat

Identifier

Mp00024: Dyck paths —to 321-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] => [2,1,3] => 10 => 01

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => 01010 => 01010

[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 00100 => 00100

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

[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => 01000 => 00010

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

[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => 01010 => 01010

[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => 00100 => 00100

[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => 10001 => 10001

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

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

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 00010 => 01000

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

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

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

[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => 01010 => 01010

[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => 00100 => 00100

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

[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => 01000 => 00010

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

[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => 01010 => 01010

[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => 00100 => 00100

[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => 10001 => 10001

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

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

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => 00010 => 01000

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

[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => 01000 => 00010

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

[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => 01010 => 01010

[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => 00100 => 00100

[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => 10001 => 10001

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

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

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => 00010 => 01000

>>> Load all 2425 entries. <<<

Map

to 321-avoiding permutation

Description

Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.

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.