- FindStat

Identifier

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => 00000 => [6]

[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 00000 => [6]

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

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

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

[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => 00001 => [5,1]

[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => 00001 => [5,1]

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

[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => 00000 => [6]

[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => 00000 => [6]

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 00000 => [6]

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

[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 00000 => [6]

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

[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => 00000 => [6]

[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => 00000 => [6]

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

[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => 00001 => [5,1]

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

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

[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => 00001 => [5,1]

[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => 01100 => [2,1,3]

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

[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => 00000 => [6]

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => 00000 => [6]

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

[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => 00000 => [6]

[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => 01100 => [2,1,3]

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

[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => 00000 => [6]

[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => 01110 => [2,1,1,2]

[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => 00110 => [3,1,2]

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

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => 00000 => [6]

>>> Load all 2119 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

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is

$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$
For

$n > 1$ it can also be described as the set of occurrences of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$
or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$
see [3].
The permutation is connected, when the connectivity set is empty.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.