- FindStat

Identifier

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000234: Permutations ⟶ ℤ (values match St000056The decomposition (or block) number of a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + 3\ q$

$F_{4} = 7 + 7\ q$

$F_{5} = 23 + 19\ q$

$F_{6} = 76 + 56\ q$

Description

The number of global ascents of a permutation.
The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$
Equivalently, by the pigeonhole principle,

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

$n > 1$ it can also be described as an occurrence 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].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives oeis:A003319.

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

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].