- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000220: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 10 + 3\ q + q^{2}$

$F_{5} = 28 + 9\ q + 4\ q^{2} + q^{3}$

$F_{6} = 84 + 28\ q + 14\ q^{2} + 5\ q^{3} + q^{4}$

Description

The number of occurrences of the pattern 132 in a permutation.

Map

Alexandersson Kebede

Description

Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation

$\pi$ of length

$n$ . The mapping looks for a smallest odd integer

$i\in[n-1]$ such that swapping the entries

$\pi(i)$ and

$\pi(i+1)$ preserves the set of right-to-left minima. Otherwise,

$\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly

$\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in

$S_n$ fixed under this map, with exactly

$k$ right-to-left minima, see Lemma 35 in [1].

Map

to 132-avoiding permutation

Description

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