- FindStat

Identifier

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 1 + q^{2}$

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

$F_{4} = 6 + 4\ q + 3\ q^{2} + q^{4}$

$F_{5} = 18 + 13\ q + 6\ q^{2} + 4\ q^{3} + q^{5}$

$F_{6} = 57 + 40\ q + 21\ q^{2} + 8\ q^{3} + 5\ q^{4} + q^{6}$

Description

The number of strong fixed points of a permutation.

$i$ is called a strong fixed point of

$\pi$ if
1.

$j < i$ implies

$\pi_j < \pi_i$ , and
2.

$j > i$ implies

$\pi_j > \pi_i$
This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3].
The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see St000314The number of left-to-right-maxima of a permutation..

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.