- FindStat

Identifier

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000711: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2$

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

$F_{4} = 4 + 8\ q + 2\ q^{2}$

$F_{5} = 5 + 20\ q + 15\ q^{2} + 2\ q^{3}$

$F_{6} = 6 + 40\ q + 60\ q^{2} + 24\ q^{3} + 2\ q^{4}$

Description

The number of big exceedences of a permutation.
A big exceedence of a permutation

$\pi$ is an index

$i$ such that

$\pi(i) - i > 1$ .
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.

Map

Inverse Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$c\pi^{-1}$ where

$c = (1,\ldots,n)$ is the long cycle.

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.