- FindStat

Identifier

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is St000213The number of weak exceedances (also weak excedences) of a permutation., the number of deficiencies is St000703The number of deficiencies 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.

Map

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.