- FindStat

Identifier

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤ

Values

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

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$F_{1} = q$

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

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

$F_{4} = 6\ q^{3} + 8\ q^{4}$

$F_{5} = 2\ q^{3} + 24\ q^{4} + 16\ q^{5}$

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

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.

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

Map

to 321-avoiding permutation (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength

$n$ in an

$n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.