- FindStat

Identifier

Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000625: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{4} = 16\ q^{3} + 8\ q^{4}$

$F_{5} = 40\ q^{4} + 80\ q^{5}$

$F_{6} = 96\ q^{5} + 384\ q^{6} + 240\ q^{7}$

$F_{7} = 224\ q^{6} + 1456\ q^{7} + 2800\ q^{8} + 560\ q^{9}$

Description

The sum of the minimal distances to a greater element.
Set

$\pi_0 = \pi_{n+1} = n+1$ , then this statistic is

$\sum_{i=1}^n \min_d(\pi_{i-d}>\pi_i\text{ or }\pi_{i+d}>\pi_i)$
This statistic appears in [1].
The generating function for the sequence of maximal values attained on

$\mathfrak S_r$ ,

$r\geq 0$ apparently coincides with [2], which satisfies the functional equation

$(x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x = 0.$

Map

cactus evacuation

Description

The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.