- FindStat

Identifier

Mp00252: Permutations —restriction⟶ Permutations
St000625: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

restriction

Description

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