- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001004: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see St000371The number of mid points of decreasing subsequences of length 3 in a permutation. and St000372The number of mid points of increasing subsequences of length 3 in a permutation..

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.