- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000326: Binary words ⟶ ℤ

Values

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

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

[1,2,3] => [1,2,3] => 00 => 3

[1,3,2] => [1,3,2] => 01 => 2

[2,1,3] => [1,3,2] => 01 => 2

[2,3,1] => [1,2,3] => 00 => 3

[3,1,2] => [1,2,3] => 00 => 3

[3,2,1] => [1,2,3] => 00 => 3

[1,2,3,4] => [1,2,3,4] => 000 => 4

[1,2,4,3] => [1,2,4,3] => 001 => 3

[1,3,2,4] => [1,3,2,4] => 010 => 2

[1,3,4,2] => [1,3,4,2] => 001 => 3

[1,4,2,3] => [1,4,2,3] => 001 => 3

[1,4,3,2] => [1,4,2,3] => 001 => 3

[2,1,3,4] => [1,3,4,2] => 001 => 3

[2,1,4,3] => [1,4,2,3] => 001 => 3

[2,3,1,4] => [1,4,2,3] => 001 => 3

[2,3,4,1] => [1,2,3,4] => 000 => 4

[2,4,1,3] => [1,3,2,4] => 010 => 2

[2,4,3,1] => [1,2,4,3] => 001 => 3

[3,1,2,4] => [1,2,4,3] => 001 => 3

[3,1,4,2] => [1,4,2,3] => 001 => 3

[3,2,1,4] => [1,4,2,3] => 001 => 3

[3,2,4,1] => [1,2,4,3] => 001 => 3

[3,4,1,2] => [1,2,3,4] => 000 => 4

[3,4,2,1] => [1,2,3,4] => 000 => 4

[4,1,2,3] => [1,2,3,4] => 000 => 4

[4,1,3,2] => [1,3,2,4] => 010 => 2

[4,2,1,3] => [1,3,2,4] => 010 => 2

[4,2,3,1] => [1,2,3,4] => 000 => 4

[4,3,1,2] => [1,2,3,4] => 000 => 4

[4,3,2,1] => [1,2,3,4] => 000 => 4

[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 5

[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 4

[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 3

[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 4

[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 4

[1,2,5,4,3] => [1,2,5,3,4] => 0001 => 4

[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 2

[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2

[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 3

[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 4

[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 4

[1,3,5,4,2] => [1,3,5,2,4] => 0001 => 4

[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 3

[1,4,2,5,3] => [1,4,2,5,3] => 0011 => 3

[1,4,3,2,5] => [1,4,2,5,3] => 0011 => 3

[1,4,3,5,2] => [1,4,2,3,5] => 0010 => 3

[1,4,5,2,3] => [1,4,5,2,3] => 0001 => 4

[1,4,5,3,2] => [1,4,5,2,3] => 0001 => 4

[1,5,2,3,4] => [1,5,2,3,4] => 0001 => 4

[1,5,2,4,3] => [1,5,2,4,3] => 0011 => 3

[1,5,3,2,4] => [1,5,2,4,3] => 0011 => 3

[1,5,3,4,2] => [1,5,2,3,4] => 0001 => 4

[1,5,4,2,3] => [1,5,2,3,4] => 0001 => 4

[1,5,4,3,2] => [1,5,2,3,4] => 0001 => 4

[2,1,3,4,5] => [1,3,4,5,2] => 0001 => 4

[2,1,3,5,4] => [1,3,5,2,4] => 0001 => 4

[2,1,4,3,5] => [1,4,2,3,5] => 0010 => 3

[2,1,4,5,3] => [1,4,5,2,3] => 0001 => 4

[2,1,5,3,4] => [1,5,2,3,4] => 0001 => 4

[2,1,5,4,3] => [1,5,2,3,4] => 0001 => 4

[2,3,1,4,5] => [1,4,5,2,3] => 0001 => 4

[2,3,1,5,4] => [1,5,2,3,4] => 0001 => 4

[2,3,4,1,5] => [1,5,2,3,4] => 0001 => 4

[2,3,4,5,1] => [1,2,3,4,5] => 0000 => 5

[2,3,5,1,4] => [1,4,2,3,5] => 0010 => 3

[2,3,5,4,1] => [1,2,3,5,4] => 0001 => 4

[2,4,1,3,5] => [1,3,5,2,4] => 0001 => 4

[2,4,1,5,3] => [1,5,2,4,3] => 0011 => 3

[2,4,3,1,5] => [1,5,2,4,3] => 0011 => 3

[2,4,3,5,1] => [1,2,4,3,5] => 0010 => 3

[2,4,5,1,3] => [1,3,2,4,5] => 0100 => 2

[2,4,5,3,1] => [1,2,4,5,3] => 0001 => 4

[2,5,1,3,4] => [1,3,4,2,5] => 0010 => 3

[2,5,1,4,3] => [1,4,2,5,3] => 0011 => 3

[2,5,3,1,4] => [1,4,2,5,3] => 0011 => 3

[2,5,3,4,1] => [1,2,5,3,4] => 0001 => 4

[2,5,4,1,3] => [1,3,2,5,4] => 0101 => 2

[2,5,4,3,1] => [1,2,5,3,4] => 0001 => 4

[3,1,2,4,5] => [1,2,4,5,3] => 0001 => 4

[3,1,2,5,4] => [1,2,5,3,4] => 0001 => 4

[3,1,4,2,5] => [1,4,2,5,3] => 0011 => 3

[3,1,4,5,2] => [1,4,5,2,3] => 0001 => 4

[3,1,5,2,4] => [1,5,2,4,3] => 0011 => 3

[3,1,5,4,2] => [1,5,2,3,4] => 0001 => 4

[3,2,1,4,5] => [1,4,5,2,3] => 0001 => 4

[3,2,1,5,4] => [1,5,2,3,4] => 0001 => 4

[3,2,4,1,5] => [1,5,2,4,3] => 0011 => 3

[3,2,4,5,1] => [1,2,4,5,3] => 0001 => 4

[3,2,5,1,4] => [1,4,2,5,3] => 0011 => 3

[3,2,5,4,1] => [1,2,5,3,4] => 0001 => 4

[3,4,1,2,5] => [1,2,5,3,4] => 0001 => 4

[3,4,1,5,2] => [1,5,2,3,4] => 0001 => 4

[3,4,2,1,5] => [1,5,2,3,4] => 0001 => 4

[3,4,2,5,1] => [1,2,5,3,4] => 0001 => 4

[3,4,5,1,2] => [1,2,3,4,5] => 0000 => 5

[3,4,5,2,1] => [1,2,3,4,5] => 0000 => 5

[3,5,1,2,4] => [1,2,4,3,5] => 0010 => 3

[3,5,1,4,2] => [1,4,2,3,5] => 0010 => 3

[3,5,2,1,4] => [1,4,2,3,5] => 0010 => 3

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Description

The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.

Map

descent tops

Description

The descent tops of a permutation as a binary word.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.