- FindStat

Identifier

Mp00252: Permutations —restriction⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000518: Binary words ⟶ ℤ

Values

[1,2,3] => [1,2] => 0 => 2

[1,3,2] => [1,2] => 0 => 2

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

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

[3,1,2] => [1,2] => 0 => 2

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

[1,2,3,4] => [1,2,3] => 00 => 3

[1,2,4,3] => [1,2,3] => 00 => 3

[1,3,2,4] => [1,3,2] => 01 => 4

[1,3,4,2] => [1,3,2] => 01 => 4

[1,4,2,3] => [1,2,3] => 00 => 3

[1,4,3,2] => [1,3,2] => 01 => 4

[2,1,3,4] => [2,1,3] => 10 => 4

[2,1,4,3] => [2,1,3] => 10 => 4

[2,3,1,4] => [2,3,1] => 01 => 4

[2,3,4,1] => [2,3,1] => 01 => 4

[2,4,1,3] => [2,1,3] => 10 => 4

[2,4,3,1] => [2,3,1] => 01 => 4

[3,1,2,4] => [3,1,2] => 10 => 4

[3,1,4,2] => [3,1,2] => 10 => 4

[3,2,1,4] => [3,2,1] => 11 => 3

[3,2,4,1] => [3,2,1] => 11 => 3

[3,4,1,2] => [3,1,2] => 10 => 4

[3,4,2,1] => [3,2,1] => 11 => 3

[4,1,2,3] => [1,2,3] => 00 => 3

[4,1,3,2] => [1,3,2] => 01 => 4

[4,2,1,3] => [2,1,3] => 10 => 4

[4,2,3,1] => [2,3,1] => 01 => 4

[4,3,1,2] => [3,1,2] => 10 => 4

[4,3,2,1] => [3,2,1] => 11 => 3

[1,2,3,4,5] => [1,2,3,4] => 000 => 4

[1,2,3,5,4] => [1,2,3,4] => 000 => 4

[1,2,4,3,5] => [1,2,4,3] => 001 => 6

[1,2,4,5,3] => [1,2,4,3] => 001 => 6

[1,2,5,3,4] => [1,2,3,4] => 000 => 4

[1,2,5,4,3] => [1,2,4,3] => 001 => 6

[1,3,2,4,5] => [1,3,2,4] => 010 => 7

[1,3,2,5,4] => [1,3,2,4] => 010 => 7

[1,3,4,2,5] => [1,3,4,2] => 001 => 6

[1,3,4,5,2] => [1,3,4,2] => 001 => 6

[1,3,5,2,4] => [1,3,2,4] => 010 => 7

[1,3,5,4,2] => [1,3,4,2] => 001 => 6

[1,4,2,3,5] => [1,4,2,3] => 010 => 7

[1,4,2,5,3] => [1,4,2,3] => 010 => 7

[1,4,3,2,5] => [1,4,3,2] => 011 => 6

[1,4,3,5,2] => [1,4,3,2] => 011 => 6

[1,4,5,2,3] => [1,4,2,3] => 010 => 7

[1,4,5,3,2] => [1,4,3,2] => 011 => 6

[1,5,2,3,4] => [1,2,3,4] => 000 => 4

[1,5,2,4,3] => [1,2,4,3] => 001 => 6

[1,5,3,2,4] => [1,3,2,4] => 010 => 7

[1,5,3,4,2] => [1,3,4,2] => 001 => 6

[1,5,4,2,3] => [1,4,2,3] => 010 => 7

[1,5,4,3,2] => [1,4,3,2] => 011 => 6

[2,1,3,4,5] => [2,1,3,4] => 100 => 6

[2,1,3,5,4] => [2,1,3,4] => 100 => 6

[2,1,4,3,5] => [2,1,4,3] => 101 => 7

[2,1,4,5,3] => [2,1,4,3] => 101 => 7

[2,1,5,3,4] => [2,1,3,4] => 100 => 6

[2,1,5,4,3] => [2,1,4,3] => 101 => 7

[2,3,1,4,5] => [2,3,1,4] => 010 => 7

[2,3,1,5,4] => [2,3,1,4] => 010 => 7

[2,3,4,1,5] => [2,3,4,1] => 001 => 6

[2,3,4,5,1] => [2,3,4,1] => 001 => 6

[2,3,5,1,4] => [2,3,1,4] => 010 => 7

[2,3,5,4,1] => [2,3,4,1] => 001 => 6

[2,4,1,3,5] => [2,4,1,3] => 010 => 7

[2,4,1,5,3] => [2,4,1,3] => 010 => 7

[2,4,3,1,5] => [2,4,3,1] => 011 => 6

[2,4,3,5,1] => [2,4,3,1] => 011 => 6

[2,4,5,1,3] => [2,4,1,3] => 010 => 7

[2,4,5,3,1] => [2,4,3,1] => 011 => 6

[2,5,1,3,4] => [2,1,3,4] => 100 => 6

[2,5,1,4,3] => [2,1,4,3] => 101 => 7

[2,5,3,1,4] => [2,3,1,4] => 010 => 7

[2,5,3,4,1] => [2,3,4,1] => 001 => 6

[2,5,4,1,3] => [2,4,1,3] => 010 => 7

[2,5,4,3,1] => [2,4,3,1] => 011 => 6

[3,1,2,4,5] => [3,1,2,4] => 100 => 6

[3,1,2,5,4] => [3,1,2,4] => 100 => 6

[3,1,4,2,5] => [3,1,4,2] => 101 => 7

[3,1,4,5,2] => [3,1,4,2] => 101 => 7

[3,1,5,2,4] => [3,1,2,4] => 100 => 6

[3,1,5,4,2] => [3,1,4,2] => 101 => 7

[3,2,1,4,5] => [3,2,1,4] => 110 => 6

[3,2,1,5,4] => [3,2,1,4] => 110 => 6

[3,2,4,1,5] => [3,2,4,1] => 101 => 7

[3,2,4,5,1] => [3,2,4,1] => 101 => 7

[3,2,5,1,4] => [3,2,1,4] => 110 => 6

[3,2,5,4,1] => [3,2,4,1] => 101 => 7

[3,4,1,2,5] => [3,4,1,2] => 010 => 7

[3,4,1,5,2] => [3,4,1,2] => 010 => 7

[3,4,2,1,5] => [3,4,2,1] => 011 => 6

[3,4,2,5,1] => [3,4,2,1] => 011 => 6

[3,4,5,1,2] => [3,4,1,2] => 010 => 7

[3,4,5,2,1] => [3,4,2,1] => 011 => 6

[3,5,1,2,4] => [3,1,2,4] => 100 => 6

[3,5,1,4,2] => [3,1,4,2] => 101 => 7

[3,5,2,1,4] => [3,2,1,4] => 110 => 6

[3,5,2,4,1] => [3,2,4,1] => 101 => 7

[3,5,4,1,2] => [3,4,1,2] => 010 => 7

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Description

The number of distinct subsequences in a binary word.
In contrast to the subword complexity St000294The number of distinct factors of a binary word. this is the cardinality of the set of all subsequences of not necessarily consecutive letters.

Map

restriction

Description

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

Map

descent word

Description

The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.