- FindStat

Identifier

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤ

Values

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

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

[1,2,3] => 11 => 00 => 0

[1,3,2] => 10 => 01 => 1

[2,1,3] => 01 => 10 => 1

[2,3,1] => 00 => 11 => 1

[3,1,2] => 00 => 11 => 1

[3,2,1] => 00 => 11 => 1

[1,2,3,4] => 111 => 000 => 0

[1,2,4,3] => 110 => 001 => 1

[1,3,2,4] => 101 => 010 => 1

[1,3,4,2] => 100 => 011 => 1

[1,4,2,3] => 100 => 011 => 1

[1,4,3,2] => 100 => 011 => 1

[2,1,3,4] => 011 => 100 => 1

[2,1,4,3] => 010 => 101 => 2

[2,3,1,4] => 001 => 110 => 1

[2,3,4,1] => 000 => 111 => 1

[2,4,1,3] => 000 => 111 => 1

[2,4,3,1] => 000 => 111 => 1

[3,1,2,4] => 001 => 110 => 1

[3,1,4,2] => 000 => 111 => 1

[3,2,1,4] => 001 => 110 => 1

[3,2,4,1] => 000 => 111 => 1

[3,4,1,2] => 000 => 111 => 1

[3,4,2,1] => 000 => 111 => 1

[4,1,2,3] => 000 => 111 => 1

[4,1,3,2] => 000 => 111 => 1

[4,2,1,3] => 000 => 111 => 1

[4,2,3,1] => 000 => 111 => 1

[4,3,1,2] => 000 => 111 => 1

[4,3,2,1] => 000 => 111 => 1

[1,2,3,4,5] => 1111 => 0000 => 0

[1,2,3,5,4] => 1110 => 0001 => 1

[1,2,4,3,5] => 1101 => 0010 => 1

[1,2,4,5,3] => 1100 => 0011 => 1

[1,2,5,3,4] => 1100 => 0011 => 1

[1,2,5,4,3] => 1100 => 0011 => 1

[1,3,2,4,5] => 1011 => 0100 => 1

[1,3,2,5,4] => 1010 => 0101 => 2

[1,3,4,2,5] => 1001 => 0110 => 1

[1,3,4,5,2] => 1000 => 0111 => 1

[1,3,5,2,4] => 1000 => 0111 => 1

[1,3,5,4,2] => 1000 => 0111 => 1

[1,4,2,3,5] => 1001 => 0110 => 1

[1,4,2,5,3] => 1000 => 0111 => 1

[1,4,3,2,5] => 1001 => 0110 => 1

[1,4,3,5,2] => 1000 => 0111 => 1

[1,4,5,2,3] => 1000 => 0111 => 1

[1,4,5,3,2] => 1000 => 0111 => 1

[1,5,2,3,4] => 1000 => 0111 => 1

[1,5,2,4,3] => 1000 => 0111 => 1

[1,5,3,2,4] => 1000 => 0111 => 1

[1,5,3,4,2] => 1000 => 0111 => 1

[1,5,4,2,3] => 1000 => 0111 => 1

[1,5,4,3,2] => 1000 => 0111 => 1

[2,1,3,4,5] => 0111 => 1000 => 1

[2,1,3,5,4] => 0110 => 1001 => 2

[2,1,4,3,5] => 0101 => 1010 => 2

[2,1,4,5,3] => 0100 => 1011 => 2

[2,1,5,3,4] => 0100 => 1011 => 2

[2,1,5,4,3] => 0100 => 1011 => 2

[2,3,1,4,5] => 0011 => 1100 => 1

[2,3,1,5,4] => 0010 => 1101 => 2

[2,3,4,1,5] => 0001 => 1110 => 1

[2,3,4,5,1] => 0000 => 1111 => 1

[2,3,5,1,4] => 0000 => 1111 => 1

[2,3,5,4,1] => 0000 => 1111 => 1

[2,4,1,3,5] => 0001 => 1110 => 1

[2,4,1,5,3] => 0000 => 1111 => 1

[2,4,3,1,5] => 0001 => 1110 => 1

[2,4,3,5,1] => 0000 => 1111 => 1

[2,4,5,1,3] => 0000 => 1111 => 1

[2,4,5,3,1] => 0000 => 1111 => 1

[2,5,1,3,4] => 0000 => 1111 => 1

[2,5,1,4,3] => 0000 => 1111 => 1

[2,5,3,1,4] => 0000 => 1111 => 1

[2,5,3,4,1] => 0000 => 1111 => 1

[2,5,4,1,3] => 0000 => 1111 => 1

[2,5,4,3,1] => 0000 => 1111 => 1

[3,1,2,4,5] => 0011 => 1100 => 1

[3,1,2,5,4] => 0010 => 1101 => 2

[3,1,4,2,5] => 0001 => 1110 => 1

[3,1,4,5,2] => 0000 => 1111 => 1

[3,1,5,2,4] => 0000 => 1111 => 1

[3,1,5,4,2] => 0000 => 1111 => 1

[3,2,1,4,5] => 0011 => 1100 => 1

[3,2,1,5,4] => 0010 => 1101 => 2

[3,2,4,1,5] => 0001 => 1110 => 1

[3,2,4,5,1] => 0000 => 1111 => 1

[3,2,5,1,4] => 0000 => 1111 => 1

[3,2,5,4,1] => 0000 => 1111 => 1

[3,4,1,2,5] => 0001 => 1110 => 1

[3,4,1,5,2] => 0000 => 1111 => 1

[3,4,2,1,5] => 0001 => 1110 => 1

[3,4,2,5,1] => 0000 => 1111 => 1

[3,4,5,1,2] => 0000 => 1111 => 1

[3,4,5,2,1] => 0000 => 1111 => 1

[3,5,1,2,4] => 0000 => 1111 => 1

[3,5,1,4,2] => 0000 => 1111 => 1

[3,5,2,1,4] => 0000 => 1111 => 1

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$F_{2} = 1 + q$

$F_{3} = 1 + 5\ q$

$F_{4} = 1 + 22\ q + q^{2}$

$F_{5} = 1 + 110\ q + 9\ q^{2}$

$F_{6} = 1 + 659\ q + 59\ q^{2} + q^{3}$

$F_{7} = 1 + 4655\ q + 371\ q^{2} + 13\ q^{3}$

Description

The number of runs of ones in a binary word.

Map

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is

$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$
For

$n > 1$ it can also be described as the set of occurrences of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$
or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$
see [3].
The permutation is connected, when the connectivity set is empty.

Map

complement

Description

Send a binary word to the word obtained by interchanging the two letters.