- FindStat

Identifier

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,6,5] => 11110 => [4,1] => [1,1] => 3

[1,2,3,5,4,6] => 11101 => [3,1,1] => [1,2] => 4

[1,2,3,5,6,4] => 11100 => [3,2] => [1,1] => 3

[1,2,3,6,4,5] => 11100 => [3,2] => [1,1] => 3

[1,2,3,6,5,4] => 11100 => [3,2] => [1,1] => 3

[1,2,4,3,5,6] => 11011 => [2,1,2] => [1,1,1] => 4

[1,2,4,3,6,5] => 11010 => [2,1,1,1] => [1,3] => 5

[1,2,4,5,3,6] => 11001 => [2,2,1] => [2,1] => 4

[1,2,4,5,6,3] => 11000 => [2,3] => [1,1] => 3

[1,2,4,6,3,5] => 11000 => [2,3] => [1,1] => 3

[1,2,4,6,5,3] => 11000 => [2,3] => [1,1] => 3

[1,2,5,3,4,6] => 11001 => [2,2,1] => [2,1] => 4

[1,2,5,3,6,4] => 11000 => [2,3] => [1,1] => 3

[1,2,5,4,3,6] => 11001 => [2,2,1] => [2,1] => 4

[1,2,5,4,6,3] => 11000 => [2,3] => [1,1] => 3

[1,2,5,6,3,4] => 11000 => [2,3] => [1,1] => 3

[1,2,5,6,4,3] => 11000 => [2,3] => [1,1] => 3

[1,2,6,3,4,5] => 11000 => [2,3] => [1,1] => 3

[1,2,6,3,5,4] => 11000 => [2,3] => [1,1] => 3

[1,2,6,4,3,5] => 11000 => [2,3] => [1,1] => 3

[1,2,6,4,5,3] => 11000 => [2,3] => [1,1] => 3

[1,2,6,5,3,4] => 11000 => [2,3] => [1,1] => 3

[1,2,6,5,4,3] => 11000 => [2,3] => [1,1] => 3

[1,3,2,4,5,6] => 10111 => [1,1,3] => [2,1] => 4

[1,3,2,4,6,5] => 10110 => [1,1,2,1] => [2,1,1] => 5

[1,3,2,5,4,6] => 10101 => [1,1,1,1,1] => [5] => 6

[1,3,2,5,6,4] => 10100 => [1,1,1,2] => [3,1] => 5

[1,3,2,6,4,5] => 10100 => [1,1,1,2] => [3,1] => 5

[1,3,2,6,5,4] => 10100 => [1,1,1,2] => [3,1] => 5

[1,3,4,2,5,6] => 10011 => [1,2,2] => [1,2] => 4

[1,3,4,2,6,5] => 10010 => [1,2,1,1] => [1,1,2] => 5

[1,3,4,5,2,6] => 10001 => [1,3,1] => [1,1,1] => 4

[1,3,4,5,6,2] => 10000 => [1,4] => [1,1] => 3

[1,3,4,6,2,5] => 10000 => [1,4] => [1,1] => 3

[1,3,4,6,5,2] => 10000 => [1,4] => [1,1] => 3

[1,3,5,2,4,6] => 10001 => [1,3,1] => [1,1,1] => 4

[1,3,5,2,6,4] => 10000 => [1,4] => [1,1] => 3

[1,3,5,4,2,6] => 10001 => [1,3,1] => [1,1,1] => 4

[1,3,5,4,6,2] => 10000 => [1,4] => [1,1] => 3

[1,3,5,6,2,4] => 10000 => [1,4] => [1,1] => 3

[1,3,5,6,4,2] => 10000 => [1,4] => [1,1] => 3

>>> Load all 9494 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition $C$ is the compositions composed of the lengths of consecutive runs of the same integer in $C$.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.

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.