- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
St000807: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0] => [1,1,1,1] => [4] => [4] => 0

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,0] => [4] => [1] => [1] => 0

[1,1,0,1,1,0,0,0] => [4] => [1] => [1] => 0

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

[1,1,1,0,0,1,0,0] => [4] => [1] => [1] => 0

[1,1,1,0,1,0,0,0] => [4] => [1] => [1] => 0

[1,1,1,1,0,0,0,0] => [4] => [1] => [1] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [5] => [5] => 0

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

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

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

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

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

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

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

[1,0,1,1,0,1,0,1,0,0] => [1,4] => [1,1] => [1,1] => 0

[1,0,1,1,0,1,1,0,0,0] => [1,4] => [1,1] => [1,1] => 0

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

[1,0,1,1,1,0,0,1,0,0] => [1,4] => [1,1] => [1,1] => 0

[1,0,1,1,1,0,1,0,0,0] => [1,4] => [1,1] => [1,1] => 0

[1,0,1,1,1,1,0,0,0,0] => [1,4] => [1,1] => [1,1] => 0

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,0,1,0] => [4,1] => [1,1] => [1,1] => 0

[1,1,0,1,0,1,0,1,0,0] => [5] => [1] => [1] => 0

[1,1,0,1,0,1,1,0,0,0] => [5] => [1] => [1] => 0

[1,1,0,1,1,0,0,0,1,0] => [4,1] => [1,1] => [1,1] => 0

[1,1,0,1,1,0,0,1,0,0] => [5] => [1] => [1] => 0

[1,1,0,1,1,0,1,0,0,0] => [5] => [1] => [1] => 0

[1,1,0,1,1,1,0,0,0,0] => [5] => [1] => [1] => 0

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

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

[1,1,1,0,0,1,0,0,1,0] => [4,1] => [1,1] => [1,1] => 0

[1,1,1,0,0,1,0,1,0,0] => [5] => [1] => [1] => 0

[1,1,1,0,0,1,1,0,0,0] => [5] => [1] => [1] => 0

[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,1] => [1,1] => 0

[1,1,1,0,1,0,0,1,0,0] => [5] => [1] => [1] => 0

[1,1,1,0,1,0,1,0,0,0] => [5] => [1] => [1] => 0

[1,1,1,0,1,1,0,0,0,0] => [5] => [1] => [1] => 0

[1,1,1,1,0,0,0,0,1,0] => [4,1] => [1,1] => [1,1] => 0

[1,1,1,1,0,0,0,1,0,0] => [5] => [1] => [1] => 0

[1,1,1,1,0,0,1,0,0,0] => [5] => [1] => [1] => 0

[1,1,1,1,0,1,0,0,0,0] => [5] => [1] => [1] => 0

[1,1,1,1,1,0,0,0,0,0] => [5] => [1] => [1] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [6] => [6] => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [4,1] => [4,1] => 0

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

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

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

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

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

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

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => [2,1] => [2,1] => 0

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => [2,1] => [2,1] => 0

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

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => [2,1] => [2,1] => 0

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => [2,1] => [2,1] => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [2,1] => [2,1] => 0

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

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

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

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

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

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

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

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,4,1] => [1,1,1] => [1,1,1] => 0

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => [1,1] => [1,1] => 0

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => [1,1] => [1,1] => 0

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,4,1] => [1,1,1] => [1,1,1] => 0

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => [1,1] => [1,1] => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => [1,1] => [1,1] => 0

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => [1,1] => [1,1] => 0

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

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

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,1] => [1,1,1] => [1,1,1] => 0

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => [1,1] => [1,1] => 0

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => [1,1] => [1,1] => 0

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,1] => [1,1,1] => [1,1,1] => 0

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => [1,1] => [1,1] => 0

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => [1,1] => [1,1] => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => [1,1] => [1,1] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 14$

$F_{5} = 42$

$F_{6} = 131 + q$

$F_{7} = 426 + 3\ q$

$F_{8} = 1420 + 10\ q$

Description

The sum of the heights of the valleys of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.

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

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

Map

Foata bijection

Description

The Foata bijection for compositions.
The Foata bijection

$\phi$ is a bijection on the set of words whose letters are positive integers. It can be defined by induction on the size of the word:
Given a word

$w_1 w_2 ... w_n$ , compute the image inductively by starting with

$\phi(w_1) = w_1$ .
At the

$i$ -th step, if

$\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$ , define

$\phi(w_1 w_2 ... w_i w_{i+1})$ by placing

$w_{i+1}$ on the end of the word

$v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.

If $w_{i+1} \geq v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$ .
If $w_{i+1} < v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$ .

In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute

$\phi([1,4,2,5,3])$ , the sequence of words is

$1$
$|1|4 \to 14$
$|14|2 \to 412$
$|4|1|2|5 \to 4125$
$|4|125|3 \to 45123.$

In total, this gives

$\phi([1,4,2,5,3]) = [4,5,1,2,3]$ .
This bijection sends the major index St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..