- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤ (values match St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$ .)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2\ q^{2}$

Description

The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ .

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

Foata bijection

Description

Sends a permutation to its image under the Foata bijection.
The Foata bijection

$\phi$ is a bijection on the set of words with no two equal letters. 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 (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).