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Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000028: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 4\ q + 7\ q^{2} + 12\ q^{3}$

$F_{5} = 1 + 7\ q + 17\ q^{2} + 35\ q^{3} + 60\ q^{4}$

$F_{6} = 1 + 12\ q + 44\ q^{2} + 93\ q^{3} + 210\ q^{4} + 360\ q^{5}$

Description

The number of stack-sorts needed to sort a permutation.
A permutation is (West)

$t$ -stack sortable if it is sortable using

$t$ stacks in series.
Let

$W_t(n,k)$ be the number of permutations of size

$n$
with

$k$ descents which are

$t$ -stack sortable. Then the polynomials

$W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have

$W_{n,1}(x) = A_n(x)$ , the Eulerian polynomials. One can show that

$W_{n,1}(x)$ and

$W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern

$231$ have statistic at most

$1$ , see [3]. These are counted by

$\frac{1}{n+1}\binom{2n}{n}$ (OEIS:A000108). Precisely the permutations that avoid the pattern

$2341$ and the barred pattern

$3\bar 5241$ have statistic at most

$2$ , see [4]. These are counted by

$\frac{2(3n)!}{(n+1)!(2n+1)!}$ (OEIS:A000139).

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.).

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

Map

Demazure product with inverse

Description

This map sends a permutation

$\pi$ to

$\pi^{-1} \star \pi$ where

$\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations

$\pi$ for which

$\pi = \pi^{-1}$ .