- FindStat

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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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

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

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.