- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000028: Permutations ⟶ ℤ

Values

[1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4367 entries. <<<

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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

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.