- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000028: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 18248 entries. <<<

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$F_{1} = 1$

$F_{2} = 2 + q$

$F_{3} = 6 + 7\ q + 3\ q^{2}$

$F_{4} = 24 + 51\ q + 34\ q^{2} + 16\ q^{3}$

$F_{5} = 120 + 421\ q + 365\ q^{2} + 255\ q^{3} + 135\ q^{4}$

$F_{6} = 720 + 3963\ q + 4376\ q^{2} + 3461\ q^{3} + 2751\ q^{4} + 1536\ q^{5}$

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

to Dyck path

Description

Sends a parking function to its support Dyck word.

Map

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..