- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000028: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,5] => 0

[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,3,4] => 1

[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,2,3,5] => 1

[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,2,5,3,4] => 1

[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,2,5,3] => 2

[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,2,4,5] => 1

[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,5,2,4] => 2

[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,4,2,3,5] => 1

[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,5,2,3,4] => 1

[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,4,2,3] => 1

[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,4,2,5] => 2

[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,1,5,2,4] => 2

[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,4,5,2,3] => 2

[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,4,5,2] => 3

[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,1,3,4,5] => 1

[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,5,1,3,4] => 2

[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,4,1,3,5] => 2

[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,1,5,3,4] => 1

[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,4,1,5,3] => 2

[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,2,4,5] => 1

[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,3,1,2,4] => 1

[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,2,3,5] => 1

[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,1,2,3,4] => 1

[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,4,1,2,3] => 1

[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,2,5] => 1

[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,1,5,2,4] => 2

[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,1,5,2,3] => 2

[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,3,1,5,2] => 3

[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,1,4,5] => 2

[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,3,5,1,4] => 3

[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,4,1,3,5] => 2

[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,5,1,3,4] => 2

[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [2,5,4,1,3] => 2

[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,4,1,2,5] => 2

[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,5,1,2,4] => 2

[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,4,1,2,3] => 1

[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,3,4,1,2] => 2

[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,3,4,1,5] => 3

[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,3,5,1,4] => 3

[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,4,5,1,3] => 3

[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,4,5,1,2] => 3

[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,3,4,5,1] => 4

[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,6] => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,4,5] => 1

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [5,1,2,3,4,6] => 1

[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [1,2,3,6,4,5] => 1

[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [5,1,2,3,6,4] => 2

[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [4,1,2,3,5,6] => 1

[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [4,1,2,6,3,5] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,2,5,3,4,6] => 1

[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,2,6,3,4,5] => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [6,1,2,5,3,4] => 1

[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [4,1,2,5,3,6] => 2

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [4,1,2,6,3,5] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,2,5,6,3,4] => 2

[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [4,1,2,5,6,3] => 3

[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [3,1,2,4,5,6] => 1

[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [3,1,6,2,4,5] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [3,1,5,2,4,6] => 2

[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [3,1,2,6,4,5] => 1

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [3,1,5,2,6,4] => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,4,2,3,5,6] => 1

[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [6,1,4,2,3,5] => 1

[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,5,2,3,4,6] => 1

[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [6,1,2,3,4,5] => 1

[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [6,1,5,2,3,4] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [5,1,4,2,3,6] => 1

[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,4,2,6,3,5] => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [1,5,2,6,3,4] => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [5,1,4,2,6,3] => 3

[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [3,1,4,2,5,6] => 2

[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [3,1,4,6,2,5] => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [3,1,5,2,4,6] => 2

[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [3,1,6,2,4,5] => 2

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [3,1,6,5,2,4] => 2

[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [1,4,5,2,3,6] => 2

[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [1,4,6,2,3,5] => 2

[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,6,5,2,3,4] => 1

[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [6,1,4,5,2,3] => 2

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + 16\ q + 17\ q^{2} + 7\ q^{3} + q^{4}$

$F_{6} = 1 + 36\ q + 54\ q^{2} + 31\ q^{3} + 9\ q^{4} + q^{5}$

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

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.