- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001744: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

$F_{4} = 15 + 8\ q + q^{2}$

$F_{5} = 52 + 51\ q + 16\ q^{2} + q^{3}$

$F_{6} = 203 + 312\ q + 172\ q^{2} + 32\ q^{3} + q^{4}$

Description

The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let

$\nu$ be a (partial) permutation of

$[k]$ with

$m$ letters together with dashes between some of its letters. An occurrence of

$\nu$ in a permutation

$\tau$ is a subsequence

$\tau_{a_1},\dots,\tau_{a_m}$
such that

$a_i + 1 = a_{i+1}$ whenever there is a dash between the

$i$ -th and the

$(i+1)$ -st letter of

$\nu$ , which is order isomorphic to

$\nu$ .
Thus,

$\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size

$k$ consists of such a generalized vincular pattern

$\nu$ and arrows

$b_1\to c_1, b_2\to c_2,\dots$ , such that precisely the numbers

$1,\dots,k$ appear in the vincular pattern and the arrows.
Let

$\Phi$ be the map Mp00087inverse first fundamental transformation. Let

$\tau$ be a permutation and

$\sigma = \Phi(\tau)$ . Then a subsequence

$w = (x_{a_1},\dots,x_{a_m})$ of

$\tau$ is an occurrence of the arrow pattern if

$w$ is an occurrence of

$\nu$ , for each arrow

$b\to c$ we have

$\sigma(x_b) = x_c$ and

$x_1 < x_2 < \dots < x_k$ .

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation

$\sigma$ is inverse fireworks if its inverse avoids the vincular pattern

$3-12$ . The inverse fireworks map sends any permutation

$\sigma$ to an inverse fireworks permutation that is below

$\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..