- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001744: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 22 + 2\ q$

$F_{5} = 92 + 26\ q + 2\ q^{2}$

$F_{6} = 426 + 242\ q + 50\ q^{2} + 2\ q^{3}$

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

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.

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.