- FindStat

Identifier

Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000842: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{4} = 24\ q^{2}$

$F_{5} = 110\ q^{2} + 10\ q^{3}$

$F_{6} = 660\ q^{2} + 60\ q^{3}$

$F_{7} = 4662\ q^{2} + 378\ q^{3}$

Description

The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of

$\pi$ :

$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$
According to [1, Def.1.3], a permutation

$\pi$ is

$k$ -prolific, if the set of permutations obtained from

$\pi$ by deleting any

$k$ elements and standardising has maximal cardinality, i.e.,

$\binom{n}{k}$ .
By [1, Thm.2.22], a permutation is

$k$ -prolific if and only if its breath is at least

$k+2$ .
By [1, Cor.4.3], the smallest permutations that are

$k$ -prolific have size

$\lceil k^2+2k+1\rceil$ , and by [1, Thm.4.4], there are

$k$ -prolific permutations of any size larger than this.
According to [2] the proportion of

$k$ -prolific permutations in the set of all permutations is asymptotically equal to

$\exp(-k^2-k)$ .

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

Lehmer-code to major-code bijection

Description

Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation

$\sigma$ is given by

$L(\sigma) = l_1 \ldots l_n$ with

$l_i = \# \{ j > i : \sigma_j < \sigma_i \}$ . In particular,

$l_i$ is the number of boxes in the

$i$ -th column of the Rothe diagram. For example, the Lehmer code of

$\sigma = [4,3,1,5,2]$ is

$32010$ . The Lehmer code

$L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size

$n$ and sequences

$l_1\ldots l_n \in \mathbf{N}^n$ with

$l_i \leq i$ .
* The major code

$M(\sigma)$ of a permutation

$\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence

$m_1 m_2 \ldots m_n$ with

$m_i \geq i$ . To define

$m_i$ , let

$\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all

$\sigma_j < i$ from the one-line notation of

$\sigma$ . The

$i$ -th index is then given by

$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$
For example, the permutation

$[9,3,5,7,2,1,4,6,8]$ has major code

$[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since

$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$

$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$
Observe that the sum of the major code of

$\sigma$ equals the major index of

$\sigma$ .

Map

restriction

Description

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