- FindStat

Identifier

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,4,3,5,1] => [5,2,4,3,1] => [5,2,4,3,1] => [1,2,1,1] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,2,4,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => [1,2,1,1] => 8

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

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

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

[3,4,1,5,2] => [5,4,3,1,2] => [4,5,3,2,1] => [2,1,1,1] => 9

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

[3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1] => 10

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

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

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

[3,5,1,4,2] => [4,5,3,1,2] => [4,1,5,3,2] => [1,2,1,1] => 8

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 3\ q + 5\ q^{2} + 6\ q^{3} + 5\ q^{4} + 3\ q^{5} + q^{6}$

$F_{5} = 1 + 4\ q + 9\ q^{2} + 15\ q^{3} + 20\ q^{4} + 22\ q^{5} + 20\ q^{6} + 15\ q^{7} + 9\ q^{8} + 4\ q^{9} + q^{10}$

$F_{6} = 1 + 5\ q + 14\ q^{2} + 29\ q^{3} + 49\ q^{4} + 71\ q^{5} + 90\ q^{6} + 101\ q^{7} + 101\ q^{8} + 90\ q^{9} + 71\ q^{10} + 49\ q^{11} + 29\ q^{12} + 14\ q^{13} + 5\ q^{14} + q^{15}$

$F_{7} = 1 + 6\ q + 20\ q^{2} + 49\ q^{3} + 98\ q^{4} + 169\ q^{5} + 259\ q^{6} + 359\ q^{7} + 455\ q^{8} + 531\ q^{9} + 573\ q^{10} + 573\ q^{11} + 531\ q^{12} + 455\ q^{13} + 359\ q^{14} + 259\ q^{15} + 169\ q^{16} + 98\ q^{17} + 49\ q^{18} + 20\ q^{19} + 6\ q^{20} + q^{21}$

Description

The major index of the composition.
The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.

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

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation

$\pi$ of length

$n$ is the integer composition of

$n$ whose descent set equals the descent set of

$\pi$ . The descent set of a permutation

$\pi$ is

$\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$ . The descent set of a composition

$c = (i_1, i_2, \ldots, i_k)$ is the set

$\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$ .

Map

Clarke-Steingrimsson-Zeng

Description

The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map

$\Phi$ in [1, sec.3]. In particular, it satisfies

$(des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi),$
where

$des$ is the number of descents, St000021The number of descents of a permutation.,
$exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
$Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
$Ebot$ is the sum of the excedance bottoms,
$Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
$Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
$Res$ is the sum of the (right) embracing numbers,
$Ine$ is the sum of the side numbers.