- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001551: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of restricted non-inversions between exceedances where the rightmost exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \wedge \sigma^{-1}(j) < j \}.$$

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

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].

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.