Identifier
-
Mp00159:
Permutations
—Demazure product with inverse⟶
Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed 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] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[2,4,1,5,3] => [3,5,1,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,5,1,3,4] => [3,5,1,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,5,1,4,3] => [3,5,1,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
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Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
An index is a right descent if it is a left descent of the inverse signed permutation.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
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 $\sigma$.
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
to signed permutation
Description
The signed permutation with all signs positive.
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