- FindStat

Identifier

Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000405: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$

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.

Map

upper permutation

Description

The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $v$.

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 $\sigma$.