- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001550: 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] => [2,3,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

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

Map

ones to leading

Description

The unique permutation obtained by applying the Foata-Riordan map to obtain a Prüfer code, then prepending zero and cyclically shifting.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.

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

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].