Identifier
Values
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [2,3,1] => [3,1,2] => 0
[1,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [3,2,1] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 0
[1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 0
[1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 1
[1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
[2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 0
[2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 0
[2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [3,4,2,1] => [4,1,3,2] => 1
[3,1,4,2] => [3,1,2,4] => [3,2,1,4] => 1
[3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1
[3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 2
[3,4,2,1] => [1,4,2,3] => [1,4,3,2] => 2
[4,1,2,3] => [3,4,1,2] => [3,1,4,2] => 1
[4,1,3,2] => [3,1,4,2] => [4,2,1,3] => 1
[4,2,1,3] => [4,3,1,2] => [4,2,3,1] => 1
[4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 1
[4,3,1,2] => [4,1,3,2] => [3,4,2,1] => 1
[4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,2,4,3,5] => [2,3,5,4,1] => [4,5,1,2,3] => 0
[1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,2,5,3,4] => [2,3,5,1,4] => [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,3,1,5,4] => [3,1,2,5,4] => 1
[1,3,2,4,5] => [2,4,3,5,1] => [3,5,1,2,4] => 0
[1,3,2,5,4] => [2,4,3,1,5] => [3,4,1,2,5] => 0
[1,3,4,2,5] => [2,5,3,4,1] => [3,4,5,1,2] => 0
[1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,3,5,2,4] => [2,5,3,1,4] => [3,5,4,1,2] => 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,4,2,3,5] => [2,4,5,3,1] => [5,1,2,4,3] => 1
[1,4,2,5,3] => [2,4,1,3,5] => [4,3,1,2,5] => 1
[1,4,3,2,5] => [2,5,4,3,1] => [4,3,5,1,2] => 1
[1,4,3,5,2] => [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,4,5,2,3] => [2,5,1,3,4] => [5,4,3,1,2] => 2
[1,4,5,3,2] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,5,2,3,4] => [2,4,5,1,3] => [4,1,2,5,3] => 1
[1,5,2,4,3] => [2,4,1,5,3] => [5,3,1,2,4] => 1
[1,5,3,2,4] => [2,5,4,1,3] => [5,3,4,1,2] => 1
[1,5,3,4,2] => [2,1,4,5,3] => [2,1,5,3,4] => 1
[1,5,4,2,3] => [2,5,1,4,3] => [4,5,3,1,2] => 1
[1,5,4,3,2] => [2,1,5,4,3] => [2,1,4,5,3] => 1
[2,1,3,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => 0
[2,1,3,5,4] => [3,2,4,1,5] => [2,4,1,3,5] => 0
[2,1,4,3,5] => [3,2,5,4,1] => [2,4,5,1,3] => 0
[2,1,4,5,3] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[2,1,5,3,4] => [3,2,5,1,4] => [2,5,4,1,3] => 1
[2,1,5,4,3] => [3,2,1,5,4] => [2,3,1,5,4] => 1
[2,3,1,4,5] => [4,2,3,5,1] => [2,3,5,1,4] => 0
[2,3,1,5,4] => [4,2,3,1,5] => [2,3,4,1,5] => 0
[2,3,4,1,5] => [5,2,3,4,1] => [2,3,4,5,1] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,4,1] => 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[2,4,1,3,5] => [4,2,5,3,1] => [2,5,1,4,3] => 1
[2,4,1,5,3] => [4,2,1,3,5] => [2,4,3,1,5] => 1
[2,4,3,1,5] => [5,2,4,3,1] => [2,4,3,5,1] => 1
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[2,4,5,1,3] => [5,2,1,3,4] => [2,5,4,3,1] => 2
[2,4,5,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[2,5,1,3,4] => [4,2,5,1,3] => [2,4,1,5,3] => 1
[2,5,1,4,3] => [4,2,1,5,3] => [2,5,3,1,4] => 1
[2,5,3,1,4] => [5,2,4,1,3] => [2,5,3,4,1] => 1
[2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[2,5,4,1,3] => [5,2,1,4,3] => [2,4,5,3,1] => 1
[2,5,4,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[3,1,2,4,5] => [3,4,2,5,1] => [5,1,3,2,4] => 1
[3,1,2,5,4] => [3,4,2,1,5] => [4,1,3,2,5] => 1
[3,1,4,2,5] => [3,5,2,4,1] => [4,5,1,3,2] => 1
[3,1,4,5,2] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[3,1,5,2,4] => [3,5,2,1,4] => [5,4,1,3,2] => 2
[3,1,5,4,2] => [3,1,2,5,4] => [3,2,1,5,4] => 2
[3,2,1,4,5] => [4,3,2,5,1] => [3,2,5,1,4] => 1
[3,2,1,5,4] => [4,3,2,1,5] => [3,2,4,1,5] => 1
[3,2,4,1,5] => [5,3,2,4,1] => [3,2,4,5,1] => 1
[3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[3,2,5,1,4] => [5,3,2,1,4] => [3,2,5,4,1] => 2
[3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[3,4,1,2,5] => [4,5,2,3,1] => [5,1,4,3,2] => 2
[3,4,1,5,2] => [4,1,2,3,5] => [4,3,2,1,5] => 2
[3,4,2,1,5] => [5,4,2,3,1] => [4,3,2,5,1] => 2
[3,4,2,5,1] => [1,4,2,3,5] => [1,4,3,2,5] => 2
[3,4,5,1,2] => [5,1,2,3,4] => [5,4,3,2,1] => 3
[3,4,5,2,1] => [1,5,2,3,4] => [1,5,4,3,2] => 3
[3,5,1,2,4] => [4,5,2,1,3] => [4,1,5,3,2] => 2
[3,5,1,4,2] => [4,1,2,5,3] => [5,3,2,1,4] => 2
[3,5,2,1,4] => [5,4,2,1,3] => [5,3,2,4,1] => 2
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [1,4,2,5,3] => [1,5,3,2,4] => 2
[3,5,4,1,2] => [5,1,2,4,3] => [4,5,3,2,1] => 2
[3,5,4,2,1] => [1,5,2,4,3] => [1,4,5,3,2] => 2
[4,1,2,3,5] => [3,4,5,2,1] => [4,2,5,1,3] => 1
[4,1,2,5,3] => [3,4,1,2,5] => [3,1,4,2,5] => 1
[4,1,3,2,5] => [3,5,4,2,1] => [5,1,3,4,2] => 1
[4,1,3,5,2] => [3,1,4,2,5] => [4,2,1,3,5] => 1
[4,1,5,2,3] => [3,5,1,2,4] => [3,1,5,4,2] => 2
[4,1,5,3,2] => [3,1,5,2,4] => [5,4,2,1,3] => 2
[4,2,1,3,5] => [4,3,5,2,1] => [5,1,4,2,3] => 1
[4,2,1,5,3] => [4,3,1,2,5] => [4,2,3,1,5] => 1
[4,2,3,1,5] => [5,3,4,2,1] => [4,2,3,5,1] => 1
[4,2,3,5,1] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[4,2,5,1,3] => [5,3,1,2,4] => [5,4,2,3,1] => 2
[4,2,5,3,1] => [1,3,5,2,4] => [1,5,4,2,3] => 2
[4,3,1,2,5] => [4,5,3,2,1] => [3,5,1,4,2] => 1
[4,3,1,5,2] => [4,1,3,2,5] => [3,4,2,1,5] => 1
[4,3,2,1,5] => [5,4,3,2,1] => [3,4,2,5,1] => 1
[4,3,2,5,1] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[4,3,5,1,2] => [5,1,3,2,4] => [3,5,4,2,1] => 2
[4,3,5,2,1] => [1,5,3,2,4] => [1,3,5,4,2] => 2
[4,5,1,2,3] => [4,5,1,2,3] => [5,3,1,4,2] => 2
[4,5,1,3,2] => [4,1,5,2,3] => [4,2,1,5,3] => 2
[4,5,2,1,3] => [5,4,1,2,3] => [4,2,5,3,1] => 2
[4,5,2,3,1] => [1,4,5,2,3] => [1,4,2,5,3] => 2
[4,5,3,1,2] => [5,1,4,2,3] => [5,3,4,2,1] => 2
[4,5,3,2,1] => [1,5,4,2,3] => [1,5,3,4,2] => 2
[5,1,2,3,4] => [3,4,5,1,2] => [5,2,4,1,3] => 1
[5,1,2,4,3] => [3,4,1,5,2] => [3,1,5,2,4] => 1
[5,1,3,2,4] => [3,5,4,1,2] => [4,1,3,5,2] => 1
[5,1,3,4,2] => [3,1,4,5,2] => [5,2,1,3,4] => 1
[5,1,4,2,3] => [3,5,1,4,2] => [3,1,4,5,2] => 1
[5,1,4,3,2] => [3,1,5,4,2] => [4,5,2,1,3] => 1
[5,2,1,3,4] => [4,3,5,1,2] => [4,1,5,2,3] => 1
[5,2,1,4,3] => [4,3,1,5,2] => [5,2,3,1,4] => 1
[5,2,3,1,4] => [5,3,4,1,2] => [5,2,3,4,1] => 1
[5,2,3,4,1] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[5,2,4,1,3] => [5,3,1,4,2] => [4,5,2,3,1] => 1
[5,2,4,3,1] => [1,3,5,4,2] => [1,4,5,2,3] => 1
[5,3,1,2,4] => [4,5,3,1,2] => [3,4,1,5,2] => 1
[5,3,1,4,2] => [4,1,3,5,2] => [3,5,2,1,4] => 1
[5,3,2,1,4] => [5,4,3,1,2] => [3,5,2,4,1] => 1
[5,3,2,4,1] => [1,4,3,5,2] => [1,3,5,2,4] => 1
[5,3,4,1,2] => [5,1,3,4,2] => [3,4,5,2,1] => 1
[5,3,4,2,1] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[5,4,1,2,3] => [4,5,1,3,2] => [4,3,1,5,2] => 2
[5,4,1,3,2] => [4,1,5,3,2] => [5,2,1,4,3] => 2
[5,4,2,1,3] => [5,4,1,3,2] => [5,2,4,3,1] => 2
[5,4,2,3,1] => [1,4,5,3,2] => [1,5,2,4,3] => 2
[5,4,3,1,2] => [5,1,4,3,2] => [4,3,5,2,1] => 2
[5,4,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 2
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Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
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
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.