Identifier
Values
[(1,2)] => [2,1] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 1
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => 1
[(1,4),(2,3)] => [3,4,2,1] => [4,1,3,2] => 0
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,2,6,5] => 1
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,1,3,2,6,5] => 1
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [5,1,3,2,6,4] => 1
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,1,3,2,5,4] => 0
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,1,4,2,5,3] => 0
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [5,1,4,2,6,3] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => 1
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,3,5,4] => 1
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,1,6,2,5,4] => 1
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [4,1,6,2,5,3] => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,2,5,1,4,3] => 0
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [5,2,6,1,4,3] => 1
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Description
The rix statistic of a permutation.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
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.