Identifier
Values
[(1,2)] => [2,1] => [2,1] => [1,2] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 0
[(1,4),(2,3)] => [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => 3
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,2,6,5] => [5,6,2,4,1,3] => 1
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [3,2,4,1,6,5] => [5,6,1,4,2,3] => 2
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [3,2,5,1,6,4] => [4,6,1,5,2,3] => 1
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,2,5,4,6,1] => [1,6,4,5,2,3] => 2
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,2,5,3,6,1] => [1,6,3,5,2,4] => 0
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [4,2,5,1,6,3] => [3,6,1,5,2,4] => 0
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [3,6,2,5,1,4] => 0
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => 0
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => [4,6,3,5,1,2] => 1
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => [3,6,4,5,1,2] => 2
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => [2,6,4,5,1,3] => 1
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => [2,6,3,5,1,4] => 0
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [4,3,5,1,6,2] => [2,6,1,5,3,4] => 1
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => [1,6,2,5,3,4] => 1
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => 4
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => [6,8,1,7,2,5,3,4] => 1
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => [2,8,1,7,5,6,3,4] => 2
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => [4,8,5,7,1,6,2,3] => 1
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => [3,8,6,7,1,5,2,4] => 1
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => 5
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Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see St000441The number of successions of a permutation., and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the 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.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.