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3 Permutation Encoding

A permutation \(\pi=\pi_{1} \ldots \pi_{n}\) has rank encoding \(p_{1} \ldots p_{n}\) where \( p_{i}= |\{j : j \geq i, \pi_{j} \leq \pi_{i} \} | \). In other words the rank encoded permutation is a sequence of \(p_{i}\) with \(1\leq i\leq n\), where \(p_{i}\) is the rank of \(\pi_{i}\) in \(\{\pi_{i},\pi_{i+1},\ldots ,\pi_{n}\}\). [AAR03]

The encoding of the permutation 3 2 5 1 6 7 4 8 9 is done as follows:

Decoding a permutation is done in a similar fashion, taking the sequence \(p_{1} \ldots p_{n}\) and using the reverse process will lead to the permutation \(\pi=\pi_{1} \ldots \pi_{n}\), where \(\pi_{i}\) is determined by finding the number that has rank \(p_{i}\) in \(\{\pi_{i}, \pi_{i+1}, \ldots , \pi_{n}\}\).

The sequence 3 2 3 1 2 2 1 1 1 is decoded as:

3.1 Encoding and Decoding

3.1-1 RankEncoding

Returns: A list that represents the rank encoding of the permutation p.

Using the algorithm above RankEncoding turns the permutation p into a list of integers.

gap> RankEncoding([3, 2, 5, 1, 6, 7, 4, 8, 9]);
[ 3, 2, 3, 1, 2, 2, 1, 1, 1 ]
gap> RankEncoding([ 4, 2, 3, 5, 1 ]);                 
[ 4, 2, 2, 2, 1 ]
gap> 

3.1-2 RankDecoding

Returns: A permutation in list form.

A rank encoded permutation is decoded by using the reversed process from encoding, which is also explained above.

gap> RankDecoding([ 3, 2, 3, 1, 2, 2, 1, 1, 1 ]);
[ 3, 2, 5, 1, 6, 7, 4, 8, 9 ]
gap> RankDecoding([ 4, 2, 2, 2, 1 ]);
[ 4, 2, 3, 5, 1 ]
gap> 

3.1-3 SequencesToRatExp

Returns: A rational expression that describes all the words in list.

A list of sequences is turned into a rational expression by concatenating each sequence and unifying all of them.

gap> SequencesToRatExp([[ 1, 1, 1, 1, 1 ],[ 2, 1, 2, 2, 1 ],[ 3, 2, 1, 2, 1 ],
> [ 4, 2, 3, 2, 1 ]]);
11111U21221U32121U42321
gap> 

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