GAP (PatternClass) - Chapter 7: Some Permutation Essentials

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7 Some Permutation Essentials
7.1 Complement
7.1-1 PermComplement
7.2 Rank Encoding
7.2-1 IsRankEncoding

### 7 Some Permutation Essentials

In this chapter we mention a couple functions that are fairly basic but useful tools to work with.

#### 7.1 Complement

##### 7.1-1 PermComplement

`‣ PermComplement`( perm ) | ( function )

Returns: The permutation that is the complement of `perm`.

The complement of a permutation \(\tau=\tau_{1}\ldots\tau_{n}\) is the permutation

\[\tau^{C}=(n+1)-\tau_{1}\ \ (n+1)-\tau_{2}\ldots (n+1)-\tau_{n}\]

.

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

#### 7.2 Rank Encoding

##### 7.2-1 IsRankEncoding

`‣ IsRankEncoding`( perm ) | ( function )

Returns: `true` if `perm` is a valid rank encoding of a permutation.

`IsRankEncoding` checkes whether the input list `perm` is a valid rank encoding by checking whether it is accepted by the bounded class automaton, with the highest rank being set by the highest element in `perm`.

    gap> IsRankEncoding([3,2,6,4,4,1,2,1]);
    true
    gap> IsRankEncoding([3,2,6,4,5,1,2,1]);
    false
    gap>

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