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<h3 class="section">9.8 Permutations in cyclic form</h3>
<p>A permutation can be represented in both <dfn>linear</dfn> and <dfn>cyclic</dfn>
notations. The functions described in this section convert between the
two forms. The linear notation is an index mapping, and has already
been described above. The cyclic notation expresses a permutation as a
series of circular rearrangements of groups of elements, or
<dfn>cycles</dfn>.
<p>For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced
by 3 and 3 is replaced by 1 in a circular fashion. Cycles of different
sets of elements can be combined independently, for example (1 2 3) (4
5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange
of elements 4 and 5. A cycle of length one represents an element which
is unchanged by the permutation and is referred to as a <dfn>singleton</dfn>.
<p>It can be shown that every permutation can be decomposed into
combinations of cycles. The decomposition is not unique, but can always
be rearranged into a standard <dfn>canonical form</dfn> by a reordering of
elements. The library uses the canonical form defined in Knuth's
<cite>Art of Computer Programming</cite> (Vol 1, 3rd Ed, 1997) Section 1.3.3,
p.178.
<p>The procedure for obtaining the canonical form given by Knuth is,
<ol type=1 start=1>
<li>Write all singleton cycles explicitly
<li>Within each cycle, put the smallest number first
<li>Order the cycles in decreasing order of the first number in the cycle.
</ol>
<p class="noindent">For example, the linear representation (2 4 3 0 1) is represented as (1
4) (0 2 3) in canonical form. The permutation corresponds to an
exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
<p>The important property of the canonical form is that it can be
reconstructed from the contents of each cycle without the brackets. In
addition, by removing the brackets it can be considered as a linear
representation of a different permutation. In the example given above
the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping has
many applications in the theory of permutations.
<div class="defun">
— Function: int <b>gsl_permutation_linear_to_canonical</b> (<var>gsl_permutation * q, const gsl_permutation * p</var>)<var><a name="index-gsl_005fpermutation_005flinear_005fto_005fcanonical-1024"></a></var><br>
<blockquote><p>This function computes the canonical form of the permutation <var>p</var> and
stores it in the output argument <var>q</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_permutation_canonical_to_linear</b> (<var>gsl_permutation * p, const gsl_permutation * q</var>)<var><a name="index-gsl_005fpermutation_005fcanonical_005fto_005flinear-1025"></a></var><br>
<blockquote><p>This function converts a permutation <var>q</var> in canonical form back into
linear form storing it in the output argument <var>p</var>.
</p></blockquote></div>
<div class="defun">
— Function: size_t <b>gsl_permutation_inversions</b> (<var>const gsl_permutation * p</var>)<var><a name="index-gsl_005fpermutation_005finversions-1026"></a></var><br>
<blockquote><p>This function counts the number of inversions in the permutation
<var>p</var>. An inversion is any pair of elements that are not in order.
For example, the permutation 2031 has three inversions, corresponding to
the pairs (2,0) (2,1) and (3,1). The identity permutation has no
inversions.
</p></blockquote></div>
<div class="defun">
— Function: size_t <b>gsl_permutation_linear_cycles</b> (<var>const gsl_permutation * p</var>)<var><a name="index-gsl_005fpermutation_005flinear_005fcycles-1027"></a></var><br>
<blockquote><p>This function counts the number of cycles in the permutation <var>p</var>, given in linear form.
</p></blockquote></div>
<div class="defun">
— Function: size_t <b>gsl_permutation_canonical_cycles</b> (<var>const gsl_permutation * q</var>)<var><a name="index-gsl_005fpermutation_005fcanonical_005fcycles-1028"></a></var><br>
<blockquote><p>This function counts the number of cycles in the permutation <var>q</var>, given in canonical form.
</p></blockquote></div>
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