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Next: <a href="maxima_198.html#Functions-and-Variables-for-Combinatorics" accesskey="n" rel="next">Functions and Variables for Combinatorics</a>, Previous: <a href="maxima_196.html#combinatorics_002dpkg" accesskey="p" rel="previous">combinatorics-pkg</a>, Up: <a href="maxima_196.html#combinatorics_002dpkg" accesskey="u" rel="up">combinatorics-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_368.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Package-combinatorics-1"></a>
<h3 class="section">48.1 Package combinatorics</h3>
<p>The <code>combinatorics</code> package provides several functions to work with
permutations and to permute elements of a list. The permutations of
degree <em>n</em> are all the <em>n</em>! possible orderings of the first
<em>n</em> positive integers, 1, 2, …, <em>n</em>. The functions in this
packages expect a permutation to be represented by a list of those
integers.
</p>
<p>Cycles are represented as a list of two or more integers <em>i_1</em>,
<em>i_2</em>, …, <em>i_m</em>, all different. Such a list represents a permutation
where the integer <em>i_2</em> appears in the <em>i_1</em>th position, the
integer <em>i_3</em> appears in the <em>i_2</em>th position and so on, until
the integer <em>i_1</em>, which appears in the <em>i_m</em>th position.
</p>
<p>For instance, [4, 2, 1, 3] is one of the 24 permutations of degree four,
which can also be represented by the cycle [1, 4, 3]. The functions
where cycles are used to represent permutations also require the
order of the permutation to avoid ambiguity. For instance, the same
cycle [1, 4, 3] could refer to the permutation of order 6: [4, 2, 1, 3, 5,
6]. A product of cycles must be represented by a list of cycles; the
cycles at the end of the list are applied first. For example, [[2,
4], [1, 3, 6, 5]] is equivalent to the permutation [3, 4, 6, 2, 1, 5].
</p>
<p>A cycle can be written in several ways. for instance, [1, 3, 6, 5], [3,
6, 5, 1] and [6, 5, 1, 3] are all equivalent. The canonical form used in
the package is the one that places the lowest index in the first
place. A cycle with only two indices is also called a transposition and
if the two indices are consecutive, it is called an adjacent
transposition.
</p>
<p>To run an interactive tutorial, use the command <code>demo
(combinatorics)</code>. Since this is an additional package, it must be loaded
with the command <code>load("combinatorics")</code>.
</p>
<div class=categorybox>·<p>Categories: <a href="maxima_369.html#Category_003a-Share-packages">Share packages</a>
·<a href="maxima_369.html#Category_003a-Package-combinatorics">Package combinatorics</a>
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Next: <a href="maxima_198.html#Functions-and-Variables-for-Combinatorics" accesskey="n" rel="next">Functions and Variables for Combinatorics</a>, Previous: <a href="maxima_196.html#combinatorics_002dpkg" accesskey="p" rel="previous">combinatorics-pkg</a>, Up: <a href="maxima_196.html#combinatorics_002dpkg" accesskey="u" rel="up">combinatorics-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_368.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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