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<html>
<head>
<title>
UNICYCLE - Permutations with a Single Cycle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
UNICYCLE <br> Permutations with a Single Cycle
</h1>
<hr>
<p>
<b>UNICYCLE</b>
is a MATLAB library which
carries out some operations on permutations with a single cycle.
</p>
<p>
A permutation with a single cycle is a permutation P of N objects
with the property that, for every object, it takes exactly N applications
of P to restore an object to its original value.
</p>
<p>
Another way to think of this is that a permutation with a single cycle
can be symbolized by a bracelet with N beads; the action of the permutation
is to rotate the bracelet one position.
</p>
<p>
A permutation with a single cycle can be written in "sequence" form.
Assuming the objects are labeled 1 through N, we start with 1, followed
by P(1), followed by P(P(1)), and so on. Thus, the sequence (1,4,2,5,3)
indicates the permutation which maps 1->4, 2->5, 3->1, 4->2, and 5->3.
</p>
<p>
A permutation with a single cycle is sometimes called a "cyclic permutation",
but this term is also used with other meanings. Hence, we will affectionately
call these objects "unicycles".
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this
web page are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>UNICYCLE</b> is available in
<a href = "../../c_src/unicycle/unicycle.html">a C version</a> and
<a href = "../../cpp_src/unicycle/unicycle.html">a C++ version</a> and
<a href = "../../f77_src/unicycle/unicycle.html">a FORTRAN77 version</a> and
<a href = "../../f_src/unicycle/unicycle.html">a FORTRAN90 version</a> and
<a href = "../../m_src/unicycle/unicycle.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/combo/combo.html">
COMBO</a>,
a MATLAB library which
includes many combinatorial routines.
</p>
<p>
<a href = "../../m_src/subset/subset.html">
SUBSET</a>,
a MATLAB library which
generates, ranks and unranks various combinatorial objects.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "i4_factorial.m">i4_factorial.m</a>,
computes the factorial of N.
</li>
<li>
<a href = "i4_modp.m">i4_modp.m</a>,
returns the nonnegative remainder of I4 division.
</li>
<li>
<a href = "i4_uniform.m">i4_uniform.m</a>,
returns a scaled pseudorandom I4.
</li>
<li>
<a href = "i4_wrap.m">i4_wrap.m</a>,
forces an I4 to lie between given limits by wrapping.
</li>
<li>
<a href = "i4vec_indicator.m">i4vec_indicator.m</a>,
sets an I4VEC to the indicator vector.
</li>
<li>
<a href = "i4vec_reverse.m">i4vec_reverse.m</a>,
reverses the elements of an I4VEC.
</li>
<li>
<a href = "perm_check.m">perm_check.m</a>,
checks a representation of a permutation.
</li>
<li>
<a href = "perm_enum.m">perm_enum.m</a>,
enumerates the permutations on N digits.
</li>
<li>
<a href = "perm_inverse.m">perm_inverse.m</a>,
computes the inverse of a permutation.
</li>
<li>
<a href = "perm_is_unicycle.m">perm_is_unicycle.m</a>,
is TRUE if a permutation is a unicycle.
</li>
<li>
<a href = "perm_lex_next.m">perm_lex_next.m</a>,
computes the lexicographic permutation successor.
</li>
<li>
<a href = "perm_lex_rank.m">perm_lex_rank.m</a>,
computes the lexicographic rank of a permutation.
</li>
<li>
<a href = "perm_lex_unrank.m">perm_lex_unrank.m</a>,
computes the permutation of given lexicographic rank.
</li>
<li>
<a href = "perm_print.m">perm_print.m</a>,
prints a permutation.
</li>
<li>
<a href = "perm_random.m">perm_random.m</a>,
selects a random permutation of N objects.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a time stamp.
</li>
<li>
<a href = "unicycle_check.m">unicycle_check.m</a>,
checks that a vector represents a unicycle.
</li>
<li>
<a href = "unicycle_enum.m">unicycle_enum.m</a>,
enumerates the unicycles.
</li>
<li>
<a href = "unicycle_index.m">unicycle_index.m</a>,
returns the index form of a unicycle.
</li>
<li>
<a href = "unicycle_index_print.m">unicycle_index_print.m</a>,
prints a unicycle given in index form.
</li>
<li>
<a href = "unicycle_index_to_sequence.m">unicycle_index_to_sequence.m</a>,
converts a unicycle from index o sequence form.
</li>
<li>
<a href = "unicycle_inverse.m">unicycle_inverse.m</a>,
returns the inverse of a unicycle.
</li>
<li>
<a href = "unicycle_next.m">unicycle_next.m</a>,
generates unicycles in lexical order, one at a time.
</li>
<li>
<a href = "unicycle_print.m">unicycle_print.m</a>,
prints a unicycle given in sequence form.
</li>
<li>
<a href = "unicycle_random.m">unicycle_random.m</a>,
selects a random unicycle of N objects.
</li>
<li>
<a href = "unicycle_rank.m">unicycle_rank.m</a>,
computes the rank of a unicycle.
</li>
<li>
<a href = "unicycle_unrank.m">unicycle_unrank.m</a>,
"unranks" a unicycle.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "unicycle_test.m">unicycle_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "unicycle_test_output.txt">unicycle_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "unicycle_test01.m">unicycle_test01.m</a>,
tests PERM_IS_UNICYCLE.
</li>
<li>
<a href = "unicycle_test02.m">unicycle_test02.m</a>,
tests UNICYCLE_ENUM.
</li>
<li>
<a href = "unicycle_test03.m">unicycle_test03.m</a>,
tests UNICYCLE_INVERSE;
</li>
<li>
<a href = "unicycle_test04.m">unicycle_test04.m</a>,
tests UNICYCLE_NEXT.
</li>
<li>
<a href = "unicycle_test05.m">unicycle_test05.m</a>,
tests UNICYCLE_RANDOM;
</li>
<li>
<a href = "unicycle_test06.m">unicycle_test06.m</a>,
tests UNICYCLE_RANK.
</li>
<li>
<a href = "unicycle_test07.m">unicycle_test07.m</a>,
tests UNICYCLE_UNRANK.
</li>
<li>
<a href = "unicycle_test08.m">unicycle_test08.m</a>,
tests UNICYCLE_INDEX, UNICYCLE_INDEX_TO_SEQUENCE.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 17 June 2012.
</i>
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