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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 FORTRAN90 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 = "../../f_src/combo/combo.html">
COMBO</a>,
a FORTRAN90 library which
includes many combinatorial routines.
</p>
<p>
<a href = "../../f_src/select/select.html">
SELECT</a>,
a FORTRAN90 library which
generates various combinatorial objects.
</p>
<p>
<a href = "../../f_src/subset/subset.html">
SUBSET</a>,
a FORTRAN90 library which
generates, ranks and unranks various combinatorial objects.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "unicycle.f90">unicycle.f90</a>, the source code.
</li>
<li>
<a href = "unicycle.sh">unicycle.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "unicycle_prb.f90">unicycle_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "unicycle_prb.sh">unicycle_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "unicycle_prb_output.txt">unicycle_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>I4_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_UNIFORM</b> returns a scaled pseudorandom I4.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>I4VEC_INDICATOR</b> sets an I4VEC to the indicator vector.
</li>
<li>
<b>I4VEC_REVERSE</b> reverses the elements of an I4VEC.
</li>
<li>
<b>PERM_CHECK</b> checks a representation of a permutation.
</li>
<li>
<b>PERM_ENUM</b> enumerates the permutations on N digits.
</li>
<li>
<b>PERM_INVERSE</b> computes the inverse of a permutation.
</li>
<li>
<b>PERM_IS_UNICYCLE</b> is TRUE if a permutation is a unicycle.
</li>
<li>
<b>PERM_LEX_NEXT</b> computes the lexicographic permutation successor.
</li>
<li>
<b>PERM_LEX_RANK</b> computes the lexicographic rank of a permutation.
</li>
<li>
<b>PERM_LEX_UNRANK</b> computes the permutation of given lexicographic rank.
</li>
<li>
<b>PERM_PRINT</b> prints a permutation.
</li>
<li>
<b>PERM_RANDOM</b> selects a random permutation of N objects.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>UNICYCLE_CHECK</b> checks that a vector represents a unicycle.
</li>
<li>
<b>UNICYCLE_ENUM</b> enumerates the unicycles.
</li>
<li>
<b>UNICYCLE_INDEX</b> returns the index form of a unicycle.
</li>
<li>
<b>UNICYCLE_INDEX_PRINT</b> prints a unicycle given in index form.
</li>
<li>
<b>UNICYCLE_INDEX_TO_SEQUENCE</b> converts a unicycle from index o sequence form.
</li>
<li>
<b>UNICYCLE_INVERSE</b> returns the inverse of a unicycle.
</li>
<li>
<b>UNICYCLE_NEXT</b> generates unicycles in lexical order, one at a time.
</li>
<li>
<b>UNICYCLE_PRINT</b> prints a unicycle given in sequence form.
</li>
<li>
<b>UNICYCLE_RANDOM</b> selects a random unicycle of N objects.
</li>
<li>
<b>UNICYCLE_RANK</b> computes the rank of a unicycle.
</li>
<li>
<b>UNICYCLE_UNRANK</b> "unranks" a unicycle.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 14 June 2012.
</i>
<!-- John Burkardt -->
</body>
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</html>