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<html>
<head>
<title>
COMBO - Kreher and Stinson Combinatorial Routines
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
COMBO <br> Kreher and Stinson Combinatorial Routines
</h1>
<hr>
<p>
<b>COMBO</b>
is a MATLAB library which
includes routines for ranking, unranking, enumerating and
randomly selecting balanced sequences, cycles, graphs, Gray codes,
subsets, partitions, permutations, restricted growth functions,
Pruefer codes and trees.
</p>
<p>
Routines are available to count, list, rank and unrank such objects
<ul>
<li>
<b>BAL</b>, balanced sequences;
</li>
<li>
<b>CYCLE</b>, permutations of the first N integers in cycle form;
</li>
<li>
<b>GRAPH</b>, graphs stored as a list of edges.
</li>
<li>
<b>GRAY</b>, Gray codes;
</li>
<li>
<b>KNAPSACK</b>, optimally filling a knapsack of given size using
a set of smaller objects;
</li>
<li>
<b>KSUBSET</b>, subsets of size exactly K from a set of N objects;
</li>
<li>
<b>NPART</b>, partitions of an integer having exactly N parts;
</li>
<li>
<b>PART</b>, partitions of an integer;
</li>
<li>
<b>PERM</b>, permutations of the first N integers in standard form;
</li>
<li>
<b>PRUEFER</b>, Pruefer codes;
</li>
<li>
<b>RGF</b>, restricted growth functions;
</li>
<li>
<b>SETPART</b>, partitions of a set;
</li>
<li>
<b>SUBSET</b>, subsets of a set of N objects;
</li>
<li>
<b>TABLEAU</b>, tableaus;
</li>
<li>
<b>TREE</b>, trees;
</li>
</ul>
</p>
<p>
Some of these sets of objects can be ordered in several different
ways, and in some cases, a separate set of ranking, unranking, and
successor routines are available for the various orderings
(lexical, colexical, revolving door, Trotter-Johnson).
</p>
<p>
Kreher and Stinson provide C source-code for the routines,
as well as other information, at
<a href = "http://www.math.mtu.edu/~kreher/cages.html">
their web site.</a>
</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>COMBO</b> is available in
<a href = "../../c_src/combo/combo.html">a C version</a> and
<a href = "../../cpp_src/combo/combo.html">a C++ version</a> and
<a href = "../../f77_src/combo/combo.html">a FORTRAN77 version</a> and
<a href = "../../f_src/combo/combo.html">a FORTRAN90 version</a> and
<a href = "../../m_src/combo/combo.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/change_making/change_making.html">
CHANGE_MAKING</a>,
a MATLAB library which
considers the change making problem,
in which a given sum is to be formed using coins of various denominations.
</p>
<p>
<a href = "../../m_src/combination_lock/combination_lock.html">
COMBINATION_LOCK</a>,
a MATLAB program which
simulates the process of determining the secret combination of a lock.
</p>
<p>
<a href = "../../m_src/floyd/floyd.html">
FLOYD</a>,
a MATLAB library which
implements Floyd's algorithm for finding the shortest distance between
pairs of nodes on a directed graph.
</p>
<p>
<a href = "../../m_src/gray_code_display/gray_code_display.html">
GRAY_CODE_DISPLAY</a>,
a MATLAB program which
computes the Hamming distance tables for both the binary and Gray codes,
and displays 3D plots that illustrate how the Gray code does a better job
of providing nearby representations for nearby numbers.
</p>
<p>
<a href = "../../m_src/monomial/monomial.html">
MONOMIAL</a>,
a MATLAB library which
enumerates, lists, ranks, unranks and randomizes multivariate monomials
in a space of M dimensions, with total degree less than N,
equal to N, or lying within a given range.
</p>
<p>
<a href = "../../m_src/partition_problem/partition_problem.html">
PARTITION_PROBLEM</a>,
a MATLAB library which
seeks solutions of the partition problem, splitting a set of integers into
two subsets with equal sum.
</p>
<p>
<a href = "../../m_src/polynomial/polynomial.html">
POLYNOMIAL</a>,
a MATLAB library which
adds, multiplies, differentiates, evaluates and prints multivariate
polynomials in a space of M dimensions.
</p>
<p>
<a href = "../../m_src/set_theory/set_theory.html">
SET_THEORY</a>,
a MATLAB library which
demonstrates MATLAB commands that implement various set theoretic operations.
</p>
<p>
<a href = "../../m_src/subset/subset.html">
SUBSET</a>,
a MATLAB library which
generates, ranks and unranks various combinatorial objects.
</p>
<p>
<a href = "../../m_src/subset_sum/subset_sum.html">
SUBSET_SUM</a>,
a MATLAB library which
seeks solutions of the subset sum problem.
</p>
<p>
<a href = "../../m_src/unicycle/unicycle.html">
UNICYCLE</a>,
a MATLAB library which
considers permutations containing a single cycle, sometimes called
cyclic permutations.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673,<br>
LC: QA76.9.C65.B73.
</li>
<li>
William Cody, Kenneth Hillstrom,<br>
Chebyshev Approximations for the Natural Logarithm of the
Gamma Function,
Mathematics of Computation,<br>
Volume 21, Number 98, April 1967, pages 198-203.
</li>
<li>
Robert Fenichel,<br>
Algorithm 329:
Distribution of Indistinguishable Objects into
Distinguishable Slots,<br>
Communications of the ACM,<br>
Volume 11, Number 6, June 1968, page 430.
</li>
<li>
Bennett Fox,<br>
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,<br>
ACM Transactions on Mathematical Software,<br>
Volume 12, Number 4, December 1986, pages 362-376.
</li>
<li>
John Hart, Ward Cheney, Charles Lawson, Hans Maehly,
Charles Mesztenyi, John Rice, Henry Thacher,
Christoph Witzgall,<br>
Computer Approximations,<br>
Wiley, 1968,<br>
LC: QA297.C64.
</li>
<li>
Brian Hayes,<br>
The Easiest Hard Problem,<br>
American Scientist,<br>
Volume 90, Number 2, March-April 2002, pages 113-117.
</li>
<li>
Donald Kreher, Douglas Simpson,<br>
Combinatorial Algorithms,<br>
CRC Press, 1998,<br>
ISBN: 0-8493-3988-X,<br>
LC: QA164.K73.
</li>
<li>
Pierre LEcuyer,<br>
Random Number Generation,<br>
in Handbook of Simulation,<br>
edited by Jerry Banks,<br>
Wiley, 1998,<br>
ISBN: 0471134031,<br>
LC: T57.62.H37.
</li>
<li>
Peter Lewis, Allen Goodman, James Miller,<br>
A Pseudo-Random Number Generator for the System/360,<br>
IBM Systems Journal,<br>
Volume 8, 1969, pages 136-143.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
Robert Sedgewick,<br>
Algorithms in C,<br>
Addison-Wesley, 1990,<br>
ISBN: 0-201-51425-7,<br>
LC: QA76.73.C15S43.
</li>
<li>
Jack vanLint, Richard Wilson,<br>
A Course in Combinatorics,<br>
Cambridge, 1992,<br>
ISBN: 0-521-42260-4,<br>
LC: QA164.L56.
</li>
<li>
ML Wolfson, HV Wright,<br>
Algorithm 160:
Combinatorial of M Things Taken N at a Time,<br>
Communications of the ACM,<br>
Volume 6, Number 4, April 1963, page 161.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "backtrack.m">backtrack.m</a>,
supervises a backtrack search.
</li>
<li>
<a href = "bal_seq_check.m">bal_seq_check.m</a>,
checks a balanced sequence.
</li>
<li>
<a href = "bal_seq_enum.m">bal_seq_enum.m</a>,
enumerates the balanced sequences.
</li>
<li>
<a href = "bal_seq_rank.m">bal_seq_rank.m</a>,
ranks a balanced sequence.
</li>
<li>
<a href = "bal_seq_successor.m">bal_seq_successor.m</a>,
returns the successor of a balanced sequence.
</li>
<li>
<a href = "bal_seq_to_tableau.m">bal_seq_to_tableau.m</a>,
converts a balanced sequence to a tableau.
</li>
<li>
<a href = "bal_seq_unrank.m">bal_seq_unrank.m</a>,
unranks a balanced sequence.
</li>
<li>
<a href = "bell_numbers.m">bell_numbers.m</a>,
computes the Bell numbers.
</li>
<li>
<a href = "bell_values.m">bell_values.m</a>,
returns some values of the Bell numbers.
</li>
<li>
<a href = "combin.m">combin.m</a>,
computes the binomial coefficient C(N,K).
</li>
<li>
<a href = "cycle_check.m">cycle_check.m</a>,
checks a permutation in cycle form.
</li>
<li>
<a href = "cycle_to_perm.m">cycle_to_perm.m</a>,
converts a permutation from cycle to array form.
</li>
<li>
<a href = "dist_enum.m">dist_enum.m</a>,
enumerates the number of distributions of indistinguishable objects.
</li>
<li>
<a href = "dist_next.m">dist_next.m</a>,
returns the next distribution of indistinguishable objects.
</li>
<li>
<a href = "edge_check.m">edge_check.m</a>,
checks a graph described by an edge list.
</li>
<li>
<a href = "edge_degree.m">edge_degree.m</a>,
returns the degree of each node in a graph described by an edge list.
</li>
<li>
<a href = "edge_enum.m">edge_enum.m</a>,
enumerates the maximum number of edges in a graph with a given
number of nodes.
</li>
<li>
<a href = "fall.m">fall.m</a>,
evaluates the falling factorial function.
</li>
<li>
<a href = "gray_code_check.m">gray_code_check.m</a>,
checks a Gray code element.
</li>
<li>
<a href = "gray_code_enum.m">gray_code_enum.m</a>,
enumerates the Gray codes on N digits.
</li>
<li>
<a href = "gray_code_rank.m">gray_code_rank.m</a>,
computes the rank of a Gray code element.
</li>
<li>
<a href = "gray_code_successor.m">gray_code_successor.m</a>,
computes the binary reflected Gray code successor.
</li>
<li>
<a href = "gray_code_unrank.m">gray_code_unrank.m</a>,
computes the Gray code element of given rank.
</li>
<li>
<a href = "i4_choose.m">i4_choose.m</a>,
computes the binomial coefficient C(N,K).
</li>
<li>
<a href = "i4_factorial.m">i4_factorial.m</a>,
computes the factorial function.
</li>
<li>
<a href = "i4_factorial_values.m">i4_factorial_values.m</a>,
returns some values of the factorial function.
</li>
<li>
<a href = "i4_huge.m">i4_huge.m</a>,
returns a huge I4;
</li>
<li>
<a href = "i4_uniform.m">i4_uniform.m</a>,
returns a pseudorandom I4 in a given range;
</li>
<li>
<a href = "i4vec_backtrack.m">i4vec_backtrack.m</a>,
supervises a backtrack search for an I4VEC.
</li>
<li>
<a href = "i4vec_indicator.m">i4vec_indicator.m</a>,
sets an I4VEC to the indicator vector.
</li>
<li>
<a href = "i4vec_part1.m">i4vec_part1.m</a>,
partitions an integer N into NPART parts.
</li>
<li>
<a href = "i4vec_part2.m">i4vec_part2.m</a>,
partitions an integer N into NPART nearly equal parts.
</li>
<li>
<a href = "i4vec_print.m">i4vec_print.m</a>,
prints an I4VEC.
</li>
<li>
<a href = "i4vec_reverse.m">i4vec_reverse.m</a>,
reverses the order of the entries of an I4VEC;
</li>
<li>
<a href = "i4vec_search_binary_a.m">i4vec_search_binary_a.m</a>,
searches the ascending sorted I4VEC for a value.
</li>
<li>
<a href = "i4vec_search_binary_d.m">i4vec_search_binary_d.m</a>,
searches the descending sorted I4VEC for a value.
</li>
<li>
<a href = "i4vec_sort_insert_a.m">i4vec_sort_insert_a.m</a>,
uses an ascending insertion sort on an I4VEC.
</li>
<li>
<a href = "i4vec_sort_insert_d.m">i4vec_sort_insert_d.m</a>,
uses a descending insertion sort on an I4VEC.
</li>
<li>
<a href = "knapsack_01.m">knapsack_01.m</a>,
solves the 0,1 knapsack problem.
</li>
<li>
<a href = "knapsack_rational.m">knapsack_rational.m</a>,
solves the rational knapsack problem.
</li>
<li>
<a href = "knapsack_reorder.m">knapsack_reorder.m</a>,
reorders the knapsack data by "profit density".
</li>
<li>
<a href = "ksubset_colex_check.m">ksubset_colex_check.m</a>,
checks a K subset in colex form.
</li>
<li>
<a href = "ksubset_colex_rank.m">ksubset_colex_rank.m</a>,
computes the colex rank of a K subset.
</li>
<li>
<a href = "ksubset_colex_successor.m">ksubset_colex_successor.m</a>,
computes the K subset colex successor.
</li>
<li>
<a href = "ksubset_colex_unrank.m">ksubset_colex_unrank.m</a>,
computes the K subset of given colex rank.
</li>
<li>
<a href = "ksubset_enum.m">ksubset_enum.m</a>,
enumerates the K element subsets of an N set.
</li>
<li>
<a href = "ksubset_lex_check.m">ksubset_lex_check.m</a>,
checks a K subset in lex form.
</li>
<li>
<a href = "ksubset_lex_rank.m">ksubset_lex_rank.m</a>,
computes the lexicographic rank of a K subset.
</li>
<li>
<a href = "ksubset_lex_successor.m">ksubset_lex_successor.m</a>,
computes the K subset lexicographic successor.
</li>
<li>
<a href = "ksubset_lex_unrank.m">ksubset_lex_unrank.m</a>,
computes the K subset of given lexicographic rank.
</li>
<li>
<a href = "ksubset_revdoor_rank.m">ksubset_revdoor_rank.m</a>,
computes the revolving door rank of a K subset.
</li>
<li>
<a href = "ksubset_revdoor_successor.m">ksubset_revdoor_successor.m</a>,
computes the K subset revolving door successor.
</li>
<li>
<a href = "ksubset_revdoor_unrank.m">ksubset_revdoor_unrank.m</a>,
computes the K subset of given revolving door rank.
</li>
<li>
<a href = "marriage.m">marriage.m</a>,
finds a stable set of marriages for given preferences.
</li>
<li>
<a href = "mountain.m">mountain.m</a>,
enumerates the mountains.
</li>
<li>
<a href = "npart_enum.m">npart_enum.m</a>,
enumerates the number of partitions of N with NPART parts.
</li>
<li>
<a href = "npart_rsf_lex_random.m">npart_rsf_lex_random.m</a>,
returns a random RSF NPART partition.
</li>
<li>
<a href = "npart_rsf_lex_rank.m">npart_rsf_lex_rank.m</a>,
computes the lex rank of an RSF NPART partition.
</li>
<li>
<a href = "npart_rsf_lex_successor.m">npart_rsf_lex_successor.m</a>,
computes the RSF lex successor for NPART partitions.
</li>
<li>
<a href = "npart_rsf_lex_unrank.m">npart_rsf_lex_unrank.m</a>,
unranks an RSF NPART partition in the lex ordering.
</li>
<li>
<a href = "npart_sf_lex_successor.m">npart_sf_lex_successor.m</a>,
computes SF NPART partition.
</li>
<li>
<a href = "npart_table.m">npart_table.m</a>,
tabulates the number of partitions of N having NPART parts.
</li>
<li>
<a href = "part_enum.m">part_enum.m</a>,
enumerates the partitions of an integer.
</li>
<li>
<a href = "part_rsf_check.m">part_rsf_check.m</a>,
checks a reverse standard form partition of an integer.
</li>
<li>
<a href = "part_sf_check.m">part_sf_check.m</a>,
checks a standard form partition of an integer.
</li>
<li>
<a href = "part_sf_conjugate.m">part_sf_conjugate.m</a>,
computes the conjugate of a partition.
</li>
<li>
<a href = "part_sf_majorize.m">part_sf_majorize.m</a>,
determines if partition A majorizes partition B.
</li>
<li>
<a href = "part_successor.m">part_successor.m</a>,
computes the lexicographic partition successor.
</li>
<li>
<a href = "part_table.m">part_table.m</a>,
tabulates the number of partitions of N.
</li>
<li>
<a href = "partition_greedy.m">partition_greedy.m</a>,
attacks the partition problem with a greedy algorithm.
</li>
<li>
<a href = "partn_enum.m">partn_enum.m</a>,
enumerates the partitions of N with maximum element NMAX.
</li>
<li>
<a href = "partn_sf_check.m">partn_sf_check.m</a>,
checks an SF partition of an integer with largest entry NMAX.
</li>
<li>
<a href = "partn_successor.m">partn_successor.m</a>,
computes partitions whose largest part is NMAX.
</li>
<li>
<a href = "perm_check.m">perm_check.m</a>,
checks that a vector represents a permutation.
</li>
<li>
<a href = "perm_enum.m">perm_enum.m</a>,
enumerates the permutations on N digits.
</li>
<li>
<a href = "perm_inv.m">perm_inv.m</a>,
computes the inverse of a permutation.
</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_successor.m">perm_lex_successor.m</a>,
computes the lexicographic permutation successor.
</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_mul.m">perm_mul.m</a>,
computes the product of two permutations.
</li>
<li>
<a href = "perm_parity.m">perm_parity.m</a>,
computes the parity of a permutation.
</li>
<li>
<a href = "perm_print.m">perm_print.m</a>,
prints a permutation.
</li>
<li>
<a href = "perm_tj_rank.m">perm_tj_rank.m</a>,
computes the Trotter-Johnson rank of a permutation.
</li>
<li>
<a href = "perm_tj_successor.m">perm_tj_successor.m</a>,
computes the Trotter-Johnson permutation successor.
</li>
<li>
<a href = "perm_tj_unrank.m">perm_tj_unrank.m</a>,
computes the permutation of given Trotter-Johnson rank.
</li>
<li>
<a href = "perm_to_cycle.m">perm_to_cycle.m</a>,
converts a permutation from array to cycle form.
</li>
<li>
<a href = "pruefer_check.m">pruefer_check.m</a>,
checks a Pruefer code.
</li>
<li>
<a href = "pruefer_enum.m">pruefer_enum.m</a>,
enumerates the Pruefer codes on N-2 digits.
</li>
<li>
<a href = "pruefer_rank.m">pruefer_rank.m</a>,
ranks a Pruefer code.
</li>
<li>
<a href = "pruefer_successor.m">pruefer_successor.m</a>,
computes the lexical Pruefer sequence successor.
</li>
<li>
<a href = "pruefer_to_tree.m">pruefer_to_tree.m</a>,
converts a Pruefer code to a tree.
</li>
<li>
<a href = "pruefer_unrank.m">pruefer_unrank.m</a>,
unranks a Pruefer code.
</li>
<li>
<a href = "queens.m">queens.m</a>,
finds possible positions for the K-th nonattacking queen.
</li>
<li>
<a href = "r8_gamma_log.m">r8_gamma_log.m</a>,
returns the logarithm of the gamma function.
</li>
<li>
<a href = "r8vec_backtrack.m">r8vec_backtrack.m</a>,
supervises a backtrack search for an R8VEC.
</li>
<li>
<a href = "rgf_check.m">rgf_check.m</a>,
checks a restricted growth function.
</li>
<li>
<a href = "rgf_enum.m">rgf_enum.m</a>,
enumerates the restricted growth functions on M.
</li>
<li>
<a href = "rgf_g_enum.m">rgf_g_enum.m</a>,
enumerates the generalized restricted growth functions.
</li>
<li>
<a href = "rgf_rank.m">rgf_rank.m</a>,
ranks a restricted growth function.
</li>
<li>
<a href = "rgf_successor.m">rgf_successor.m</a>,
generates the next restricted growth function.
</li>
<li>
<a href = "rgf_to_setpart.m">rgf_to_setpart.m</a>,
converts a restricted growth function to a set partition.
</li>
<li>
<a href = "rgf_unrank.m">rgf_unrank.m</a>,
returns the restricted growth function of a given rank.
</li>
<li>
<a href = "setpart_check.m">setpart_check.m</a>,
checks a set partition.
</li>
<li>
<a href = "setpart_enum.m">setpart_enum.m</a>,
enumerates the partitions of a set of M elements.
</li>
<li>
<a href = "setpart_to_rgf.m">setpart_to_rgf.m</a>,
converts a set partition to a restricted growth function.
</li>
<li>
<a href = "stirling_numbers1.m">stirling_numbers1.m</a>,
computes the Stirling numbers of the first kind.
</li>
<li>
<a href = "stirling_numbers2.m">stirling_numbers2.m</a>,
computes the Stirling numbers of the second kind.
</li>
<li>
<a href = "subset_check.m">subset_check.m</a>,
checks a subset.
</li>
<li>
<a href = "subset_colex_rank.m">subset_colex_rank.m</a>,
computes the colexicographic rank of a subset.
</li>
<li>
<a href = "subset_colex_successor.m">subset_colex_successor.m</a>,
computes the subset colexicographic successor.
</li>
<li>
<a href = "subset_colex_unrank.m">subset_colex_unrank.m</a>,
computes the subset of given colexicographic rank.
</li>
<li>
<a href = "subset_complement.m">subset_complement.m</a>,
computes the complement of a set.
</li>
<li>
<a href = "subset_dist.m">subset_dist.m</a>,
computes the Hamming distance between two sets.
</li>
<li>
<a href = "subset_enum.m">subset_enum.m</a>,
enumerates the subsets of a set with N elements.
</li>
<li>
<a href = "subset_intersect.m">subset_intersect.m</a>,
computes the intersection of two sets.
</li>
<li>
<a href = "subset_lex_rank.m">subset_lex_rank.m</a>,
computes the rank of a given lexicographic subset.
</li>
<li>
<a href = "subset_lex_successor.m">subset_lex_successor.m</a>,
computes the lexicographic successor of a subset.
</li>
<li>
<a href = "subset_lex_unrank.m">subset_lex_unrank.m</a>,
computes the subset of a given lexicographic rank.
</li>
<li>
<a href = "subset_union.m">subset_union.m</a>,
computes the union of two sets.
</li>
<li>
<a href = "subset_weight.m">subset_weight.m</a>,
computes the Hamming weight of a set.
</li>
<li>
<a href = "subset_xor.m">subset_xor.m</a>,
computes the symmetric difference of two sets.
</li>
<li>
<a href = "subsetsum_swap.m">subsetsum_swap.m</a>,
seeks a solution of the subset sum problem by swapping.
</li>
<li>
<a href = "tableau_check.m">tableau_check.m</a>,
checks a 2 by N tableau.
</li>
<li>
<a href = "tableau_enum.m">tableau_enum.m</a>,
enumerates the 2 by N standard tableaus.
</li>
<li>
<a href = "tableau_to_bal_seq.m">tableau_to_bal_seq.m</a>,
converts a 2 by N tableau to a balanced sequence.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a timestamp.
</li>
<li>
<a href = "tree_check.m">tree_check.m</a>,
checks a tree.
</li>
<li>
<a href = "tree_enum.m">tree_enum.m</a>,
enumerates the trees on N nodes.
</li>
<li>
<a href = "tree_rank.m">tree_rank.m</a>,
ranks a tree.
</li>
<li>
<a href = "tree_successor.m">tree_successor.m</a>,
returns the successor of a tree.
</li>
<li>
<a href = "tree_to_pruefer.m">tree_to_pruefer.m</a>,
converts a tree to a Pruefer code.
</li>
<li>
<a href = "tree_unrank.m">tree_unrank.m</a>,
unranks a tree.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "combo_test.m">combo_test.m</a>, the calling program;
</li>
<li>
<a href = "combo_test_output.txt">combo_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "combo_test01.m">combo_test01.m</a>
tests BAL_SEQ_ENUM, BAL_SEQ_RANK, BAL_SEQ_SUCCESSOR, BAL_SEQ_UNRANK.
</li>
<li>
<a href = "combo_test02.m">combo_test02.m</a>
tests BAL_SEQ_TO_TABLEAU and TABLEAU_TO_BAL_SEQ.
</li>
<li>
<a href = "combo_test03.m">combo_test03.m</a>
tests BELL_NUMBERS.
</li>
<li>
<a href = "combo_test04.m">combo_test04.m</a>
tests I4_CHOOSE.
</li>
<li>
<a href = "combo_test05.m">combo_test05.m</a>
tests CYCLE_TO_PERM, PERM_TO_CYCLE.
</li>
<li>
<a href = "combo_test06.m">combo_test06.m</a>
tests DIST_ENUM, DIST_NEXT.
</li>
<li>
<a href = "combo_test07.m">combo_test07.m</a>
tests I4_FACTORIAL and I4_FACTORIAL_VALUES.
</li>
<li>
<a href = "combo_test08.m">combo_test08.m</a>
tests GRAY_CODE_*.
</li>
<li>
<a href = "combo_test09.m">combo_test09.m</a>
tests I4VEC_SEARCH_BINARY_A and I4VEC_SORT_INSERT_A.
</li>
<li>
<a href = "combo_test10.m">combo_test10.m</a>
tests I4VEC_SEARCH_BINARY_D and I4VEC_SORT_INSERT_D.
</li>
<li>
<a href = "combo_test11.m">combo_test11.m</a>
tests KNAPSACK_REORDER and KNAPSACK_01.
</li>
<li>
<a href = "combo_test12.m">combo_test12.m</a>
tests KNAPSACK_REORDER and KNAPSACK_RATIONAL.
</li>
<li>
<a href = "combo_test13.m">combo_test13.m</a>
tests KSUBSET_COLEX_RANK, _SUCCESSOR, _UNRANK, _ENUM.
</li>
<li>
<a href = "combo_test14.m">combo_test14.m</a>
tests KSUBSET_ENUM, _LEX_RANK, _LEX_SUCCESSOR, _LEX_UNRANK.
</li>
<li>
<a href = "combo_test15.m">combo_test15.m</a>
tests KSUBSET_*.
</li>
<li>
<a href = "combo_test16.m">combo_test16.m</a>
tests MARRIAGE.
</li>
<li>
<a href = "combo_test17.m">combo_test17.m</a>
tests MOUNTAIN.
</li>
<li>
<a href = "combo_test18.m">combo_test18.m</a>
tests NPART_ENUM, _RSF_LEX_RANK, _RSF_LEX_SUCCESSOR, _RSF_LEX_UNRANK.
</li>
<li>
<a href = "combo_test19.m">combo_test19.m</a>
tests NPART_RSF_LEX_RANDOM;
</li>
<li>
<a href = "combo_test20.m">combo_test20.m</a>
tests NPART_ENUM and NPART_SF_SUCCESSOR;
</li>
<li>
<a href = "combo_test21.m">combo_test21.m</a>
tests NPART_TABLE and PART_TABLE.
</li>
<li>
<a href = "combo_test22.m">combo_test22.m</a>
tests PART_SUCCESSOR and PART_ENUM.
</li>
<li>
<a href = "combo_test23.m">combo_test23.m</a>
tests PART_SUCCESSOR and PART_SF_CONJUGATE.
</li>
<li>
<a href = "combo_test24.m">combo_test24.m</a>
tests PART_SF_MAJORIZE.
</li>
<li>
<a href = "combo_test25.m">combo_test25.m</a>
tests PARTITION_GREEDY.
</li>
<li>
<a href = "combo_test26.m">combo_test26.m</a>
tests PARTN_ENUM, PARTN_SUCCESSOR and PART_SF_CONJUGATE.
</li>
<li>
<a href = "combo_test27.m">combo_test27.m</a>
tests PERM_INV and PERM_MUL.
</li>
<li>
<a href = "combo_test28.m">combo_test28.m</a>
tests PERM_ENUM, PERM_LEX_RANK, PERM_LEX_SUCCESSOR, PERM_LEX_UNRANK.
</li>
<li>
<a href = "combo_test29.m">combo_test29.m</a>
tests PERM_ENUM, PERM_TJ_RANK, PERM_TJ_SUCCESSOR, PERM_TJ_UNRANK.
</li>
<li>
<a href = "combo_test30.m">combo_test30.m</a>
tests PRUEFER_*.
</li>
<li>
<a href = "combo_test31.m">combo_test31.m</a>
tests PRUEFER_TO_TREE and TREE_TO_PRUEFER.
</li>
<li>
<a href = "combo_test32.m">combo_test32.m</a>
tests QUEENS and BACKTRACK.
</li>
<li>
<a href = "combo_test33.m">combo_test33.m</a>
tests RGF_G_ENUM.
</li>