SUBSET
Combinatorial Routines

SUBSET is a FORTRAN90 library which enumerates, generates, randomizes, ranks and unranks combinatorial objects including combinations, compositions, Gray codes, index sets, partitions, permutations, polynomials, subsets, and Young tables. Backtracking routines are included to solve some combinatorial problems. Some routines for continued fractions are included.

These include the enumeration, generation, random selection, ranking and unranking of

• COMP, compositions of an integer N into K parts;
• COMPNZ, compositions of an integer N into K parts, with no zero parts;
• EQUIV's, partitions of a set of N objects;
• I4_PARTITION's, partitions of an integer;
• I4POLY's, integer polynomials in factorial, Newton, power sum, or Taylor form;
• I4VEC's, integer vectors;
• KSUB's, subsets of size K, from a set of N objects;
• MULTIPERM's, permutations of the N objects, some of which are indistinguishable.
• PERM's, permutations of the first N integers;
• R8POLY's, real polynomials in factorial, Newton, power sum, or Taylor form;
• KSUB's, subsets of a set of N objects;
• vectors whose entries range from 1 to N;
• YTB's, Young tables;

Other objects considered include

• the Bell numbers,
• Catalan numbers,
• congruence equations.
• continued fractions,
• DEC's, decimal numbers represented as a mantissa and a power of 10;
• DERANGE's, derangements (permutations that leave no element in place),
• DVEC's, decimal numbers represented as a vector of digits;
• falling factorials (20*19*18...),
• GRAY, Gray codes,
• matrix permanents (similar to determinants, but harder to compute, if you can believe that),
• Morse-Thue numbers,
• pentagonal numbers,
• Pythagorean triples,
• RAT's, rational numbers represented as a pair of integers;
• rising factorials (7*8*9...).

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

SUBSET is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BACKTRACK_BINARY_RC, a FORTRAN90 library which carries out a backtrack search for a set of binary decisions, using reverse communication.

CHANGE_MAKING, a FORTRAN90 library which considers the change making problem, in which a given sum is to be formed using coins of various denominations.

COMBINATION_LOCK, a FORTRAN90 program which simulates the process of determining the secret combination of a lock.

COMBO, a FORTRAN90 library which includes many combinatorial routines.

FLOYD, a FORTRAN90 library which implements Floyd's algorithm for finding the shortest distance between pairs of nodes on a directed graph.

GRAFPACK, a FORTRAN90 library which carries out operations on abstract graphs.

KNAPSACK_01, a FORTRAN90 library which uses brute force to solve small versions of the 0/1 knapsack problem;

LAMP, a FORTRAN77 library which solves linear assignment and matching problems.

LEGENDRE_PRODUCT_POLYNOMIAL, a FORTRAN90 library which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.

MONOMIAL, a FORTRAN90 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.

PARTIAL_DIGEST, a FORTRAN90 library which solves the partial digest problem.

PARTITION_PROBLEM, a FORTRAN90 library which seeks solutions of the partition problem, splitting a set of integers into two subsets with equal sum.

POLYNOMIAL, a FORTRAN90 library which adds, multiplies, differentiates, evaluates and prints multivariate polynomials in a space of M dimensions.

SET_THEORY, a FORTRAN90 library which demonstrates various set theoretic operations using several models of a set.

SELECT, a FORTRAN90 library which generates various combinatorial objects.

SUBSET_SUM, a FORTRAN90 library which seeks solutions of the subset sum problem.

TOMS515, a FORTRAN90 library which can select subsets of size K from a set of size N. This is a version of ACM TOMS Algorithm 515, by Bill Buckles, Matthew Lybanon.

TREEPACK, a FORTRAN90 library which carries out computations on trees, a simple kind of graph that is minimally connected.

UNICYCLE, a FORTRAN90 library which considers permutations containing a single cycle, sometimes called cyclic permutations.

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Source Code:

• subset.f90, the source code;

Examples and Tests:

• subset_test.f90, the calling program;
• subset_test.txt, the output file.

List of Routines:

• ASM_ENUM returns the number of alternating sign matrices of a given order.
• ASM_TRIANGLE returns a row of the alternating sign matrix triangle.
• BELL returns the Bell numbers from 0 to N.
• BELL_VALUES returns some values of the Bell numbers for testing.
• CATALAN computes the Catalan numbers, from C(0) to C(N).
• CATALAN_ROW_NEXT computes row N of Catalan's triangle.
• CATALAN_VALUES returns some values of the Catalan numbers for testing.
• CFRAC_TO_RAT converts a monic continued fraction to an ordinary fraction.
• CFRAC_TO_RFRAC converts polynomial fractions from continued to rational form.
• CH_CAP capitalizes a single character.
• CHANGE_GREEDY makes change for a given total using the biggest coins first.
• CHANGE_NEXT computes the next set of change for a given sum.
• CHINESE_CHECK checks the Chinese remainder moduluses.
• CHINESE_TO_I4 converts a set of Chinese remainders to an equivalent integer.
• COMB_NEXT computes combinations of K things out of N.
• COMB_ROW_NEXT computes the next row of Pascal's triangle.
• COMB_UNRANK returns the RANK-th combination of N things out of M.
• COMP_ENUM returns the number of compositions of the integer N into K parts.
• COMP_NEXT computes the compositions of the integer N into K parts.
• COMP_NEXT_GRLEX returns the next composition in grlex order.
• COMP_RANDOM selects a random composition of the integer N into K parts.
• COMP_RANDOM_GRLEX: random composition with degree less than or equal to NC.
• COMP_RANK_GRLEX computes the graded lexicographic rank of a composition.
• COMP_TO_KSUB converts a composition to a K-subset.
• COMP_UNRANK_GRLEX computes the composition of given grlex rank.
• COMPNZ_ENUM returns the number of nonzero compositions of the N into K parts.
• COMPNZ_NEXT computes the compositions of the integer N into K nonzero parts.
• COMPNZ_RANDOM selects a random composition of N into K nonzero parts.
• COMPNZ_TO_KSUB converts a nonzero composition to a K-subset.
• CONGRUENCE solves a congruence of the form A * X = C ( mod B ).
• COUNT_POSE_RANDOM poses a problem for the game "The Count is Good"
• DEBRUIJN constructs a de Bruijn sequence.
• DEC_ADD adds two decimal quantities.
• DEC_DIV divides two decimal values.
• DEC_MUL multiplies two decimals.
• DEC_ROUND rounds a decimal fraction to a given number of digits.
• DEC_TO_R8 converts a decimal to an R8.
• DEC_TO_RAT converts a decimal to a rational representation.
• DEC_TO_S returns a string representation of a decimal.
• DEC_WIDTH returns the "width" of a decimal number.
• DECMAT_DET finds the determinant of an N by N matrix of decimal entries.
• DECMAT_PRINT prints out decimal vectors and matrices.
• DERANGE_BACK_CANDIDATE finds values for the K-th entry of a derangement.
• DERANGE_BACK_NEXT returns the next derangement of N items.
• DERANGE_CHECK determines whether a permutation is a derangement.
• DERANGE_ENUM returns the number of derangements of N objects.
• DERANGE_ENUM2 returns the number of derangements of 0 through N objects.
• DERANGE_ENUM3 returns the number of derangements of 0 through N objects.
• DERANGE_WEED_NEXT computes all derangements of N objects, one at a time.
• DIGIT_TO_CH returns the character representation of a decimal digit.
• DIGRAPH_ARC_EULER returns an Euler circuit in a digraph.
• DIGRAPH_ARC_PRINT prints out a digraph from an edge list.
• DIOPHANTINE solves a Diophantine equation A * X + B * Y = C.
• DIOPHANTINE_SOLUTION_MINIMIZE: minimal solution of a Diophantine equation.
• DVEC_ADD adds two (signed) DVEC's.
• DVEC_COMPLEMENTX computes the ten's complement of a DVEC.
• DVEC_MUL computes the product of two DVEC's.
• DVEC_PRINT prints a DVEC, with an optional title.
• DVEC_SUB subtracts two DVEC's.
• DVEC_TO_I4 makes an integer from a (signed) DVEC.
• EQUIV_NEXT computes the partitions of a set one at a time.
• EQUIV_NEXT2 computes, one at a time, the partitions of a set.
• EQUIV_PRINT prints a partition of a set.
• EQUIV_PRINT2 prints a partition of a set.
• EQUIV_RANDOM selects a random partition of a set.
• EULER returns the N-th row of Euler's triangle.
• FROBENIUS_NUMBER_ORDER2 returns the Frobenius number for order 2.
• FROBENIUS_NUMBER_ORDER2_VALUES: values of the order 2 Frobenius number.
• GAMMA_LOG_VALUES returns some values of the Log Gamma function.
• GET_SEED returns a seed for the random number generator.
• GRAY_NEXT generates the next Gray code by switching one item at a time.
• GRAY_RANK ranks a Gray code.
• GRAY_RANK2 ranks a Gray code.
• GRAY_UNRANK unranks a Gray code.
• GRAY_UNRANK2 unranks a Gray code.
• I4_BCLR returns a copy of an I4 in which the POS-th bit is set to 0.
• I4_BSET returns a copy of an I4 in which the POS-th bit is set to 1.
• I4_BTEST returns TRUE if the POS-th bit of an I4 is 1.
• I4_CHOOSE computes the binomial coefficient C(N,K) as an I4.
• I4_FACTOR factors an I4 into prime factors.
• I4_FACTORIAL computes the factorial of N.
• I4_FALL computes the falling factorial function [X]_N.
• I4_FALL_VALUES returns values of the integer falling factorial function.
• I4_GCD finds the greatest common divisor of two I4's.
• I4_HUGE returns a "huge" I4.
• I4_LOG_10 returns the integer part of the logarithm base 10 of an I4.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_MOEBIUS returns the value of MU(N), the Moebius function of N.
• I4_PARTITION_CONJ computes the conjugate of a partition.
• I4_PARTITION_COUNT computes the number of partitions of an I4.
• I4_PARTITION_COUNT_VALUES returns some values of the integer partition count.
• I4_PARTITION_COUNT2 computes the number of partitions of an I4.
• I4_PARTITION_NEXT generates the partitions of an I4, one at a time.
• I4_PARTITION_NEXT2 computes the partitions of the integer N one at a time.
• I4_PARTITION_PRINT prints a partition of an I4.
• I4_PARTITION_RANDOM selects a random partition of the integer N.
• I4_PARTITIONS_NEXT: next partition into S parts.
• I4_RISE computes the rising factorial function [X]^N.
• I4_RISE_VALUES returns values of the integer rising factorial function.
• I4_SQRT finds the integer square root of N by solving N = Q*Q + R.
• I4_SQRT_CF: continued fraction representation of a square root of an integer.
• I4_SWAP switches two I4's.
• I4_TO_CHINESE converts an I4 to its Chinese remainder form.
• I4_TO_DVEC makes a signed DVEC from an I4.
• I4_TO_I4POLY converts an I4 to an I4POLY in a given base.
• I4_TO_S_LEFT converts an I4 to a left-justified string.
• I4_TO_VAN_DER_CORPUT computes an element of a van der Corput sequence.
• I4_UNIFORM_AB returns a scaled pseudorandom I4 between A and B.
• I4MAT_01_ROWCOLSUM creates a 0/1 I4MAT with given row and column sums.
• I4MAT_01_ROWCOLSUM2 creates a 0/1 I4MAT with given row and column sums.
• I4MAT_PERM permutes the rows and columns of a square I4MAT.
• I4MAT_PERM2 permutes the rows and columns of a rectangular I4MAT.
• I4MAT_PRINT prints an I4MAT.
• I4MAT_PRINT_SOME prints some of an I4MAT.
• I4MAT_U1_INVERSE inverts a unit upper triangular I4MAT.
• I4POLY performs operations on I4POLY's in power or factorial form.
• I4POLY_ADD adds two I4POLY's.
• I4POLY_CYCLO computes a cyclotomic polynomial.
• I4POLY_DEGREE returns the degree of an I4POLY.
• I4POLY_DIF differentiates an I4POLY.
• I4POLY_DIV computes the quotient and remainder of two I4POLY's.
• I4POLY_MUL computes the product of two I4POLY's.
• I4POLY_PRINT prints an I4POLY.
• I4POLY_TO_I4 evaluates an I4POLY.
• I4VEC_ASCENDS determines if an I4VEC is (weakly) ascending.
• I4VEC_BACKTRACK supervises a backtrack search for an I4VEC.
• I4VEC_DESCENDS determines if an I4VEC is decreasing.
• I4VEC_FRAC searches for the K-th smallest element in an I4VEC.
• I4VEC_HEAP_D reorders an I4VEC into an descending heap.
• I4VEC_INDICATOR sets an I4VEC to the indicator vector.
• I4VEC_INDEX returns the location of the first occurrence of a given value.
• I4VEC_MAXLOC_LAST returns the index of the last maximal I4VEC entry.
• I4VEC_PAIRWISE_PRIME checks whether an I4VEC is pairwise prime.
• I4VEC_PRINT prints an I4VEC, with an optional title.
• I4VEC_PRINT_SOME prints "some" of an I4VEC.
• I4VEC_REVERSE reverses the elements of an I4VEC.
• I4VEC_SORT_BUBBLE_A ascending sorts an I4VEC using bubble sort.
• I4VEC_SORT_HEAP_A ascending sorts an I4VEC using heap sort.
• I4VEC_SORT_HEAP_INDEX_D does an indexed heap descending sort of an I4VEC.
• I4VEC_SUM returns the sum of the entries of an I4VEC.
• I4VEC_TRANSPOSE_PRINT prints an I4VEC "transposed".
• I4VEC_UNIFORM_AB returns a scaled pseudorandom I4VEC.
• INDEX_BOX2_NEXT_2D produces index vectors on the surface of a box in 2D.
• INDEX_BOX2_NEXT_3D produces index vectors on the surface of a box in 3D.
• INDEX_BOX_NEXT_2D produces index vectors on the surface of a box in 2D.
• INDEX_BOX_NEXT_3D produces index vectors on the surface of a box in 3D.
• INDEX_NEXT0 generates all index vectors within given upper limits.
• INDEX_NEXT1 generates all index vectors within given upper limits.
• INDEX_NEXT2 generates all index vectors within given lower and upper limits.
• INDEX_RANK0 ranks an index vector within given upper limits.
• INDEX_RANK1 ranks an index vector within given upper limits.
• INDEX_RANK2 ranks an index vector within given lower and upper limits.
• INDEX_UNRANK0 unranks an index vector within given upper limits.
• INDEX_UNRANK1 unranks an index vector within given upper limits.
• INDEX_UNRANK2 unranks an index vector within given lower and upper limits.
• INS_PERM computes a permutation from its inversion sequence.
• INVERSE_MOD_N computes the inverse of B mod N.
• INVOLUTE_ENUM enumerates the involutions of N objects.
• JFRAC_TO_RFRAC converts a J-fraction into a rational polynomial fraction.
• JOSEPHUS returns the position X of the K-th man to be executed.
• KSUB_NEXT generates the subsets of size K from a set of size N.
• KSUB_NEXT2 generates the subsets of size K from a set of size N.
• KSUB_NEXT3 generates the subsets of size K from a set of size N.
• KSUB_NEXT4 generates the subsets of size K from a set of size N.
• KSUB_RANDOM selects a random subset of size K from a set of size N.
• KSUB_RANDOM2 selects a random subset of size K from a set of size N.
• KSUB_RANDOM3 selects a random subset of size K from a set of size N.
• KSUB_RANDOM4 selects a random subset of size K from a set of size N.
• KSUB_RANDOM5 selects a random subset of size K from a set of size N.
• KSUB_RANK computes the rank of a K subset of an N set.
• KSUB_TO_COMP converts a K-subset to a composition.
• KSUB_TO_COMPNZ converts a K-subset to a nonzero composition.
• KSUB_UNRANK returns the subset of a given rank.
• L4VEC_NEXT generates the next logical vector.
• MATRIX_PRODUCT_OPT determines the optimal cost of a matrix product.
• MOEBIUS_MATRIX finds the Moebius matrix from a covering relation.
• MONOMIAL_COUNT counts the number of monomials up to a given degree.
• MONOMIAL_COUNTS counts the number of monomials up to a given degree.
• MORSE_THUE generates a Morse_Thue number.
• MULTINOMIAL_COEF1 computes a multinomial coefficient.
• MULTINOMIAL_COEF2 computes a multinomial coefficient.
• MULTIPERM_ENUM enumerates multipermutations.
• MULTIPERM_NEXT returns the next multipermutation.
• NETWORK_FLOW_MAX finds the maximal flow and a minimal cut in a network.
• NIM_SUM computes the Nim sum of two integers.
• PADOVAN returns the first N values of the Padovan sequence.
• PELL_BASIC returns the fundamental solution for Pell's basic equation.
• PELL_NEXT returns the next solution of Pell's equation.
• PENT_ENUM computes the N-th pentagonal number.
• PERM_ASCEND computes the longest ascending subsequence of a permutation.
• PERM_BREAK_COUNT counts the number of "breaks" in a permutation.
• PERM_CANON_TO_CYCLE converts a permutation from canonical to cycle form.
• PERM_CHECK0 checks a 0-based permutation.
• PERM_CHECK1 checks a 1-based permutation.
• PERM_CYCLE analyzes a permutation.
• PERM_CYCLE_TO_CANON converts a permutation from cycle to canonical form.
• PERM_CYCLE_TO_INDEX converts a permutation from cycle to standard index form.
• PERM_DISTANCE computes the Ulam metric distance of two permutations.
• PERM_FIXED_ENUM enumerates the permutations of N objects with M fixed.
• PERM_FREE reports the unused items in a partial permutation.
• PERM_INDEX_TO_CYCLE converts a permutation from standard index to cycle form.
• PERM_INS computes the inversion sequence of a permutation.
• PERM_INVERSE inverts a permutation "in place".
• PERM_INVERSE2 inverts a permutation "in place".
• PERM_INVERSE3 produces the inverse of a given permutation.
• PERM_LEX_NEXT generates permutations in lexical order, one at a time.
• PERM_MUL "multiplies" two permutations.
• PERM_NEXT computes all of the permutations of N objects, one at a time.
• PERM_NEXT2 generates all the permutations of N objects.
• PERM_NEXT3 computes all of the permutations of N objects, one at a time.
• PERM_PRINT prints a permutation.
• PERM_RANDOM selects a random permutation of N objects.
• PERM_RANDOM2 selects a random permutation of N objects.
• PERM_RANDOM3 selects a random permutation of N elements.
• PERM_RANK computes the rank of a given permutation.
• PERM_SIGN returns the sign of a permutation.
• PERM_TO_EQUIV computes the partition induced by a permutation.
• PERM_TO_YTB converts a permutation to a Young table.
• PERM_UNRANK "unranks" a permutation.
• PERRIN returns the first N values of the Perrin sequence.
• PORD_CHECK checks a matrix representing a partial ordering.
• POWER_MOD computes mod ( A^N, M ).
• POWER_SERIES1 computes the power series for G(Z) = (1+F(Z))^ALPHA.
• POWER_SERIES2 computes the power series for G(Z) = exp(F(Z)) - 1.
• POWER_SERIES3 computes the power series for H(Z) = G(F(Z)).
• POWER_SERIES4 computes the power series for H(Z) = G ( 1/F(Z) ).
• PRIME returns any of the first PRIME_MAX prime numbers.
• PYTHAG_TRIPLE_NEXT computes the next Pythagorean triple.
• R8_AGM finds the arithmetic-geometric mean of two numbers.
• R8_CHOOSE computes the combinatorial coefficient C(N,K).
• R8_FACTORIAL computes the factorial of N.
• R8_FALL computes the falling factorial function [X]_N.
• R8_FALL_VALUES returns some values of the falling factorial function.
• R8_GAMMA_LOG calculates the natural logarithm of GAMMA ( X ) for positive X.
• R8_IS_INT determines if a real number represents an integer value.
• R8_PI returns the value of pi.
• R8_RISE computes the rising factorial function [X]^N.
• R8_RISE_VALUES returns some values of the rising factorial function.
• R8_SWAP switches two real values.
• R8_TO_CFRAC converts a real value to a continued fraction.
• R8_TO_DEC converts a real quantity to a decimal representation.
• R8_TO_RAT converts a real value to a rational value.
• R8_TO_S_LEFT represents a real using 14 left_justified characters.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8_UNIFORM_AB returns a scaled pseudorandom R8.
• R8MAT_DET finds the determinant of an N by N R8MAT.
• R8MAT_PERM permutes the rows and columns of a square R8MAT.
• R8MAT_PERM2 permutes rows and columns of a rectangular R8MAT, in place.
• R8MAT_PERMANENT computes the permanent of an R8MAT.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8POLY performs operations on real polynomials in power or factorial form.
• R8POLY_ADD adds two R8POLY's.
• R8POLY_DEGREE returns the degree of a polynomial in power sum form.
• R8POLY_DIF differentiates an R8POLY.
• R8POLY_DIV computes the quotient and remainder of two polynomials.
• R8POLY_F2P converts a real polynomial from factorial form to power sum form.
• R8POLY_FVAL evaluates a real polynomial in factorial form.
• R8POLY_MUL computes the product of two real polynomials A and B.
• R8POLY_N2P converts a real polynomial from Newton form to power sum form.
• R8POLY_NVAL evaluates a real polynomial in Newton form.
• R8POLY_NX replaces one of the base points in a polynomial in Newton form.
• R8POLY_P2F converts a real polynomial from power sum form to factorial form.
• R8POLY_P2N converts a real polynomial from power sum form to Newton form.
• R8POLY_P2T converts a real polynomial from power sum form to Taylor form.
• R8POLY_POWER computes a positive integer power of a polynomial.
• R8POLY_PRINT prints out a polynomial.
• R8POLY_PVAL evaluates a real polynomial in power sum form.
• R8POLY_T2P converts a real polynomial from Taylor form to power sum form.
• R8VEC_BACKTRACK supervises a backtrack search for an R8VEC.
• R8VEC_FRAC searches for the K-th smallest entry in an R8VEC.
• R8VEC_INDICATOR sets an R8VEC to the indicator vector.
• R8VEC_MIRROR_NEXT steps through all sign variations of an R8VEC.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_UNIFORM returns a scaled pseudorandom R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• RANDOM_INITIALIZE initializes the FORTRAN 90 random number seed.
• RAT_ADD adds two rational values.
• RAT_DIV divides one rational value by another.
• RAT_FAREY computes the N-th row of the Farey fraction table.
• RAT_FAREY2 computes the next row of the Farey fraction table.
• RAT_MUL multiplies two fractions.
• RAT_NORMALIZE normalizes a rational number.
• RAT_SUM_FORMULA computes the formulas for sums of powers of integers.
• RAT_TO_CFRAC converts a rational value to a continued fraction.
• RAT_TO_DEC converts a rational to a decimal representation.
• RAT_TO_R8 converts rational values to real values.
• RAT_TO_S_LEFT returns a left-justified representation of A/B.
• RAT_WIDTH returns the "width" of a rational number.
• RATMAT_DET finds the determinant of an N by N matrix of rational entries.
• RATMAT_PRINT prints out rational vectors or matrices.
• REGRO_NEXT computes restricted growth functions one at a time.
• RFRAC_TO_CFRAC converts rational polynomial fractions to continued fractions.
• RFRAC_TO_JFRAC converts a rational polynomial fraction to a J fraction.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• S_BLANKS_DELETE replaces consecutive blanks by one blank.
• S_EQI is a case insensitive comparison of two strings for equality.
• SCHROEDER generates the Schroeder numbers.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• SUBCOMP_NEXT computes the next subcomposition of N into K parts.
• SUBCOMPNZ_NEXT computes the next subcomposition of N into K nonzero parts.
• SUBCOMPNZ2_NEXT computes the next subcomposition of N into K nonzero parts.
• SUBSET_BY_SIZE_NEXT returns all subsets of an N set, in order of size.
• SUBSET_GRAY_NEXT generates all subsets of a set of order N, one at a time.
• SUBSET_GRAY_RANK ranks a subset of an N set, using the Gray code ordering.
• SUBSET_GRAY_UNRANK produces a subset of an N set of the given Gray code rank.
• SUBSET_LEX_NEXT generates the subsets of a set of N elements, one at a time.
• SUBSET_RANDOM selects a random subset of an N-set.
• SUBTRIANGLE_NEXT computes the next subtriangle of a triangle.
• THUE_BINARY_NEXT returns the next element in a binary Thue sequence.
• THUE_TERNARY_NEXT returns the next element in a ternary Thue sequence.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANG renumbers elements in accordance with a partial ordering.
• TUPLE_NEXT computes the next element of a tuple space.
• TUPLE_NEXT_FAST computes the next element of a tuple space, "fast".
• TUPLE_NEXT_GE computes the next "nondecreasing" element of a tuple space.
• TUPLE_NEXT2 computes the next element of an integer tuple space.
• UBVEC_ADD adds two unsigned binary vectors.
• UBVEC_TO_UI4 makes an unsigned integer from an unsigned binary vector.
• UBVEC_XOR computes the exclusive OR of two unsigned binary vectors.
• UI4_TO_UBVEC makes an unsigned binary vector from an unsigned integer.
• VEC_COLEX_NEXT generates vectors in colex order.
• VEC_COLEX_NEXT2 generates vectors in colex order.
• VEC_COLEX_NEXT3 generates vectors in colex order.
• VEC_GRAY_RANK computes the rank of a product space element.
• VEC_GRAY_UNRANK computes the product space element of a given rank.
• VEC_GRAY_NEXT computes the elements of a product space.
• VEC_LEX_NEXT generates vectors in lex order.
• VEC_RANDOM selects a random N-vector of integers modulo a given base.
• VECTOR_CONSTRAINED_NEXT returns the "next" constrained vector.
• VECTOR_CONSTRAINED_NEXT2 returns the "next" constrained vector.
• VECTOR_CONSTRAINED_NEXT3 returns the "next" constrained vector.
• VECTOR_CONSTRAINED_NEXT4 returns the "next" constrained vector.
• VECTOR_CONSTRAINED_NEXT5 returns the "next" constrained vector.
• VECTOR_CONSTRAINED_NEXT6 returns the "next" constrained vector.
• VECTOR_CONSTRAINED_NEXT7 returns the "next" constrained vector.
• VECTOR_NEXT returns the "next" integer vector between two ranges.
• YTB_ENUM enumerates the Young tables of size N.
• YTB_NEXT computes the next Young table for a given shape.
• YTB_PRINT prints a Young table.
• YTB_RANDOM selects a random Young table of a given shape.

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