SUBSET
Combinatorial Routines

SUBSET is a FORTRAN77 library which performs various combinatorial operations.

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

• compositions of an integer;
• partitions of a set of N objects;
• I4_PARTITION's, partitions of an integer;
• PERM's, permutations of the first N integers;
• polynomials in factorial, Newton, power sum, or Taylor form;
• subsets of a set of N objects;
• KSUB's, subsets of size K, from a set of N objects;
• vectors whose entries range from 1 to N;
• YTB's, Young tableaus;

Other objects considered include

• the Bell numbers,
• BVEC's, binary numbers represented as a vector of 0's and 1's;
• 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 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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

COMBINATION_LOCK, a FORTRAN77 program which simulates the process of determining the secret combination of a lock.

COMBO, a FORTRAN77 library which contains a variety of combinatorial routines.

GRAFPACK, a FORTRAN90 library which contains many routines for handling abstract graphs.

PARTITION_PROBLEM, a FORTRAN77 library which seeks solutions of the partition problem, splitting a set of integers into two subsets with equal sum.

SELECT, a FORTRAN90 library which which generates various combinatorial objects.

Reference:

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Source Code:

• subset.f, the source code;
• subset.sh, commands to compile the source code;

Examples and Tests:

• subset_prb.f, the calling program;
• subset_prb.sh, commands to compile, link and run the calling program;
• subset_prb_output.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.
• BINARY_VECTOR_NEXT generates the next binary vector.
• BVEC_ADD adds two (signed) binary vectors.
• BVEC_AND computes the AND of two binary vectors.
• BVEC_CHECK checks a binary vector.
• BVEC_COMPLEMENT2 computes the two's complement of a binary vector.
• BVEC_MUL computes the product of two binary vectors.
• BVEC_NEXT generates the next binary vector.
• BVEC_NOT "negates" or takes the 1's complement of a binary vector.
• BVEC_OR computes the inclusive OR of two binary vectors.
• BVEC_PRINT prints a binary integer vector, with an optional title.
• BVEC_REVERSE reverses a binary vector.
• BVEC_SUB subtracts two binary vectors.
• BVEC_TO_I4 makes an integer from a (signed) binary vector.
• BVEC_XOR computes the exclusive OR of two binary vectors.
• 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 computes row N of Pascal's triangle.
• COMB_UNRANK returns the RANK-th combination of N things out of M.
• COMBIN computes the combinatorial coefficient C(N,K).
• COMBIN2 computes the binomial coefficient C(N,K).
• 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_RANDOM selects a random composition of the integer N into K parts.
• 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.
• 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 possible 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 of the 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 seeks a minimal solution of a Diophantine equation.
• DVEC_ADD adds two (signed) decimal vectors.
• DVEC_COMPLEMENTX computes the ten's complement of a decimal vector.
• DVEC_MUL computes the product of two decimal vectors.
• DVEC_PRINT prints a decimal integer vector, with an optional title.
• DVEC_SUB subtracts two decimal vectors.
• DVEC_TO_I4 makes an integer from a (signed) decimal vector.
• 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_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 returns values of the order 2 Frobenius number.
• GAMMA_LOG calculates the natural logarithm of GAMMA ( X ) for positive X.
• 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_FACTOR factors an I4 into prime factors.
• I4_FACTORIAL computes the factorial of N.
• I4_GCD finds the greatest common divisor of I and J.
• 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_COUNT2 computes the number of partitions of an I4.
• I4_PARTITION_COUNT_VALUES returns some values of the integer partition count.
• 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 an integer.
• I4_PARTITIONS_NEXT: next partition into S parts.
• I4_RISE computes the rising factorial function [X]^N.
• I4_SQRT finds the integer square root of N by solving N = Q**2 + R.
• I4_SQRT_CF finds the continued fraction representation of a square root of an integer.
• I4_SWAP switches two I4's.
• I4_TO_BVEC makes a signed binary vector from an I4.
• I4_TO_CHINESE converts an I4 to its Chinese remainder form.
• I4_TO_DVEC makes a signed decimal vector 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 returns a scaled pseudorandom I4.
• 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 matrix.
• I4POLY performs operations on I4POLY's in power or factorial form.
• I4POLY_CYCLO computes a cyclotomic polynomial.
• I4POLY_DEGREE returns the degree of 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_COPY copies an I4VEC.
• I4VEC_DESCENDS determines if an I4VEC is (weakly) descending.
• I4VEC_FRAC searches for the K-th smallest element in an I4VEC.
• I4VEC_INDICATOR sets an I4VEC to the indicator vector.
• I4VEC_INDEX returns the location of the first occurrence of a given value.
• I4VEC_MAX computes the maximum element of an I4VEC.
• I4VEC_MAXLOC_LAST returns the index of the last maximal I4VEC entry.
• I4VEC_MIN computes the minimum element of an I4VEC.
• I4VEC_PAIRWISE_PRIME checks whether an I4VEC's entries are pairwise prime.
• I4VEC_PRINT prints an I4VEC.
• I4VEC_PRINT_SOME prints "some" of an I4VEC.
• I4VEC_PRODUCT returns the product of the entries 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_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 returns a scaled pseudorandom I4VEC.
• I4VEC_ZERO sets the entries of an I4VEC to 0.
• 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_RANK computes the rank of a K subset of an N set.
• KSUB_UNRANK returns the subset of a given rank.
• Purpose:
• MATRIX_PRODUCT_OPT determines the optimal cost of a matrix product.
• MOEBIUS_MATRIX finds the Moebius matrix from a covering relation.
• MORSE_THUE generates a Morse_Thue number.
• MULTINOMIAL_COEF1 computes a multinomial coefficient.
• MULTINOMIAL_COEF2 computes a multinomial coefficient.
• 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_CHECK checks that a vector represents a 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_EPSILON returns the R8 roundoff unit.
• R8_FACTORIAL computes the factorial of N.
• R8_FALL computes the falling factorial function [X]_N.
• R8_HUGE returns a "huge" R8.
• R8_IS_INT determines if an R8 represents an integer value.
• R8_PI returns the value of pi as an R8.
• R8_RISE computes the rising factorial function [X]^N.
• R8_SWAP switches two R8's.
• 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 returns a scaled pseudorandom R8.
• R8_UNIFORM_01 returns a unit 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_DEGREE returns the degree of a polynomial in power sum form.
• 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.
• 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 a rational polynomial fraction to a continued fraction.
• RFRAC_TO_JFRAC converts a rational polynomial fraction to a J fraction.
• S_ADJUSTR flushes a string right.
• 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.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• SCHROEDER generates the Schroeder numbers.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• SUB_BY_SIZE_NEXT returns all subsets of an N set, in order of size.
• SUB_GRAY_NEXT generates all subsets of a set of order N, one at a time.
• SUB_GRAY_RANK ranks a subset of an N set, using the Gray code ordering.
• SUB_GRAY_UNRANK produces a subset of an N set of the given Gray code rank.
• SUB_LEX_NEXT generates the subsets of a set of N elements, one at a time.
• SUB_RANDOM selects a random subset of an N-set.
• 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.
• 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 out the current YMDHMS date as a timestamp.
• TIMESTRING writes the current YMDHMS date into a string.
• 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.
• UI4_TO_UBVEC makes an unsigned binary vector from an unsigned integer.
• VEC_NEXT generates all N-vectors of integers modulo a given base.
• VEC_NEXT_GRAY computes the elements of a product space.
• VEC_RANDOM selects a random N-vector of integers modulo a given base.
• VEC_RANK computes the rank of a product space element.
• VEC_UNRANK computes the product space element of a given rank.
• 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" 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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