186    PetscDTFactorial - Approximate n! as a real number
189 .  n - a non-negative integer
192 .  factorial - n!
198 static inline PetscErrorCode PetscDTFactorial(PetscInt n, PetscReal *factorial)  in PetscDTFactorial()  argument
204 …PetscCheck(n >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Factorial called with negative numb…  in PetscDTFactorial()
205   for (PetscInt i = 1; i < n + 1; ++i) f *= (PetscReal)i;  in PetscDTFactorial()
211    PetscDTFactorialInt - Compute n! as an integer
214 .  n - a non-negative integer
217 .  factorial - n!
222 …This is limited to `n` such that n! can be represented by `PetscInt`, which is 12 if `PetscInt` is…
226 static inline PetscErrorCode PetscDTFactorialInt(PetscInt n, PetscInt *factorial)  in PetscDTFactorialInt()  argument
232 …n >= 0 && n <= PETSC_FACTORIAL_MAX, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of elements…  in PetscDTFactorialInt()
233   if (n <= 12) {  in PetscDTFactorialInt()
234     *factorial = facLookup[n];  in PetscDTFactorialInt()
239     for (i = 13; i < n + 1; ++i) f *= i;  in PetscDTFactorialInt()
246    PetscDTBinomial - Approximate the binomial coefficient `n` choose `k`
249 +  n - a non-negative integer
250 -  k - an integer between 0 and `n`, inclusive
253 .  binomial - approximation of the binomial coefficient `n` choose `k`
259 static inline PetscErrorCode PetscDTBinomial(PetscInt n, PetscInt k, PetscReal *binomial)  in PetscDTBinomial()  argument
263 …n >= 0 && k >= 0 && k <= n, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial arguments (%" Pet…  in PetscDTBinomial()
264   if (n <= 3) {  in PetscDTBinomial()
272     *binomial = (PetscReal)binomLookup[n][k];  in PetscDTBinomial()
276     k = PetscMin(k, n - k);  in PetscDTBinomial()
277     for (PetscInt i = 0; i < k; i++) binom = (binom * (PetscReal)(n - i)) / (PetscReal)(i + 1);  in PetscDTBinomial()
284    PetscDTBinomialInt - Compute the binomial coefficient `n` choose `k`
287 +  n - a non-negative integer
288 -  k - an integer between 0 and `n`, inclusive
291 .  binomial - the binomial coefficient `n` choose `k`
302 static inline PetscErrorCode PetscDTBinomialInt(PetscInt n, PetscInt k, PetscInt *binomial)  in PetscDTBinomialInt()  argument
308 …n >= 0 && k >= 0 && k <= n, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial arguments (%" Pet…  in PetscDTBinomialInt()
309 …eck(n <= PETSC_BINOMIAL_MAX, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial elements %" Pets…  in PetscDTBinomialInt()
310   if (n <= 3) {  in PetscDTBinomialInt()
318     bin = binomLookup[n][k];  in PetscDTBinomialInt()
322     k = PetscMin(k, n - k);  in PetscDTBinomialInt()
323     for (PetscInt i = 0; i < k; i++) binom = (binom * (n - i)) / (i + 1);  in PetscDTBinomialInt()
334 …PetscDTEnumPerm - Get a permutation of `n` integers from its encoding into the integers [0, n!) as…
337 +  n - a non-negative integer (see note about limits below)
338 -  k - an integer in [0, n!)
341 +  perm  - the permuted list of the integers [0, ..., n-1]
347 …A permutation can be described by the operations that convert the lists [0, 1, ..., n-1] into the …
350    (n - i).  This sequence of n-1 differences [d_0, ..., d_{n-2}] is encoded as the number
351    (n-1)! * d_0 + (n-2)! * d_1 + ... + 1! * d_{n-2}.
353 …Limited to `n` such that `n`! can be represented by `PetscInt`, which is 12 if `PetscInt` is a sig…
357 static inline PetscErrorCode PetscDTEnumPerm(PetscInt n, PetscInt k, PetscInt *perm, PeOp PetscBool…  in PetscDTEnumPerm()  argument
366 …n >= 0 && n <= PETSC_FACTORIAL_MAX, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of elements…  in PetscDTEnumPerm()
367   if (n >= 2) {  in PetscDTEnumPerm()
368     w = &work[n - 2];  in PetscDTEnumPerm()
369     for (i = 2; i <= n; i++) {  in PetscDTEnumPerm()
374   for (i = 0; i < n; i++) perm[i] = i;  in PetscDTEnumPerm()
375   for (i = 0; i < n - 1; i++) {  in PetscDTEnumPerm()
388 …PetscDTPermIndex - Encode a permutation of n into an integer in [0, n!).  This inverts `PetscDTEnu…
391 +  n    - a non-negative integer (see note about limits below)
392 -  perm - the permuted list of the integers [0, ..., n-1]
395 +  k     - an integer in [0, n!)
401 …Limited to `n` such that `n`! can be represented by `PetscInt`, which is 12 if `PetscInt` is a sig…
405 static inline PetscErrorCode PetscDTPermIndex(PetscInt n, const PetscInt *perm, PetscInt *k, PeOp P…  in PetscDTPermIndex()  argument
415 …n >= 0 && n <= PETSC_FACTORIAL_MAX, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of elements…  in PetscDTPermIndex()
416   for (i = 0; i < n; i++) work[i] = i;  /* partial permutation */  in PetscDTPermIndex()
417   for (i = 0; i < n; i++) iwork[i] = i; /* partial permutation inverse */  in PetscDTPermIndex()
418   for (idx = 0, i = 0; i < n - 1; i++) {  in PetscDTPermIndex()
424     idx = idx * (n - i) + diff;  in PetscDTPermIndex()
438 …et - Get an ordered subset of the integers [0, ..., n - 1] from its encoding as an integers in [0,…
442 +  n - a non-negative integer (see note about limits below)
443 .  k - an integer in [0, n]
444 -  j - an index in [0, n choose k)
447 .  subset - the jth subset of size k of the integers [0, ..., n - 1]
452    Limited by arguments such that `n` choose `k` can be represented by `PetscInt`
456 static inline PetscErrorCode PetscDTEnumSubset(PetscInt n, PetscInt k, PetscInt j, PetscInt *subset)  in PetscDTEnumSubset()  argument
461   PetscCall(PetscDTBinomialInt(n, k, &Nk));  in PetscDTEnumSubset()
462   for (PetscInt i = 0, l = 0; i < n && l < k; i++) {  in PetscDTEnumSubset()
463     PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);  in PetscDTEnumSubset()
478 …n ordered subset of k integers from the set [0, ..., n - 1] to its encoding as an integers in [0, …
482 +  n      - a non-negative integer (see note about limits below)
483 .  k      - an integer in [0, n]
484 -  subset - an ordered subset of the integers [0, ..., n - 1]
492    Limited by arguments such that `n` choose `k` can be represented by `PetscInt`
496 static inline PetscErrorCode PetscDTSubsetIndex(PetscInt n, PetscInt k, const PetscInt *subset, Pet…  in PetscDTSubsetIndex()  argument
502   PetscCall(PetscDTBinomialInt(n, k, &Nk));  in PetscDTSubsetIndex()
503   for (PetscInt i = 0, l = 0; i < n && l < k; i++) {  in PetscDTSubsetIndex()
504     PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);  in PetscDTSubsetIndex()
520 …PetscDTEnumSplit - Split the integers [0, ..., n - 1] into two complementary ordered subsets, the …
523 +  n - a non-negative integer (see note about limits below)
524 .  k - an integer in [0, n]
525 -  j - an index in [0, n choose k)
528 +  perm  - the jth subset of size k of the integers [0, ..., n - 1], followed by its complementary …
534    Limited by arguments such that `n` choose `k` can be represented by `PetscInt`
539 static inline PetscErrorCode PetscDTEnumSplit(PetscInt n, PetscInt k, PetscInt j, PetscInt *perm, P…  in PetscDTEnumSplit()  argument
546   PetscCall(PetscDTBinomialInt(n, k, &Nk));  in PetscDTEnumSplit()
547   for (i = 0, l = 0, m = 0; i < n && l < k; i++) {  in PetscDTEnumSplit()
548     PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);  in PetscDTEnumSplit()
561   for (; i < n; i++) subcomp[m++] = i;  in PetscDTEnumSplit()