.. _combinatorial-functions: Combinatorial ============= This module implements various combinatorial functions. .. autoclass:: sympy.functions.combinatorial.numbers.bell :members: .. autoclass:: sympy.functions.combinatorial.numbers.bernoulli :members: .. autoclass:: sympy.functions.combinatorial.factorials.binomial :members: .. autoclass:: sympy.functions.combinatorial.numbers.catalan :members: .. autoclass:: sympy.functions.combinatorial.numbers.euler :members: .. autoclass:: sympy.functions.combinatorial.factorials.factorial :members: .. autoclass:: sympy.functions.combinatorial.factorials.subfactorial :members: .. autoclass:: sympy.functions.combinatorial.factorials.factorial2 :members: .. autoclass:: sympy.functions.combinatorial.factorials.FallingFactorial :members: .. autoclass:: sympy.functions.combinatorial.numbers.fibonacci :members: .. autoclass:: sympy.functions.combinatorial.numbers.tribonacci :members: .. autoclass:: sympy.functions.combinatorial.numbers.harmonic :members: .. autoclass:: sympy.functions.combinatorial.numbers.lucas :members: .. autoclass:: sympy.functions.combinatorial.numbers.genocchi :members: .. autoclass:: sympy.functions.combinatorial.numbers.andre :members: .. autoclass:: sympy.functions.combinatorial.numbers.partition :members: .. autoclass:: sympy.functions.combinatorial.numbers.divisor_sigma :members: .. autoclass:: sympy.functions.combinatorial.numbers.udivisor_sigma :members: .. autoclass:: sympy.functions.combinatorial.numbers.legendre_symbol :members: .. autoclass:: sympy.functions.combinatorial.numbers.jacobi_symbol :members: .. autoclass:: sympy.functions.combinatorial.numbers.kronecker_symbol :members: .. autoclass:: sympy.functions.combinatorial.numbers.mobius :members: .. autoclass:: sympy.functions.combinatorial.numbers.primenu :members: .. autoclass:: sympy.functions.combinatorial.numbers.primeomega :members: .. autoclass:: sympy.functions.combinatorial.numbers.totient :members: .. autoclass:: sympy.functions.combinatorial.numbers.reduced_totient :members: .. autoclass:: sympy.functions.combinatorial.numbers.primepi :members: .. autoclass:: sympy.functions.combinatorial.factorials.MultiFactorial :members: .. autoclass:: sympy.functions.combinatorial.factorials.RisingFactorial :members: .. autofunction:: sympy.functions.combinatorial.numbers.stirling Enumeration =========== Three functions are available. Each of them attempts to efficiently compute a given combinatorial quantity for a given set or multiset which can be entered as an integer, sequence or multiset (dictionary with elements as keys and multiplicities as values). The ``k`` parameter indicates the number of elements to pick (or the number of partitions to make). When ``k`` is None, the sum of the enumeration for all ``k`` (from 0 through the number of items represented by ``n``) is returned. A ``replacement`` parameter is recognized for combinations and permutations; this indicates that any item may appear with multiplicity as high as the number of items in the original set. >>> from sympy.functions.combinatorial.numbers import nC, nP, nT >>> items = 'baby' .. autofunction:: sympy.functions.combinatorial.numbers.nC .. autofunction:: sympy.functions.combinatorial.numbers.nP .. autofunction:: sympy.functions.combinatorial.numbers.nT