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combinatorial.rst

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Combinatorial

This module implements various combinatorial functions.

bell

.. autoclass:: diofant.functions.combinatorial.numbers.bell
   :members:

bernoulli

.. autoclass:: diofant.functions.combinatorial.numbers.bernoulli
   :members:

binomial

.. autoclass:: diofant.functions.combinatorial.factorials.binomial
   :members:

catalan

.. autoclass:: diofant.functions.combinatorial.numbers.catalan
   :members:

euler

.. autoclass:: diofant.functions.combinatorial.numbers.euler
   :members:

factorial

.. autoclass:: diofant.functions.combinatorial.factorials.factorial
   :members:

subfactorial

.. autoclass:: diofant.functions.combinatorial.factorials.subfactorial
   :members:

factorial2 / double factorial

.. autoclass:: diofant.functions.combinatorial.factorials.factorial2
   :members:

FallingFactorial

.. autoclass:: diofant.functions.combinatorial.factorials.FallingFactorial
   :members:

fibonacci

.. autoclass:: diofant.functions.combinatorial.numbers.fibonacci
   :members:

harmonic

.. autoclass:: diofant.functions.combinatorial.numbers.harmonic
   :members:

lucas

.. autoclass:: diofant.functions.combinatorial.numbers.lucas
   :members:

MultiFactorial

.. autoclass:: diofant.functions.combinatorial.factorials.MultiFactorial
   :members:

RisingFactorial

.. autoclass:: diofant.functions.combinatorial.factorials.RisingFactorial
   :members:

stirling

.. autofunction:: diofant.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 diofant.functions.combinatorial.numbers import nC, nP, nT
>>> items = 'baby'
.. autofunction:: diofant.functions.combinatorial.numbers.nC


.. autofunction:: diofant.functions.combinatorial.numbers.nP


.. autofunction:: diofant.functions.combinatorial.numbers.nT

Note that the integer for n indicates identical items for nT but indicates n different items for nC and nP.