polymerist.maths.combinatorics.numbers
======================================

.. py:module:: polymerist.maths.combinatorics.numbers

.. autoapi-nested-parse::

   Utilities for calculating fundamental combinatorial numbers



Attributes
----------

.. autoapisummary::

   polymerist.maths.combinatorics.numbers.choose


Functions
---------

.. autoapisummary::

   polymerist.maths.combinatorics.numbers.alt_sign
   polymerist.maths.combinatorics.numbers.binomial_coeff
   polymerist.maths.combinatorics.numbers.multiset_coeff
   polymerist.maths.combinatorics.numbers.multinomial_coeff
   polymerist.maths.combinatorics.numbers.multinomial_coeff_native
   polymerist.maths.combinatorics.numbers.catalan
   polymerist.maths.combinatorics.numbers.pentagonal
   polymerist.maths.combinatorics.numbers.stirling_second
   polymerist.maths.combinatorics.numbers.stirling_first
   polymerist.maths.combinatorics.numbers.bell
   polymerist.maths.combinatorics.numbers.bernoulli
   polymerist.maths.combinatorics.numbers.bernoulli_up_to
   polymerist.maths.combinatorics.numbers.count_labelled_graphs
   polymerist.maths.combinatorics.numbers.count_connected_labelled_graphs


Module Contents
---------------

.. py:function:: alt_sign(n: int) -> int

   Returns the value of -1^n (useful for alternating sums)


.. py:function:: binomial_coeff(n: complex, k: int) -> float

   Calculates a generalized binomial coefficient
   Counts the number of distinct ways to make and unordered selection of k elemnts from a set of n items


.. py:data:: choose

.. py:function:: multiset_coeff(n: int, k: int) -> float

   Calculates a multiset coefficient
   Counts the number of ways to form a multiset of cardinality n from a support of size k


.. py:function:: multinomial_coeff(multips: Iterable[int]) -> float

   Calculate the multinomial coefficient of a set of multiplicities
   Counts the number of distinct permutations of a multiset with given multiplicites


.. py:function:: multinomial_coeff_native(multips: Iterable[int]) -> float

   Calculate the multinomial coefficient of a set of multiplicities
   Counts the number of distinct permutations of a multiset with given multiplicites

   Native Python implementation of multinomial_coeff() which is slightly slower but requires no imports


.. py:function:: catalan(n: int) -> float

   Calculates the n-th Catalan number
   Counts dozens of different things, including the number of monotonic 2D random walks
   and the number of valid (i.e. fully closed) groupings of n pairs of parentheses


.. py:function:: pentagonal(n: int) -> float

   Calculates the n-th pentagonal number
   Counts the number of points in n "concentric" pentagon figures which share a corner vertex

   Related to the generating function for numbers of partitions


.. py:function:: stirling_second(n: int, k: int) -> float

   Calculates the Stirling number of the second kind S(n, k)
   Counts the number of ways to partition n objects into k non-empty pairwise-disjoint subsets

   Recurrence used as listed in NIST DLMF (https://dlmf.nist.gov/26.8)


.. py:function:: stirling_first(n: int, k: int) -> float

   Calculates the Stirling number of the first kind c(n, k)
   Counts the number of ways to to decompose a permutation of n objects into k cycles

   Recurrence used as listed in Abramowitz ans Stegun (https://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=825)


.. py:function:: bell(n: int) -> float

   Calculates the n-th Bell number
   Counts the number of ways to partition n objects into any number of non-empty pairwise-disjoint subsets


.. py:function:: bernoulli(n: int) -> int

   Calculates the n-th Bernoulli number
   Related to the sums of powers of integers (via Faulhalber's formula)


.. py:function:: bernoulli_up_to(N: int) -> list[fractions.Fraction]

   Computes a list of the first N Bernoulli numbers using the Akiyama-Tanigawa algorithm


.. py:function:: count_labelled_graphs(N: int) -> int

   Gives number of all labelled graphs on N nodes


.. py:function:: count_connected_labelled_graphs(n: int) -> float

   Gives array of number labelled graphs with one connected component on N nodes