Convert integers to and from the factoradic number system
The factorial number system (also known as factoradic) is a way of representing an integer as the sum of multiples of factorials. All integers have a unique representation in the factoradic number system. For example, the number 1337 can be represented as:
1*6! + 5*5! + 0*4! + 2*3! + 2*2! + 1*1! + 0*0!
with coefficients 1 5 0 2 2 1 0. This is the unique factoradic representation of decimal 1337.
Factoradic numbers have uses in combinatorics, particularly in the numbering of permutations. This factoradic library is useful for converting to and from factoradic number representations both in Python and from the command-line.
The factoradic package is available on the Python Package Index (PyPI):
The package supports Python 3 only. To install:
$ pip install factoradic
For full help:
>>> import factoradic >>> help(factoradic)
In the meantime, here are some highlights.
To convert from an integer to factoradic use to_factoradic():
>>> factoradic.to_factoradic(1337) [0, 1, 2, 2, 0, 5, 1]
The result is the list of coefficients where the factorial of each zero-based index gives a place value, and the item at that index is the coefficient by with the place value is to be multiplied. The elements are from least-significant to most-significant. Since the coefficient at any index must be less that or equal to the index, the coefficient at index 0 is always 0.
To convert from factoradic use from_factoradic():
>>> factoradic.from_factoradic([0, 1, 2, 2, 0, 5, 1]) 1337
There’s also a handy command-line interface. Run factoradic --help to see a list of commands:
$ factoradic --help Factoradic. Convert to and from the factorial number system. Usage: factoradic from-integer <integer> [--expression] factoradic to-integer <coefficient-0> [<coefficient-1> [<coefficient-n>...]] Options: -e --expression Show as a mathematical expression.
To convert from an integer to factoradic, use the from-integer subcommand:
$ factoradic from-integer 1729 0 1 0 0 2 2 2
The coefficients are reported from least-significant to most-significant. The see the results as a math expression, specify the --expression flag:
$ factoradic from-integer 1729 --expression 2*6! + 2*5! + 2*4! + 0*3! + 0*2! + 1*1! + 0*0!
To convert from factoradic representation use the to-integer subcommand, specifying the coefficients from least-significant to most-significant:
$ factoradic to-integer 0 1 0 0 2 2 2 1729
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