Factorial number system converter

Enter a decimal integer such as twenty-three with mode set to to_factoradic, or a colon-separated digit string such as three-colon-two-colon-one-colon-zero with mode set to from_factoradic; the tool validates every place so that digit d sub k never exceeds k, expands the sum of d sub k times k factorial, and returns both the decimal value and the classic bang notation used in permutation ranking. Negative values, fractions, scientific notation, and illegal place digits raise a clear invalid_input error instead of a silent wrong rank.

What the factorial number system (factoradic) is

If any digit violates the bound for its place, conversion stops with invalid_input—there is no best-effort parse that clamps digits or invents a different radix.

How to call the API and what comes back

Send a JSON body with one required string field value (aliases n, input, text, number, digits, and factoradic) and an optional mode field that defaults to to_factoradic. Set mode to to_factoradic, encode, or decimal when value is a non-negative base-ten integer; set mode to from_factoradic, decode, or factoradic when value is an MSB-first factoradic digit string. Digit strings may use colons, spaces, or commas as separators (for example three-colon-two-colon-one-colon-zero), or the compact bang form three-two-one-zero bang when every place fits a single character and there are at most ten places; digit arrays are also accepted for from_factoradic. Absolute place count is capped at forty so Worker and browser responses stay bounded. On success the response sets mode to the normalized direction, echoes input, sets value and decimal to the integer as a JSON number when it fits Number.MAX_SAFE_INTEGER or as a decimal digit string when it does not, decimal_string always to the full decimal text, digits and digits_msb to the MSB-first array, digits_lsb to the least-significant-first array used by the remainder algorithm, factoradic to the colon-joined form, factoradic_bang to the classic bang spelling when compact, places to an array of digit, place, factorial, contribution, and label rows, place_count to the number of places, expansion to a short sum of labels, formula to the defining sum, message to a one-line English summary, and steps to an ordered list that records parsing, validation, and the expansion. Failed validation—missing value, negatives, fractions, scientific notation, empty digit lists, more than forty places, or any digit outside zero through k at place k—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.

Practical tips, limits, and related searches

For magnitudes larger than Number.MAX_SAFE_INTEGER the value and decimal fields may be strings so no mantissa rounding can corrupt low bits—always prefer decimal_string when you need a stable textual form across languages. When teaching, pair examples: decimal twenty-three maps to factoradic three-two-one-zero, factoradic one-zero-zero-zero-zero maps to decimal twenty-four, and a digit three in the two-factorial place yields invalid_input so students see the hard error path. This landing targets people searching for a factoradic converter, factorial number system calculator, permutation rank helper, or Lehmer-code digit check; it is not a general arbitrary-base converter, not a factorial-prime-factor tool, and not a full permutation generator that emits the ranked sequence of items. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when you fuzz illegal place digits. Round-trip checks are easy: convert a known decimal to factoradic, then convert the factoradic string back with mode set to from_factoradic, and confirm that decimal_string matches the original integer, then confirm that raising one digit past its place bound flips the response to invalid_input with a message that names the bad place.