Factorial Number System¶
A mixed-radix positional representation with factorial place values and digit bound 0 through i at the i-factorial place, giving every nonnegative integer a unique expansion and a natural rank/unrank bridge to permutations.
Core Idea¶
The Factorial Number System, or factoradic representation, is a mixed-radix positional system in which the digit at place (i) has weight (i!) and is restricted to
Every nonnegative integer (N) has a unique finite representation
The (0!) digit is necessarily zero; the (1!) digit is 0 or 1; the (2!) digit is 0, 1, or 2; and so on. Thus the successive digit radices are \(1,2,3,\ldots\), not one repeated base. “Factorial base” is conventional shorthand, but factorials are place values and the system is genuinely mixed-radix.
Scope of Application¶
The main application is ranking and unranking finite permutations. There are exactly (n!) permutations of (n) distinct elements and exactly (n!) digit vectors satisfying \(0\le a_i\le i\) for \(i=0,\ldots,n-1\). Interpreting the digits as successive indices into a shrinking ordered list gives a bijection between integers \(0,\ldots,n!-1\) and permutations.
The representation also appears in combinatorial generation, exhaustive search, state indexing for puzzles, compact serialization of orderings, random-permutation selection from an integer rank, and proofs involving mixed-radix enumeration.
Clarity¶
Indexing conventions cause most confusion. In this draft, (a_i) multiplies (i!) and ranges from 0 through (i). A written digit string \(a_m:a_{m-1}:\cdots:a_0\) therefore has (a_0=0). Some treatments omit the always-zero (0!) place or describe the (i)-th position using one-based indices. Equations should accompany notation.
Manages Complexity¶
The system aligns numeric capacity with combinatorial branching. At the first choice of an (n)-element permutation there are (n) possibilities; after one selection there are (n-1); the process continues to one. Factoradic digits have exactly those successive ranges. This removes wasted codes and makes every valid digit vector correspond to one permutation.
Abstract Reasoning¶
To represent an integer:
- Set the current radix to 1 and current value to (N).
- record the remainder under the current radix as the next digit.
- replace the value with the quotient and increment the radix.
- stop after the quotient becomes zero.
- read the digits from most significant to least and verify \(0\le a_i\le i\).
Knowledge Transfer¶
The mixed-radix method transfers literally to time units, calendar fields, and other systems where successive place capacities differ. The general design principle is to match each digit radix to the branching factor remaining at that stage.
The permutation correspondence transfers to other combinatorial classes only when their recursive counts support an appropriate ranking decomposition. Combinatorial number systems for subsets and Catalan-family rankings use analogous capacity-matched digits but not factorial place values; calling them factoradic would erase their different recurrence.
Relationships to Other Abstractions¶
Current abstraction Factorial Number System Domain-specific
Parents (1) — more general patterns this builds on
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Factorial Number System is a kind of Representation Prime
Representation is the proposed immediate parent.
Hierarchy path (1) — routes to 1 parentless root
- Factorial Number System → Representation → Abstraction
Neighborhood in Abstraction Space¶
Factorial Number System sits in a sparse region of the domain-specific corpus (79th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Superpartient Ratio — 0.83
- Divisor Function — 0.83
- Multiplicative Digital Root — 0.82
- Division Algorithm — 0.82
- Giuga Number — 0.81
Computed from structural-signature embeddings · 2026-09-08