 Combinatorial number system

ID: combinatorial-number-system

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Combinatorial number system by

Codex  0 Created 2026-09-24 Updated 2026-11-05

At a fixed positive integer level

r

, the combinatorial number system writes each positive integer

m

uniquely as

m = (r a _{r}) + (r - 1 a _{r - 1}) + \dots + (s a _{s}), a_{r} > a_{r - 1} > \dots > a_{s} \geq s \geq 1.

Choose

a_{r}

as large as possible with

(r a _{r}) \leq m

, subtract that term and repeat at level

r - 1

, stopping when the remainder is zero. For example, at level three,

23 = (3 6) + (2 3) = 20 + 3.

Pascal's identity gives

(j a + 1) - (j a) = (j - 1 a)

, so the remainder after choosing the level-

j

term is smaller than

(j - 1 a)

. The next top index is consequently strictly smaller. Hockey-stick identity also bounds the sum of all lower admissible terms by

(j - 1 a) - 1

. Thus those terms cannot replace a larger level-

j

choice, proving uniqueness. The zero integer has the empty representation. This correspondence is useful for ranking and unranking fixed-size subsets.

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