Numbrilises

Numbrilise is a lossless encoding of natural numbers that represents the binary digits of a number as a run-length sequence of consecutive bits, starting with a run of ones. Each number maps to a unique sequence of positive integers that records the lengths of alternating runs of 1s and 0s in its binary representation.

Construction and decoding: To construct a numbrilise, write the number in binary without leading zeros, partition the string into runs of identical bits, and record the lengths of these runs in order. This sequence is the numbrilise. Decoding reverses the process: given a sequence [a1, a2, ..., ar], generate a binary string by appending a1 ones, then a2 zeros, and so on, alternating until the last run ends; interpret the result as a binary number and convert it to decimal.

Example: The number 13 has binary 1101, which consists of runs of two 1s, one 0, and one 1. Its numbrilise is [2, 1, 1]. Decoding [2, 1, 1] yields 1101 in binary, i.e., 13. The number 8 has binary 1000, with runs [1, 3], so its numbrilise is [1, 3].

Properties: The encoding is deterministic and reversible and provides insight into a number's binary structure, such as the number of bit transitions (one less than the number of runs). It is a form of run-length encoding applied to numeric data and can be more compact than the raw binary form when long runs occur.

Variants and use: Numbrilises have been discussed in theoretical contexts as a compact representation for teaching binary concepts or testing simple compression ideas. Variants include start-with-zero schemes, delta-encoded run lengths, and prefix-free encodings to ensure immediate decodability.