Nombroses

Nombroses are a hypothetical class of combinatorial objects used in discussions of how numerical sequences can be represented within tree-like graphs. The term was proposed in mathematical folklore to explore connections between sequences, hierarchical encodings, and graph structures. In this context, a nombrose serves as a compact, code-like container for finite sets of integer sequences.

A nombrose is defined as a finite rooted directed acyclic graph with a distinguished root and a labeling function that assigns a nonnegative integer to every vertex. Edges are directed away from the root, and leaves have no children. Each path from the root to a leaf yields a sequence obtained by recording the labels along that path. The collection of all such root-to-leaf sequences constitutes the encoded information of the nombrose. Variants may impose that internal nodes have at least one child, or that leaves carry specific terminal values, to tailor the encoding to particular families of sequences.

Two common invariants associated with nombroses are the nombro sum and the nombro height. The nombro sum is the sum of labels over all vertices, providing a scalar measure of the graph’s labeling, while the nombro height is the maximum number of edges on a root-to-leaf path, reflecting depth. Variants may consider ordered versus unordered child sequences, or allow edge- or vertex-label generalizations.

Examples illustrate the idea. A root labeled 1 with two leaves labeled 0 yields two paths, each described by the sequence (1, 0). A simple chain with labels (2 → 1 → 0) encodes the single path (2, 1, 0). Nombroses remain a theoretical construct, with potential relevance to abstract enumeration, data encoding concepts, and the study of this type of hierarchical sequence representation. See also rooted trees and sequence encodings.