2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77487This paper defines for each object $X$ that can be constructed out of a finite number of vertices and cells a vector $fX$ lying in a finite dimensional vector space. This is the flag vector of $X$. It is hoped that the quantum topological invariants of a manifold $M$ can be expressed as linear functions of the flag vector of the $i$-graph that arises from any suitable triangulation $T$ of $M$. Flag vectors are also defined for finite groups and more generally for $n$-ary relations. Some problems, and suggested connections with other constructions, particularly that of the associahedron and so on, conclude the presentation.LaTeX 2e, 8 pagesCombinatoricsGroup TheoryQuantum AlgebraRings and Algebras05;05CC65;52B05;57N;57QLtext