2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102223The $F$-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the $F$-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the "$M=F$ Conjecture" of Armstrong which predicts a surprising relation between the $F$-triangle and the Möbius function of his $m$-divisible partition poset associated to a finite root system.AmS-TeX, 30 pages; style of equation numbering changed to the style in the printed publicationCombinatoricsRepresentation TheoryPrimary 05E15; Secondary 05A05 05A10 05A15 05A19 06A07 20F55 33C05text