Some Generalizations of Classical Integer Sequences Arising in Combinatorial Representation Theory

There exists a natural correspondence between the bases for a given finite-dimensional representation of a complex semisimple Lie algebra and a certain collection of finite edge-colored ranked posets, laid out by Donnelly, et al. in, for instance, [Don03]. In this correspondence, the Serre relations on the Chevalley generators of the given Lie algebra are realized as conditions on coefficients assigned to poset edges. These conditions are the so-called diamond, crossing, and structure relations (hereinafter DCS relations.) New representation constructions of Lie algebras may thus be obtained by utilizing edge-colored ranked posets. Of particular combinatorial interest are those representations whose corresponding posets are distributive lattices. We study two families of such lattices, which we dub the generalized Fibonaccian lattices LFⁱᵇpn`1, kq and generalized Catalanian lattices LCᵃᵗpn, kq. These respectively generalize known families of lattices which are DCS-correspondent to some special families of representations of the classical Lie algebras An`₁ and Cn. We state and prove explicit formulae for the vertex cardinalities of these lattices; show existence and uniqueness of DCS-satisfactory edge coefficients for certain values of n and k; and report on the efficacy of various computational and algorithmic approaches to this problem. A Python library for computationally modeling and “solving” these lattices appears as an appendix