Explicit formulas for multiplicity of arbitrary Hibi rings

Develop explicit formulas to compute the multiplicity of an arbitrary Hibi ring by determining the number of maximal chains in the associated finite distributive lattice of join-irreducible elements. This should provide closed-form expressions beyond special families, where the multiplicity is known to equal the number of maximal chains but currently lacks general explicit formulas.

Background

In the paper, the authors paper special fiber rings of ladder determinantal modules and show, via Sagbi and Gröbner degeneration, that the initial algebra is a Hibi ring associated to a distributive lattice. For such Hibi rings, classical results identify the multiplicity with the number of maximal chains in the underlying lattice.

While the authors derive explicit multiplicity formulas for their ladder determinantal modules by translating the problem into counting standard skew Young tableaux of certain skew partitions, they note that for arbitrary Hibi rings no general explicit formulas exist to compute the number of maximal chains—and thus the multiplicity—despite the conceptual identification provided by known theorems.