2014/12/18 by Kalle Karu, Karu, Kalle
Mathematics · #05E45 #06A07 #52B05 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #math.CO #msc:05E45 #msc:06A07 #msc:52B05
paper · pdf · doi:10.48550/arxiv.1412.6048
13 pages, 3 figures
arxiv created 2014/12/18 · openalex publication_date 2014/12/18 · arxiv updated 2014/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A well-known conjecture of McMullen, proved by Billera, Lee and Stanley, describes the face numbers of simple polytopes. The necessary and sufficient condition is that the toric g-vector of the polytope is an M-vector, that is, the vector of dimensions of graded pieces of a standard graded algebra A. Recent work by Murai, Nevo and Yanagawa suggests a similar condition for the coefficients of the cd-index of a poset P. The coefficients of the cd-index are conjectured to be the dimensions of graded pieces in a standard multigraded algebra A. We prove the conjecture for simplicial spheres and we give numerical evidence for general shellable spheres. In the simplicial case we construct the multi-graded algebra A explicitly using lattice paths.