Sweet division problems: from chocolate bars to honeycomb strips and back

Abstract

We consider two division problems on narrow strips of square and hexagonal lattices. In both cases we compute the bivariate enumerating sequences and the corresponding generating functions, which allowed us to determine the asymptotic behavior of the total number of such subdivisions and the expected number of parts. For the square lattice we extend results of two recent references by establishing polynomiality of enumerating sequences forming columns and diagonals of the triangular enumerating sequence. In the hexagonal case, we find a number of new combinatorial interpretations of the Fibonacci numbers and find combinatorial proofs of some Fibonacci related identities. We also show how both cases could be treated via the transfer matrix method and discuss some directions for future research.