On partitions of a rectangle into rectangles with restricted number of cross sections

Consider partitions of a rectangle into rectangles with restricted number of cross sections. The problem has an information theoretic flavor (in the spirit of [B. Bollobás, Extremal Graph Theory, Academic Press, London, 1978]): richness of two dimensional pictures under 1–dimensional restrictions of from local information to global information. sent two approaches to upper counding the cardinality of such partitions: a combinatorial recursion and harmonic analysis.