On Cobweb Posets and Discrete F-Boxes Tilings

Abstract F -boxes defined in [6] as hyper-boxes in N∞ discrete space were applied here for the geometric description of the cobweb posetes Hasse diagrams tilings. The F -boxes edges sizes are taken to be values of terms of natural numbers’ valued sequence F . The problem of partitions of hyper-boxes represented by graphs into blocks of special form is considered and these are to be called F -tilings. The proof of such tilings’ existence for certain sub-family of admissible sequences F is delivered. The family of F -tilings which we consider here includes among others F = Natural numbers, Fibonacci numbers, Gaussian integers with their corresponding F -nomial (Binomial, Fibonomial, Gaussian) coefficients as it is persistent typical for combinatorial interpretation of such tilings originated from Kwaśniewski cobweb posets tiling problem . Extension of this tiling problem onto the general case multi F nomial coefficients is here proposed. Reformulation of the present cobweb tiling problem into a clique problem of a graph specially invented for that purpose is proposed here too. To this end we illustrate the area of our reconnaissance by means of the Venn type map of various cobweb sequences families.