  We introduce a natural partial order in structurally natural finite subsets
of the cobweb prefabs sets recently constructed by the present author. Whitney
numbers of the second kind of the corresponding subposet which constitute
Stirling like numbers triangular array are then calculated and the explicit
formula for them is provided. Next, in the second construction we endow the set
sums of prefabiants with such an another partial order that their Bell like
numbers include Fibonacci triad sequences introduced recently by the present
author in order to extend famous relation between binomial Newton coefficients
and Fibonacci numbers onto the infinity of their relatives among whom there are
also the Fibonacci triad sequences and binomial like coefficients (incidence
coefficients included). The first partial order is F sequence independent while
the second partial order is F sequence dependent where F is the so called
admissible sequence determining cobweb poset by construction. An F determined
cobweb posets Hasse diagram becomes Fibonacci tree sheathed with specific
cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the
stand-point of linear algebra of formal series these are generating functions
which stay for the so called extended coherent states of quantum physics. This
information is delivered in the last section.
