A Note on Möbius Function and Möbius Inversion Formula of Fibonacci Cobweb Poset

The explicit formula for Möbius function of Fibonacci cobweb poset P is given here for the first time by the use of Kwaśniwski’s definition of P in plane grid coordinate system [1]. 1 Fibonacci cobweb poset The Fibonacci cobweb poset P has been introduced by A.K.Kwaśniewski in [3, 4] for the purpose of finding combinatorial interpretation of fibonomial coefficients and their reccurence relation. At first the partially ordered set P (Fibonacci cobweb poset) was defined via its Hasse diagram as follows: It looks like famous rabbits grown tree but it has the specific cobweb in addition, i.e. it consists of levels labeled by Fibonacci numbers (the n-th level consist of Fn elements). Every element of n-th level (n ≥ 1, n ∈ N) is in partial order relation with every element of the (n + 1)-th level but it’s not with any element from the level in which he lies (n-th level) except from it. 2 The Incidence Algebra I(P) One can define the incidence algebra of P (locally finite partially ordered set) as follows (see [5, 6]): I(P) = {f : P × P −→ R; f(x, y) = 0 unless x ≤ y}. 1 The sum of two such functions f and g and multiplication by scalar are defined as usual. The product H = f ∗ g is defined as follows: h(x, y) = (f ∗ g)(x, y) = ∑ z∈P: x≤z≤y f(x, z) ∗ g(z, y). It is immediately verified that this is an associative algebra over the real field ( associative ring). The incidence algebra has an identity element δ(x, y), the Kronecker delta. Also the zeta function of P defined for any poset by: ζ(x, y) = { 1 for x ≤ y 0 otherwise is an element of I(P). The one for Fibonacci cobweb poset was expressed by δ in [1, 4] from where quote the result: ζ = ζ1 − ζ0 (1) where for x, y ∈ N: ζ1(x, y) = ∞ ∑