On the rank polynomial of the lattice of order ideals of fences and crowns Original Research Article

In this paper, we study the rank polynomial of the distributive lattice of order ideals of fences and crowns. In particular, we prove the unimodality of these polynomials, we find their generating functions and we show that they can be expressed in terms of Chebyshev polynomials. Moreover, we obtain combinatorially an explicit formula for the Whitney numbers. Finally, we find generating functions and recurrences for the sequences of the maxima of such numbers.