On the rank polynomial of the lattice of order ideals of fences and crowns
Given that a ranked poset \(P\) has a rank function \(r\): \(P \to N\) such that \(r(y)-r(x)=1\) for all elements \(x,y\in P\) with \(x\) covered by \(y\). When \(P\) is finite, the function \(r\) is usually chosen so that the minimal elements have rank 0. The height of \(P\) is the maximal rank, the Whitney number \(W_k(P)\) is then the number of all elements of rank \(k\) and the rank polynomial of \(P\) is the polynomial \(W(p;q)=\sum_{k\geq 0}W_k(P)q^k\). If \(Z_n\) is the \(n\)-fence: \(x_1< x_2> x_3\dots\) and \(C_n\) is the \(n\)-crown \(x_1< x_2> x_3<\cdots< x_{2n}> x_1\) then \(I(Z_n)\), the lattice of order ideals ordered by inclusion, is the Fibonacci (distributive) lattice of order \(n\) with \(|I(Z_n)|=f_n\), the \(n\)th Fibonacci number, while \(I(C_n)\) is the Lucas lattice of order \(n\) with \(|I(C_n)|=l_{2n}\), the corresponding Lucas number. Having made these observations it is clear that the rank polynomials of these lattices do not merely contain interesting information about these lattices but interesting relationships among Fibonacci and Lucas numbers also. Thus, among a number of interesting relations it is shown that \(W(I(Z_n);q)\) is rank unimodal (with mode level at \(\lfloor{n\over 2}\rfloor)\) and \(W(I(C_n);q)\) is rank unimodal (with mode level at level \(n)\) and rank symmetric for example. These observations have bearing on the distribution of the roots (zeros) of these polynomials for example.
scientific article; zbMATH DE number 1882147