Let X be a finite set having n elements. How many different labeled topologies one can define on X? Let T (n, k) be the number of topologies having k open sets. We compute T (n, k) for 2 ≤ k ≤ 12, as well as other results concerning T 0 topologies on X having n + 4 ≤ k ≤ n + 6 open sets.
Let L(n, k) = n n−k n−k k. We prove that all the zeros of the polynomial L n (x) = k≥0 L(n, k)x k are real. The sequence L(n, k) is thus strictly log-concave, and hence unimodal with at most two consecutive maxima. We determine those integers where the maximum is reached. In the last section we prove that L(n, k) satisfies a central limit theorem as well as… (More)
The independence polynomial of the graph called the centipede has only real zeros. It follows that this polynomial is log-concave, and hence unimodal. Levit and Man-drescu gave a conjecture about the mode of this polynomial. In this paper, the exact value of the mode is determined, and some central limit theorems for the sequence of the coefficients are… (More)
Let E be a set with n elements, and let T (n, k) be the number of all labeled topologies having k open sets that can be defined on E. In this paper, we compute these numbers for k ≤ 17, and arbitrary n, as well as t N 0 (n, k), the number of all unlabeled non-T 0 topologies on E with k open sets, for 3 ≤ k ≤ 8.