Extremal hexagonal chains concerning k-matchings and k-independent sets

Denote by B-n, the set of the hexagonal chains with n hexagons. For any B-n is an element of B-n, let m(k)(B-n) and i(k)(B-n) be the numbers of k-matchings and k-independent sets of B-n, respectively. In the paper, we show that for any hexagonal chain B-n is an element of B-n and for any k greater than or equal to 0, m(k)(L-n) less than or equal to m(k)(B-n) less than or equal to m(k)(Z(n)) and i(k)(L-n) greater than or equal to i(k)(B-n) greater than or equal to i(k)(Z(n)), with left equalities holding for all k only if B-n = L-n, and the right equalities holding for all k only if B-n = Z(n), where L-n and Z(n) are the linear chain and the zig-zag chain, respectively. These generalize some related results known before.