Extremal polyomino chains on k-matchings and k-independent sets

Denote by T-n the set of polyomino chains with n squares. For any T-n is an element of T-n, let m(k)(T-n) and i(k)(T-n) be the number of k-matchings and k-independent sets of T-n, respectively. In this paper, we show that for any polyomino chain T-n is an element of T-n and any k >= 0, m(k)(L-n) >= m(k) (T-n) >= m(k) (Z(n)) and i(k) (L-n) <= i(k) (T-n) <= i(k) (Z(n)), with the left equalities holding for all k only if Tn = Ln, and the right equalities holding for all k only if T-n = Z(n), where L-n and Z(n) are the linear chain and the zig-zag chain, respectively.