A semigroup S is called rectangular if every element acts as a middle identity, i.e., \(ler = lr\) holds for all \(e, l, r \in S\) . The class of rectangular semigroups naturally includes the subclasses of rectangular ideal extensions of left zero semigroups by null quotients, as well as rectangular ideal extensions of right zero semigroups by null quotients. It is shown that the number of pairwise nonisomorphic semigroups of order n arising as ideal extensions of a left (right) zero semigroup by a null quotient coincides with the number p(n) of partitions of n. We denote by \(\textrm{sie}(m,k)\) the number of pairwise nonisomorphic \((m\!+\!k)\) -element rectangular ideal extensions of a left (right) zero semigroup of order m by a null quotient semigroup, and prove that \(\textrm{sie}(k,k)= 1+ \textrm{sie}(k-1,k) = p(k)\) for every integer \(k>1\) . Moreover, we show that for each positive integer \(k\) the sequence \(\textrm{sie}(m,k)\) stabilises once \(m \ge k\) , that is, \(\textrm{sie}(m,k)=\textrm{sie}(k,k)\) for all \(m\ge k\) . Burnside-type formulas for \(\textrm{sie}(m,k)\) are derived and applied to compute these values for small parameters m and k using algorithms implemented in Python. Finally, a lower bound \(\begin{aligned} \textrm{rs}(n) \ge 2p(n) + \tau (n) - 3 \end{aligned}\) for the number \(\textrm{rs}(n)\) of all pairwise nonisomorphic rectangular semigroups of order n is established, where \(\tau (n)\) denotes the number of positive divisors of n, and it is shown that this bound is attained if and only if \(n\in \{2,3,4\}\) .
