For each \(n\geq 1\) let \(p_ n({\mathcal A})\) denote the number of essentially n-ary polynomials in \({\mathcal A}\) excluding the trivial unary polynomial \(e(x)=x\), while \(p_ 0({\mathcal A})\) is the number of constant unary polynomials in \({\mathcal A}\). A sequence of cardinals \((a_ 0,...,a_ m)\) is said to be representable if there exists an algebra \({\mathcal A}\) such that \(a_ n=p_ n({\mathcal A})\) for all \(n\leq m\). The sequence is said to have the minimal extension property (MEP) if there exists an algebra \({\mathcal A}^*\) representing the sequence such that \(p_ n({\mathcal A}^*)\leq p_ n({\mathcal A})\) for all \(n\geq 0\) and for all \({\mathcal A}\) representing the sequence. It is already known that the sequences (0,0,1) and (0,0,0,1) have MEP, further the minimal representing algebras have been characterized.
In this paper the author proves the following results: Let \({\mathcal A}_ m\) denote the sequence \((a_ 0,...,a_ m)\), where \(a_ 0=a_ 1=...=a_{m-1}=0\) and \(a_ m\) is the least integer such that the sequence is representable. If \(m>3\) then (i) \(a_ m=m\); (ii) \({\mathcal A}_ m\) has MEP; (iii) the minimal extension of \({\mathcal A}_ m\) is the sequence \(a_ n=n\cdot k^ m_ n\), \(n\geq 0\) and \(k^ m_ n\) opportune integers defined by induction; (iv) an algebra \({\mathcal A}\) represents the minimal extension of \({\mathcal A}_ m\) iff \({\mathcal A}\) is equivalent to a non trivial algebra belonging to the varieties \(K^*_ m\) or \(K^ 0_ m\) (there varieties are explicitly defined by a set of five very simple identities).