Before presenting this special case, let us first discuss a more general situation for the *p*-Cockcroft property of semidirect products of monoids. In [8, 9], by considering a similar version of the picture {\mathbb{P}}_{S,x} in Figure 2, the second author investigated the *p*-Cockcroft property by using the trivializer for the semidirect product M=K{\u22ca}_{{\theta}_{2}}A, where *K* and *A* are arbitrary monoids. (It is seen that there is a single non-spherical subpicture {\mathbb{B}}_{S,x} in {\mathbb{P}}_{S,x}. In fact, {\mathbb{B}}_{S,x} contains only *S*-discs. For an illustration, see Figure 3.) As a special case of it, let us assume that *K* is a one-relator monoid and *A* is an infinite cyclic monoid ℤ with presentations

{\mathcal{P}}_{K}=\u3008\mathbf{y};{S}_{+}={S}_{-}\u3009\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{P}}_{A}=\u3008x;:\u3009,

respectively. Suppose *ψ* is an endomorphism of *K*. Then the mapping x\mapsto \psi induces a homomorphism {\theta}_{2}:A\to End(K), and we can form the semidirect product M=K{\u22ca}_{\theta}A. By (2), this product has a presentation

{\mathcal{P}}_{M}=\u3008\mathbf{y},x;{S}_{+}={S}_{-},\mathbf{t}\u3009,

(9)

where, for all y\in \mathbf{y}, the set **t** is the set of relators

{T}_{yx}:yx=x(y{\theta}_{x})

such that the relator *S* satisfies the condition \iota ({S}_{+})\ne \iota ({S}_{-}) (or \tau ({S}_{+})\ne \tau ({S}_{-})). In [8], the necessary and sufficient conditions for {\mathcal{P}}_{M} to be efficient are determined.

In the special case above, let us take *K* as a free abelian monoid of rank two (*i.e.* K={\mathbb{Z}}^{2}) presented by {\mathcal{P}}_{K}=\u3008{y}_{1},{y}_{2};{y}_{1}{y}_{2}={y}_{2}{y}_{1}\u3009, and let *ψ* be the endomorphism {\psi}_{\mathbf{M}}, where **M** is the matrix \left[\begin{array}{cc}\alpha & {\alpha}^{\mathrm{\prime}}\\ \beta & {\beta}^{\mathrm{\prime}}\end{array}\right] (\alpha ,{\alpha}^{\mathrm{\prime}},\beta ,{\beta}^{\mathrm{\prime}}\in {\mathbb{Z}}^{+}) given by [{y}_{1}]\mapsto [{y}_{1}^{\alpha}{y}_{2}^{{\alpha}^{\mathrm{\prime}}}] and [{y}_{2}]\mapsto [{y}_{1}^{\beta}{y}_{2}^{{\beta}^{\mathrm{\prime}}}]. As a special case of the presentation in (9), we obtain

{\mathcal{P}}_{M}=\u3008{y}_{1},{y}_{2},x;{y}_{1}{y}_{2}={y}_{2}{y}_{1},{y}_{1}x=x{y}_{1}^{\alpha}{y}_{2}^{{\alpha}^{\mathrm{\prime}}},{y}_{2}x=x{y}_{1}^{\beta}{y}_{2}^{{\beta}^{\mathrm{\prime}}}\u3009

(10)

for the monoid M={\mathbb{Z}}^{2}{\u22ca}_{{\theta}_{2}}\mathbb{Z} (see [9]). Again, in the same reference, the second author figured out the efficiency of the above presentation as in the following proposition.

**Proposition 2** ([9])

*For any prime* *p*, *the presentation* {\mathcal{P}}_{M} *in* (10) *is* *p*-*Cockcroft if and only if* det\mathbf{M}\equiv 1(modp).

According to Proposition 2, in particular, {\mathcal{P}}_{M} is not efficient if det\mathbf{M}=0 or 2. Therefore the following proposition is proved in the same manner.

**Proposition 3** ([9])

*The presentation* {\mathcal{P}}_{M} *in* (10) *is minimal but inefficient if* det\mathbf{M}=2.

The proof of Proposition 3 is based on the following Pride result, which is a monoid version of Lemma 1. Although this result has not been published yet, it has been used in many papers (see, for instance, [8–10]).

**Lemma 2** (Pride)

*For any monoid* *M* *with a presentation* \mathcal{P} *as in* (1), *let* *ψ* *be a ring homomorphism from* \mathbb{Z}M *into the ring of all* m\times m-*matrices* (m\ge 1) *over some commutative ring* ℛ *with* 1, *and suppose* \psi (1)={I}_{m\times m}. *If the second Fox ideal* {I}_{2}^{(l)}(\mathcal{P}) *is contained in the kernel of* *ψ*, *then* \mathcal{P} *is minimal*.

From now on, by considering Propositions 2 and 3, we will reach our main aim of this paper for monoids.

Our first result in this section gives the connection between a monoid presentation and array polynomials. In fact the *array polynomials* {S}_{k}^{n}(x) are defined by means of the following generating function:

\frac{{({e}^{t}-1)}^{k}{e}^{tx}}{x!}=\sum _{n=0}^{\mathrm{\infty}}{S}_{k}^{n}(x)\frac{{t}^{n}}{n!},

(*cf.* [29–31]). According to the same references, array polynomials can also be defined in the form

{S}_{k}^{n}(x)=\frac{1}{k!}\sum _{j=0}^{k}{(-1)}^{k-j}\left(\genfrac{}{}{0ex}{}{k}{j}\right){(x+j)}^{n}.

(11)

Since the coefficients of array polynomials are integers, these polynomials find a very large application area, especially in engineering. Array polynomials are used, for instance, in system control (*cf.* [32]).

In fact these integer coefficients give us an opportunity to use these polynomials in our case. We should note that there also exist some other polynomials, namely Dickson, Bell, Abel, Mittag-Leffler *etc.*, which have integer coefficients. But, since array polynomials have a larger application area in science, we have preferred them. Hence, by considering Proposition 3, we obtain the following theorem as another main result.

**Theorem 2**
*Let us consider the monoid*
M={\mathbb{Z}}^{2}{\u22ca}_{{\theta}_{2}}\mathbb{Z}
*with a presentation*

{\mathcal{P}}_{M}=\u3008{b}_{1},{b}_{2},a;{b}_{1}{b}_{2}={b}_{2}{b}_{1},{b}_{1}a=a{b}_{1}^{2},{b}_{2}a=a{b}_{1}{b}_{2}\u3009.

(12)

*Then*
{\mathcal{P}}_{M}
*has a set of generating functions*

\begin{array}{l}{p}_{1}(a)={S}_{n}^{n}(a)-2{S}_{0}^{1}(a)\\ {p}_{2}({b}_{1})={S}_{n}^{n}({b}_{1})-{S}_{0}^{1}({b}_{1})\\ {p}_{3}({b}_{2})={S}_{0}^{1}({b}_{2})-{S}_{n}^{n}({b}_{2})\end{array}\},

(13)

*where* {S}_{k}^{n}(x) *is defined as in* (11).

*Proof* Let us consider the spherical picture {\mathbb{P}}_{S,a} with its non-spherical subpicture {\mathbb{B}}_{S,a} as drawn in Figure 3. In fact, by [9], this is the only picture in the trivializer of \mathcal{D}({\mathcal{P}}_{M}).

In presentation (12), let us label the relators {b}_{1}{b}_{2}={b}_{2}{b}_{1}, {b}_{1}a=a{b}_{1}^{2} and {b}_{2}a=a{b}_{1}{b}_{2} by *S*, {T}_{{b}_{1},a} and {T}_{{b}_{2},a}, respectively. It is clear that {exp}_{S}({\mathbb{P}}_{S,a})=1-2=-1, {exp}_{{T}_{{b}_{1},a}}({\mathbb{P}}_{S,a})=1-1=0 and {exp}_{{T}_{{b}_{2},a}}({\mathbb{P}}_{S,a})=1-1=0. In the calculation of these exponent sums, we included the exponent sums of *S*-discs in the non-spherical picture {\mathbb{B}}_{S,a}. Actually, a simple calculation shows that det\mathbf{M}={exp}_{S}({\mathbb{B}}_{S,a}) and so, by our assumption about {\mathcal{P}}_{M} that is not efficient, we expect {exp}_{S}({\mathbb{B}}_{S,a}) to be 2.

Now, by (4) and (5), the evaluation of {\mathbb{P}}_{S,a} is determined as follows:

{eval}^{(l)}({\mathbb{P}}_{S,a})=(1-\overline{2}\overline{a}){e}_{S}+(1-\overline{{b}_{1}}){e}_{{T}_{{b}_{1},a}}+(\overline{{b}_{2}}-1){e}_{{T}_{{b}_{2},a}}.

Therefore, by the definition, the second Fox ideal {I}_{2}^{(l)}({\mathcal{P}}_{M}) of the presentation {\mathcal{P}}_{M} in (12) is generated by the polynomial elements

1-\overline{2}\overline{a},\phantom{\rule{2em}{0ex}}1-\overline{{b}_{1}},\phantom{\rule{2em}{0ex}}\overline{{b}_{2}}-1.

For simplicity, let us replace each of \overline{2}\overline{a}, \overline{{b}_{1}} and \overline{{b}_{2}} by 2*a*, {b}_{1} and {b}_{2}, respectively. In [9], by considering Lemma 2, it has been showed that this presentation in (12) is minimal.

Now, by using (11) and keeping in our mind that the coefficients of array polynomials are integer, we clearly have

{S}_{k}^{n}(x)=\{\begin{array}{cc}{x}^{n};\hfill & k=0,\hfill \\ x;\hfill & k=0\text{and}n=1,\hfill \\ 1;\hfill & k=n\text{or}n=k=0.\hfill \end{array}

Then, by reformulating the elements of the second Fox ideal {I}_{2}^{(l)}({\mathcal{P}}_{M}), we arrive at the functions in (13) as desired. □

By considering Proposition 2, if we take det\mathbf{M}\ne 2, then we get an efficient presentation. So, for an even prime *p*, let det\mathbf{M}=3. Then one of the presentations of the similar form {\mathcal{P}}_{M} as in (12) can be taken as

{\mathcal{P}}_{M}=\u3008{b}_{1},{b}_{2},a;{b}_{1}{b}_{2}={b}_{2}{b}_{1},{b}_{1}a=a{b}_{1}^{3},{b}_{2}a=a{b}_{1}{b}_{2}\u3009,

(14)

which will be efficient. The same procedure in the proof of Theorem 2 gives us the set of generating functions of {\mathcal{P}}_{M} in (14) in the form {p}_{1}(a), {p}_{2}({b}_{1}) and {p}_{3}({b}_{2}), where {p}_{1}(a)={S}_{n}^{n}(a)-3{S}_{0}^{1}(a) and the others are defined in (13) such that {S}_{k}^{n}(x) is given in (11). Nevertheless, by induction steps, we can generalise this last presentation as follows:

{\mathcal{P}}_{M}=\u3008{b}_{1},{b}_{2},a;{b}_{1}{b}_{2}={b}_{2}{b}_{1},{b}_{1}a=a{b}_{1}^{det\mathbf{M}},{b}_{2}a=a{b}_{1}{b}_{2}\u3009.

(15)

Hence we get the following version of Theorem 2 which deals with efficient presentations.

**Theorem 3** *Let us consider the presentation* {\mathcal{P}}_{M} *in* (15) *for the monoid* M={\mathbb{Z}}^{2}{\u22ca}_{{\theta}_{2}}\mathbb{Z}. *Then* {\mathcal{P}}_{M} *has a set of generating functions*

\begin{array}{l}{p}_{1}(a)={S}_{n}^{n}(a)-det\mathbf{M}{S}_{0}^{1}(a)\\ {p}_{2}({b}_{1})={S}_{n}^{n}({b}_{1})-{S}_{0}^{1}({b}_{1})\\ {p}_{3}({b}_{2})={S}_{0}^{1}({b}_{2})-{S}_{n}^{n}({b}_{2})\end{array}\},

(16)

*where* det\mathbf{M}\ne 2 *and* {S}_{k}^{n}(x) *is defined as in* (11).

**Remark 5** According to the expression in Remark 1, presentations given in (12), (14) or (15) have a minimal number of generators. But we classified these presentations according to their efficiency status separately in Theorem 2 and Theorem 3. The aim of this separation is to find a solution for a general remark depicted in the final section about obtaining a method for a minimality test by using generating functions (see Section 4 below).

At this point, we should note that for {t}_{1}\ne {t}_{2}\in {\mathbb{R}}^{+}, \lambda \in \mathbb{C}, k\in {\mathbb{N}}_{0}, generalised array type polynomials {\mathcal{S}}_{k}^{n}(x;{t}_{1},{t}_{2};\lambda ) which are related to the non-negative real parameters have been recently developed and some elementary properties including recurrence relations of these polynomials have been obtained [30]. In fact, by setting {t}_{1}=\lambda =1 and {t}_{2}=e, the equation (11) is obtained.

**Remark 6** One can try to study the generalisation of Theorem 2 by using {\mathcal{S}}_{k}^{n}(x;{t}_{1},{t}_{2};\lambda ).

The remaining goal of this section is to establish a connection between the presentation {\mathcal{P}}_{M} in (12) or (15) and Stirling numbers of the second kind (*cf.* [3, 30, 33–36]). In fact, *Stirling numbers* of the second kind S(n,k) are defined by means of the following generating function:

\frac{{({e}^{t}-1)}^{k}}{k!}=\sum _{n=0}^{\mathrm{\infty}}S(n,k)\frac{{t}^{n}}{n!}

(see [3, 36]). According to [[30], Theorem 1, Remark 2], Stirling numbers can also be defined by

S(n,k)=\frac{1}{k!}\sum _{j=0}^{k}{(-1)}^{j}\left(\genfrac{}{}{0ex}{}{k}{j}\right){(k-j)}^{n}.

We remind that these numbers satisfy the well-known properties

S(n,k)=\{\begin{array}{cc}1;\hfill & k=1\text{or}k=n,\hfill \\ \left(\genfrac{}{}{0ex}{}{n}{2}\right);\hfill & k=n-1,\hfill \\ {\delta}_{n,0};\hfill & k=0,\hfill \end{array}

where {\delta}_{n,0} denotes the Kronecker symbol (see [3, 36]). It is known that Stirling numbers are used in combinatorics, in number theory, in discrete probability distributions for finding higher-order moments, *etc.* We finally note that since S(n,k) is the number of ways to partition a set of *n* objects into *k* groups, these numbers find an application area in combinatorics and in the theory of partitions.

In addition to the above formulas for S(n,k), by [30, 35, 36], we have

{x}^{n}=\sum _{k=0}^{n}\left(\genfrac{}{}{0ex}{}{x}{k}\right)k!S(n,k)

(17)

as a formula for Stirling numbers. Therefore, in equation (17) by replacing *x* with *a*, {b}_{1} and {b}_{2}, respectively, and taking n=1, n=0, the polynomial elements of the second Fox ideal {I}_{2}^{(l)}({\mathcal{P}}_{M}) of the presentation {\mathcal{P}}_{M} in (12) can be restated as follows:

\begin{array}{l}{a}^{0}-2{a}^{1}=\sum _{k=0}^{0}\left(\genfrac{}{}{0ex}{}{a}{k}\right)k!S(0,k)-2\sum _{k=0}^{1}\left(\genfrac{}{}{0ex}{}{a}{k}\right)k!S(1,k)\\ {b}_{1}^{0}-{b}_{1}^{1}=\sum _{k=0}^{0}\left(\genfrac{}{}{0ex}{}{{b}_{1}}{k}\right)k!S(0,k)-\sum _{k=0}^{1}\left(\genfrac{}{}{0ex}{}{{b}_{1}}{k}\right)k!S(1,k)\\ {b}_{2}^{1}-{b}_{2}^{0}=\sum _{k=0}^{1}\left(\genfrac{}{}{0ex}{}{{b}_{2}}{k}\right)k!S(1,k)-\sum _{k=0}^{0}\left(\genfrac{}{}{0ex}{}{{b}_{2}}{k}\right)k!S(0,k)\end{array}\}.

(18)

As a different version of Theorem 2, we express the following corollary.

**Corollary 3** *The presentation* {\mathcal{P}}_{M} *in* (12) *has a set of generating functions in terms of Stirling numbers as given in* (18).

We note that the above corollary can also be stated for the presentation {\mathcal{P}}_{M} in (15).

Furthermore, in a recent work, Simsek [30] has constructed the *generalised* *λ-Stirling numbers of the second kind* \mathcal{S}(n,v;a,b;\lambda ) related to non-negative real parameters (a,b\in {\mathbb{R}}^{+}, a\ne b, *λ* is a complex number and v\in {\mathbb{N}}_{0}). In fact, this new generalisation is defined by the generating function

{f}_{S,v}(t;a,b;\lambda )=\frac{{(\lambda {b}^{t}-{a}^{t})}^{v}}{v!}=\sum _{n=0}^{\mathrm{\infty}}\mathcal{S}(n,v;a,b;\lambda )\frac{{t}^{n}}{n!}.

(19)

By setting a=1 and b=e in (19), one can obtain the *λ*-Stirling numbers of the second kind S(n,v;\lambda ) which are defined by the generating function

\frac{{(\lambda {e}^{t}-1)}^{v}}{v!}=\sum _{n=0}^{\mathrm{\infty}}S(n,v;\lambda )\frac{{t}^{n}}{n!}

(see [3, 36]). By substituting \lambda =1 into the above equation, the Stirling numbers of the second kind S(n,v)are obtained.

By considering this new generalisation \mathcal{S}(n,v;a,b;\lambda ), in [[30], Theorem 1], it has been obtained that

\mathcal{S}(n,v;a,b;\lambda )=\frac{1}{v!}\sum _{j=0}^{n}{(-1)}^{j}\left(\genfrac{}{}{0ex}{}{v}{j}\right){\lambda}^{v-j}{(jlna+(v-j)lnb)}^{n},

(20)

for *λ*-Stirling numbers of the second kind. In fact, by setting a=1 and b=e in (20), one can get the following equality on *λ*-Stirling numbers:

S(n,v;\lambda )=\frac{1}{v!}\sum _{j=0}^{v}\left(\genfrac{}{}{0ex}{}{v}{j}\right){\lambda}^{(v-j)}{(-1)}^{j}{(v-j)}^{n}

(21)

(see [3, 36]).

Hence we can present the following notes about this section:

**Remark 7** It is clearly seen that in Theorems 2, 3 and Corollary 3, only Stirling numbers are considered. However, one can also study the *λ*-Stirling numbers S(n,v;\lambda ) defined in (21) and generalised *λ*-Stirling numbers \mathcal{S}(n,v;a,b;\lambda ) defined in (20) as stated in these theorems and corollaries.

**Remark 8** For a suitable {\mathbf{M}}_{n\times n} matrix, it is possible to define the presentation {\mathcal{P}}_{M} in (9) (or in (10)) for the monoid {\mathbb{Z}}^{n}{\u22ca}_{{\theta}_{2}}\mathbb{Z}. Thus one can try to transform all studies in Section 3 to this general case.