In fact, this is the main section of the paper and it will be given as two subsections under the names of Part I and Part II. Since we will define generating functions by considering the exponent sums of the generating pictures over the presentation of this semidirect product, the first subsection is aimed to define these generating pictures and the related results about them.
2.1 Part I: generating pictures
In this subsection, we will mainly present the efficiency (equivalently, pCockcroft property for a prime p by Theorem 1.1) for the semidirect products of finite cyclic monoids.
Let A and K be two finite cyclic monoids with presentations
{\mathcal{P}}_{A}=[x;{x}^{\mu}={x}^{\lambda}]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{P}}_{K}=[y;{y}^{k}={y}^{l}]
(4)
respectively, where l,k,\lambda ,\mu \in {\mathbb{Z}}^{+} such that l<k and \lambda <\mu, or equivalently,
\mu =\lambda +r\phantom{\rule{1em}{0ex}}(1\le r\le \mu 1)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}k=l+\omega \phantom{\rule{1em}{0ex}}(1\le \omega \le k1).
(5)
Due to [10], a trivializer set {\mathbf{X}}_{\mathbf{K}} (and similarly {\mathbf{X}}_{\mathbf{A}}) of the Squier complex \mathcal{D}({\mathcal{P}}_{K}) (and similarly \mathcal{D}({\mathcal{P}}_{A})) is given by the pictures {\mathbb{P}}_{k,l}^{m} (1\le m\le k1), as in Figure 1.
Let {\psi}_{i} (0\le i\le k1) be an endomorphism of K. Then we have a mapping x\u27f6End(K), x\mapsto {\psi}_{i}. In fact this induces a homomorphism \theta :A\u27f6End(K), x\mapsto {\psi}_{i} if and only if {\psi}_{i}^{\mu}={\psi}_{i}^{\lambda}. Since {\psi}_{i}^{\mu} and {\psi}_{i}^{\lambda} are equal if and only if they agree on the generator y of K, we must have
\left[{y}^{{i}^{\mu}}\right]=\left[{y}^{{i}^{\lambda}}\right].
(6)
We then have the semidirect product M=K{\u22ca}_{\theta}A and, by [10], a standard presentation
{\mathcal{P}}_{M}=[y,x;S,R,{T}_{yx}],
(7)
as in (3), for the monoid M where
S:{y}^{k}={y}^{l},\phantom{\rule{2em}{0ex}}R:{x}^{\mu}={x}^{\lambda}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{T}_{yx}:yx=x{y}^{i}.
In the rest of the paper, we will assume that the equality in Equation (6) holds when we talk about the semidirect product M of K by A.
The subpicture {\mathbb{B}}_{S,x} can be drawn as in Figure 2(a), and in fact, by considering this subpicture, we clearly have
{exp}_{S}({\mathbb{B}}_{S,x})=i.
As it is seen in Figure 2(b), we also have the subpicture {\mathbb{A}}_{{R}_{+},y} (and similarly {\mathbb{A}}_{{R}_{},y}) with
{exp}_{{T}_{yx}}({\mathbb{A}}_{{R}_{+},y})=1+i+{i}^{2}+\cdots +{i}^{\mu 1}=\frac{{i}^{\mu}1}{i1}
and
{exp}_{{T}_{yx}}({\mathbb{A}}_{{R}_{},y})=1+i+{i}^{2}+\cdots +{i}^{\lambda 1}=\frac{{i}^{\lambda}1}{i1}.
By equality (6), we must have [{y}^{{i}^{\mu}}]=[{y}^{{i}^{\lambda}}]. Hence, by [10], the subpicture {\mathbb{C}}_{y,{\theta}_{R}} with
\iota ({\mathbb{C}}_{y,{\theta}_{R}})={y}^{{i}^{\mu}},\phantom{\rule{2em}{0ex}}\tau ({\mathbb{C}}_{y,{\theta}_{R}})={y}^{{i}^{\lambda}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{exp}_{S}({\mathbb{C}}_{y,{\theta}_{R}})=\frac{{i}^{\mu}{i}^{\lambda}}{kl}
can be depicted as in Figure 3.
After all, the whole generating pictures {\mathbb{P}}_{S,x} and {\mathbb{P}}_{R,y} can be drawn as in Figure 4.
The following result states necessary and sufficient conditions for the presentation of the split extension of two finite monogenic monoids to be efficient.
Proposition 2.1 ([18])
Let p be a prime. Suppose that K{\u22ca}_{\theta}A is a monoid with the associated monoid presentation {\mathcal{P}}_{M}, as in (7). Then {\mathcal{P}}_{M} is pCockcroft (equivalently efficient) if and only if
p\mid kl,\phantom{\rule{2em}{0ex}}p\mid i1,\phantom{\rule{2em}{0ex}}p\frac{{i}^{\mu}{i}^{\lambda}}{kl},\phantom{\rule{2em}{0ex}}p\frac{{i}^{\mu}{i}^{\lambda}}{i1}.
Remark 2.2 To be an example of Proposition 2.1, one can take

k=10, l=6, \mu =4, \lambda =2 or k=6, l=2, \mu =5, \lambda =3 while p=2 and i=3 to get 2Cockcroft property for the presentation {\mathcal{P}}_{M} in (7), or more generally

for any prime p, k=(p+1)[\frac{{(p+1)}^{p}1}{p}]+1, l=1, \mu =p+1=i and \lambda =1 to get pCockcroft property for the presentation {\mathcal{P}}_{M} in (7).
Considering Theorem 1.1, one can say that the monoid presentation {\mathcal{P}}_{M}, as in (7), is efficient if and only if there is a prime p such that
In particular, if we choose {exp}_{S}({\mathbb{B}}_{S,x})=i=0 or 2, then {\mathcal{P}}_{M} will be inefficient.
Recall that, by the meaning of finite cyclic monoids, {exp}_{y}(S)=kl cannot be equal to 0. We also note that a similar proof for the following result about minimal but inefficiency of {\mathcal{P}}_{M} can be found in [18].
Proposition 2.3 Let M be the semidirect product of K by A, and let {\mathcal{P}}_{M}, as in (7), be the presentation for M where l,k,\lambda ,\mu ,i\in {\mathbb{Z}}^{+} and l<k, \lambda <\mu. If i=2 and the subtraction kl is not even and not equal to 1, then {\mathcal{P}}_{M} is minimal but inefficient.
Remark 2.4 To be an example of Proposition 2.3, we can consider the following:

For an odd positive integer t, one can take k=4t, l=t, \mu =3t, \lambda =t and i=2 in the presentation given in (7). Since kl=3t\ne 2n (n\in {\mathbb{Z}}^{+}), the presentation is minimal but inefficient.

For all s,t\in {\mathbb{Z}}^{+} such that s<t, one can take k=2t+1, l=2s, \mu =kl, \lambda =1 and i=2 in the presentation given in (7). Since kl=2(ts)+1\ne 2n (n\in {\mathbb{Z}}^{+}), the presentation is minimal while it is inefficient.
2.2 Part II: generating functions
By considering the pictures defined in the previous section and also the evaluations obtained from them, we will define the related generating functions. In another words, by taking into account Propositions 2.1 and 2.3, we will reach our main aim over monoids of this paper.
We firstly recall that, as noted in [[4], Remark 1.1], if a monoid presentation satisfies efficiency or inefficiency (while it is minimal), then it always has a minimal number of generators. Working with the minimal number of elements gives a great opportunity to define related generating functions over this presentation. This will be one of the key points in our results.
Our first result of this section is related to the connection of the monoid presentation in (7) with array polynomials. In fact array polynomials {S}_{a}^{n}(x) are defined by means of the generating function
\frac{{({e}^{t}1)}^{a}{e}^{tx}}{x!}=\sum _{n=0}^{\mathrm{\infty}}{S}_{a}^{n}(x)\frac{{t}^{n}}{n!}
(cf. [19–21]). According to the same references, array polynomials can also be defined as the form
{S}_{a}^{n}(x)=\frac{1}{a!}\sum _{j=0}^{a}{(1)}^{aj}\left(\genfrac{}{}{0ex}{}{a}{j}\right){(x+j)}^{n}.
(8)
Since the coefficients of array polynomials are integers, they find very large application area, especially in system control (cf. [22]). In fact, these integer coefficients give us the opportunity to use these polynomials in our case. We should note that there also exist some other polynomials, namely Dickson, Bell, Abel, MittagLeffler etc., which have integer coefficients which will not be handled in this paper.
From (5), we know that \mu =\lambda +r, where 1\le r\le \mu 1. Hence, by considering the meaning and conditions of Proposition 2.1, we obtain the following theorem as one of the main results of this paper.
Theorem 2.5 The efficient presentation {\mathcal{P}}_{M} defined in (7) has a set of generating functions
\begin{array}{l}{p}_{1}(x)={S}_{n}^{n}(x)i{S}_{0}^{1}(x),\phantom{\rule{2em}{0ex}}{p}_{2}(y)=(kl){S}_{n}^{n}(y),\\ {p}_{3}(x)=\frac{{i}^{\lambda}({i}^{r}1)}{i1}{S}_{n}^{n}(x),\phantom{\rule{2em}{0ex}}{p}_{4}(y)=\frac{{i}^{\lambda}({i}^{r}1)}{kl}{S}_{n}^{n}(y),\end{array}\}
(9)
where {S}_{a}^{n}(x) and {S}_{a}^{n}(y) are defined as in (8).
Proof Let us consider the generating pictures {\mathbb{P}}_{S,x}, {\mathbb{P}}_{R,y} (in Figure 4) with their nonspherical subpictures defined in Figures 2 and 3, and the generating pictures of finite monogenic monoids defined in Figure 1. Recall that by counting the exponent sums of the discs R, S and {T}_{yx} in the related pictures, the conditions of Proposition 2.1 have been obtained [18]. (For more similar results and applications, one can see the papers [10, 11].)
To reach our aim in the proof, we first need to calculate {eval}^{(l)}({\mathbb{P}}_{S,x}), {eval}^{(l)}({\mathbb{P}}_{R,y}), {eval}^{(l)}({\mathbb{P}}_{k,l}^{m}) (1\le m\le k1) and {eval}^{(l)}({\mathbb{P}}_{\lambda +r,\lambda}^{n}) (1\le n\le (\lambda +r)1). By Equations (1) and (2), we have
where \frac{\partial}{\partial y} denotes the Fox derivation [23]. Also, for each 1\le m\le k1 and 1\le n\le (\lambda +r)1,
{eval}^{(l)}\left({\mathbb{P}}_{k,l}^{m}\right)=(1{y}^{km}){e}_{S}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{eval}^{(l)}\left({\mathbb{P}}_{\lambda +r,\lambda}^{n}\right)=(1{x}^{(\lambda +r)n}){e}_{R}.
Therefore, by the definition, the second Fox ideal {I}_{2}^{(l)}({\mathcal{P}}_{M}) of the presentation {\mathcal{P}}_{M} in (7) is generated by the polynomial elements
\begin{array}{l}1x({eval}^{(l)}({\mathbb{B}}_{S,x})),\phantom{\rule{2em}{0ex}}\frac{{\partial}^{M}S}{\partial y},\\ {eval}^{(l)}({\mathbb{A}}_{{R}_{+},x}){eval}^{(l)}({\mathbb{A}}_{{R}_{},x}),\phantom{\rule{2em}{0ex}}{eval}^{(l)}({\mathbb{C}}_{y,{\theta}_{R}}),\\ 1{y}^{k1},1{y}^{k2},\dots ,1y,\phantom{\rule{2em}{0ex}}1{x}^{(\lambda +r)1},1{x}^{(\lambda +r)2},\dots ,1x.\end{array}\}
(10)
We need to keep our calculations going to other evaluations in the above polynomial elements. To do that, one can consider the augmentation map aug:\mathbb{Z}M\u27f6\mathbb{Z}, b\mapsto 1. Under this map, it is easy to see that
\begin{array}{l}aug({eval}^{(l)}({\mathbb{B}}_{S,x}))={exp}_{S}({\mathbb{B}}_{S,x})=i,\\ aug(\frac{{\partial}^{M}S}{\partial y})={exp}_{y}(S)=kl,\\ aug({eval}^{(l)}({\mathbb{A}}_{{R}_{+},y}){eval}^{(l)}({\mathbb{A}}_{{R}_{},y}))={exp}_{{T}_{yx}}({\mathbb{P}}_{R,y})=\frac{{i}^{\lambda +r}{i}^{\lambda}}{i1},\\ aug({eval}^{(l)}({\mathbb{C}}_{y,{\theta}_{R}}))={exp}_{S}({\mathbb{P}}_{R,y})=\frac{{i}^{\lambda +r}{i}^{\lambda}}{kl}\end{array}\}
(11)
and for each 1\le m\le k1 and 1\le n\le (\lambda +r)1,
aug\left({eval}^{(l)}\left({\mathbb{P}}_{k,l}^{m}\right)\right)=0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}aug\left({eval}^{(l)}\left({\mathbb{P}}_{\lambda +r,\lambda}^{n}\right)\right)=0.
Now, by using (8) and keeping in our mind the coefficients of array polynomials are integer, we clearly have
{S}_{a}^{n}(b)=\{\begin{array}{cc}{b}^{n};\hfill & a=0,\hfill \\ b;\hfill & a=0\text{and}n=1,\hfill \\ 1;\hfill & k=n\text{or}n=a=0.\hfill \end{array}
Then, by reformulating the elements in (10) and (11) of the second Fox ideal {I}_{2}^{(l)}({\mathcal{P}}_{M}), we arrive at the functions in (9) as desired. □
Considering Remark 2.2, we obtain the following corollary as a consequence of Theorem 2.5.
Corollary 2.6 For any prime p, the presentation
{\mathcal{P}}_{M}=[y,x;{y}^{(p+1)[\frac{{(p+1)}^{p}1}{p}]+1}=y,{x}^{p+1}=x,yx=x{y}^{p+1}]
has a set of generating functions
In Proposition 2.3, the minimality (while satisfying inefficiency) of the presentation {\mathcal{P}}_{M} was expressed in (7). Thus, by considering the meaning and conditions of Proposition 2.3, we obtain the following theorem as another main result of this paper. Since the proof is quite similar to the proof of Theorem 2.5, we omit it.
Theorem 2.7 The inefficient but minimal presentation {\mathcal{P}}_{M} defined in (7) has a set of generating functions
where kl is an odd integer and {S}_{a}^{n}(x) and {S}_{a}^{n}(y) are defined as in (8).
By considering Remark 2.4, we can have the following consequences of Theorem 2.7.
Corollary 2.8 For an odd positive integer t, the presentation
{\mathcal{P}}_{M}=[y,x;{y}^{4t}={y}^{t},{x}^{3t}={x}^{t},yx=x{y}^{2}]
has a set of generating functions
Corollary 2.9 For any positive integers s and t with the condition s<t, the presentation
{\mathcal{P}}_{M}=[y,x;{y}^{2t+1}={y}^{2s},{x}^{2(ts)+1}=x,yx=x{y}^{2}]
has a set of generating functions
Remark 2.10 Since both presentations in Propositions 2.1 and 2.3 have the minimal number of generators because of their efficiency or inefficiency (but minimal) status, this situation affected very positively the number and type of generating functions defined on them.
At this point, we should note that for {t}_{1}\ne {t}_{2}\in {\mathbb{R}}^{+}, \gamma \in \mathbb{C}, a\in {\mathbb{N}}_{0}, generalized array type polynomials {\mathcal{S}}_{a}^{n}(x;{t}_{1},{t}_{2};\gamma ) related to the nonnegative real parameters have been recently developed (in [20]) and some elementary properties including recurrence relations of these polynomials have been derived. In fact, by setting {t}_{1}=\gamma =1 and {t}_{2}=e, Equation (8) is obtained.
Remark 2.11 For a future project, one can study the generalization of Theorems 2.5 and 2.7 by using {\mathcal{S}}_{a}^{n}(x;{t}_{1},{t}_{2};\gamma ).
The remaining goal of this section is to make a connection between the presentation {\mathcal{P}}_{M} in (7) and Stirling numbers of the second kind (cf. [20, 24–28] and the references of these papers). In fact, Stirling numbers of the second kind S(n,a) are defined by means of the generating function
\frac{{({e}^{t}1)}^{a}}{a!}=\sum _{n=0}^{\mathrm{\infty}}S(n,a)\frac{{t}^{n}}{n!}
(see [27, 28]). According to [[20], Theorem 1, Remark 2], Stirling numbers can also be defined as the form
S(n,a)=\frac{1}{a!}\sum _{j=0}^{a}{(1)}^{j}\left(\genfrac{}{}{0ex}{}{a}{j}\right){(kj)}^{n}.
We remind that these numbers satisfy the wellknown properties
S(n,a)=\{\begin{array}{cc}1;\hfill & a=1\text{or}a=n,\hfill \\ \left(\begin{array}{c}n\\ 2\end{array}\right);\hfill & a=n1,\hfill \\ {\delta}_{n,0};\hfill & a=0,\hfill \end{array}
where {\delta}_{n,0} denotes the Kronecker symbol (see [27, 28]). It is known that Stirling numbers are used in combinatorics, in number theory, in discrete probability distributions for finding higherorder moments, etc. We finally note that since S(n,a) 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 theory of partitions.
In addition to the above formulas, for S(n,a), by [20, 26, 27], we also have
{x}^{n}=\sum _{a=0}^{n}\left(\genfrac{}{}{0ex}{}{x}{a}\right)a!S(n,a)
(12)
as a formula for Stirling numbers. Therefore, by taking n=1 and n=0 in Equation (12), the polynomial elements of the second Fox ideal {I}_{2}^{(l)}({\mathcal{P}}_{M}) of the presentation {\mathcal{P}}_{M} in (7) can be restated as follows:
\begin{array}{l}{x}^{0}i{x}^{1}={\sum}_{a=0}^{0}\left(\begin{array}{c}x\\ a\end{array}\right)a!S(0,a)i{\sum}_{a=0}^{1}\left(\begin{array}{c}x\\ a\end{array}\right)a!S(1,a),\\ (kl){y}^{0}=(kl){\sum}_{a=0}^{0}\left(\begin{array}{c}y\\ a\end{array}\right)a!S(0,a),\\ \frac{{i}^{\lambda}({i}^{r}1)}{i1}{x}^{0}=\frac{{i}^{\lambda}({i}^{r}1)}{i1}{\sum}_{a=0}^{0}\left(\begin{array}{c}x\\ a\end{array}\right)a!S(0,a),\\ \frac{{i}^{\lambda}({i}^{r}1)}{kl}{y}^{0}=\frac{{i}^{\lambda}({i}^{r}1)}{kl}{\sum}_{a=0}^{0}\left(\begin{array}{c}y\\ a\end{array}\right)a!S(0,a).\end{array}\}
(13)
After that, as a different version of Theorem 2.5 (and so Theorem 2.7), we present the following result.
Theorem 2.12 The efficient presentation {\mathcal{P}}_{M} in (7) has a set of generating functions in terms of Stirling numbers as given in (13). By taking i=2 and kl is an odd positive integer, we get a set of generating functions in terms of Stirling numbers for the inefficient but minimal presentation of the form as defined in (7).
Furthermore, in a recent work, Simsek [20] has constructed the generalized γStirling numbers of the second kind \mathcal{S}(n,v;a,b;\gamma ) related to nonnegative real parameters (a,b\in {\mathbb{R}}^{+}, a\ne b, a complex number γ and v\in {\mathbb{N}}_{0}). In fact, this new generalization is defined by the generating function as the equality
{f}_{S,v}(t;a,b;\gamma )=\frac{{(\gamma {b}^{t}{a}^{t})}^{v}}{v!}=\sum _{n=0}^{\mathrm{\infty}}\mathcal{S}(n,v;a,b;\gamma )\frac{{t}^{n}}{n!}.
(14)
By setting a=1 and b=e in (14), one can obtain the γStirling numbers of the second kind S(n,v;\gamma ) which are defined by the generating function
\frac{{(\gamma {e}^{t}1)}^{v}}{v!}=\sum _{n=0}^{\mathrm{\infty}}S(n,v;\gamma )\frac{{t}^{n}}{n!}
(see [27, 28]). According to the same references, by substituting \gamma =1 into the above equation, the Stirling numbers of the second kind S(n,v) are obtained.
By considering this new generalization \mathcal{S}(n,v;a,b;\gamma ), in [[20], Theorem 1], the equality
\mathcal{S}(n,v;a,b;\gamma )=\frac{1}{v!}\sum _{j=0}^{n}{(1)}^{j}\left(\genfrac{}{}{0ex}{}{v}{j}\right){\gamma}^{vj}{(jlna+(vj)lnb)}^{n}
(15)
has also been obtained for γStirling numbers of the second kind. In fact, by setting a=1 and b=e in (15), one can get the following equality on γStirling numbers:
S(n,v;\gamma )=\frac{1}{v!}\sum _{j=0}^{v}\left(\genfrac{}{}{0ex}{}{v}{j}\right){\lambda}^{(vj)}{(1)}^{j}{(vj)}^{n}
(16)
(see [27, 28]).
Hence, we can present the following note.
Remark 2.13 It is clearly seen that Stirling numbers have been only considered in Theorems 2.5 and 2.7 (and the corollaries about them). However, one can also study the γStirling numbers S(n,v;\gamma ) defined in (16) and generalized γStirling numbers \mathcal{S}(n,v;a,b;\gamma ) defined in (15) to obtain different types of generating functions.