Coxeter groups are named after a British born Canadian geometer Harold Scott MacDonald Coxeter (1907–2003). Coxeter groups were introduced by Coxeter in 1934 as abstract form of reflection groups and defined as the groups with the generators *a*_{i}, *i* ∈ *I* and relations $\begin{array}{}{a}_{i}^{2}\end{array}$ = 1 and *a*_{i}*a*_{j}*a*_{i} = *a*_{j}*a*_{i}*a*_{j}, *i*, *j* ∈ *I*. Coxeter classified these groups into two categories, finite and infinite, in 1935. If the relation $\begin{array}{}{a}_{i}^{2}\end{array}$ = 1 from the presentation of Coxeter group is removed then we get the presentation of Artin groups. Thus, we can say that Coxeter groups are quotient groups of the Artin groups. A Coxeter group is called finite Coxeter group if it has the presentation as a discrete, properly acting group of reflections of the sphere [1]. Therefore these groups are also called spherical Coxeter groups. In the list of spherical Coxeter groups, *A*_{n} is the first. The Artin group associated to *A*_{n} is the braid group. A Coxeter group is affine if it is infinite. It is generated by reflections in affine spaces and has the presentation as discrete, properly acting, affine reflection group [1]. Cardinality is invariant of graded algebraic structures. Hilbert series deals with the cardinality of elements in the graded algebraic structures.

In [2] Saito computed the growth series of the Artin monoids. In [3] Parry computed the growth series of the Coxeter groups. In [4] Mairesse and Mathéus gave the growth series of Artin groups of dihedral type. In [5] Iqbal gave a linear system for the reducible and irreducible words of the braid monoid *MB*_{n}, which leads to compute the Hilbert series of *MB*_{n} and in [6] computed the Hilbert series of the braid monoid *MB*_{4} in band generators. In [7] Berceanu and Iqbal proved that the growth rate of all the spherical Artin monoids is less than 4. In the present paper we study the affine-type Coxeter group $\begin{array}{}{D}_{n}^{\mathrm{\infty}},\end{array}$ and find the Hilbert series (or spherical growth series) of the associated right-angled affine Artin monoid $\begin{array}{}M({D}_{n}^{\mathrm{\infty}})\end{array}$. We also discuss its recurrence relations and the growth rate. In [8, 9, 10, 11, 12, 13] authors presented some new ways to compute different solutions methods.

Let *S* be a set. A *Coxeter matrix* over *S* is a square matrix *M* = (*m*_{st})_{s,t∈S} such that

*m*_{ss} = 1 for all *s* ∈ *S*;

*m*_{st} = *m*_{ts} ∈ {2, 3, 4, …, ∞} for all *s*, *t* ∈ *S*, *s* ≠ *t*.

A *Coxeter graph* *Γ* is a labeled graph defined by the following data:

*S* is a set of vertices of *Γ*.

Two vertices *s*, *t* ∈ *S*, *s* ≠ *t* are joined by an edge if *m*_{st} ≥ 3. This edge is labeled by *m*_{st} if *m*_{st} ≥ 4.

#### Definition 1.2

Let *M* = (*m*_{st})_{s,t∈S} be the Coxeter matrix and *Γ*(*M*) its Coxeter graph. Then the group

$$\begin{array}{}{\displaystyle W=\u3008s\in S\mid {s}^{2}=1\phantom{\rule{thinmathspace}{0ex}},(st{)}^{{m}_{st}}=1,s,t\in S,\phantom{\rule{thinmathspace}{0ex}}s\ne t\u3009}\end{array}$$

is called the Coxeter group (of type *Γ*(*M*)).

In a simple way, we can write $\begin{array}{}W\phantom{\rule{negativethinmathspace}{0ex}}=\phantom{\rule{negativethinmathspace}{0ex}}\u3008s\in S\mid {s}^{2}\phantom{\rule{negativethinmathspace}{0ex}}=\phantom{\rule{negativethinmathspace}{0ex}}1,\underset{{m}_{st}\phantom{\rule{thinmathspace}{0ex}}\text{factors}}{\underset{\u23df}{sts\cdots}}\phantom{\rule{negativethinmathspace}{0ex}}=\phantom{\rule{negativethinmathspace}{0ex}}\underset{{m}_{st}\text{factors}}{\underset{\u23df}{tst\cdots}\phantom{\rule{thinmathspace}{0ex}}},s,t\in S,\phantom{\rule{thinmathspace}{0ex}}s\ne t\u3009.\end{array}$ We call *Γ* to be of *spherical type* if *W* is finite.

An *Artin spherical monoid (or group)* is given by a finite union of connected Coxeter graphs from the well known classical list of Coxeter diagrams (see [1, 14]).

Figure 1 Spherical Coxeter graphs

By convention *m*_{ij} is the label of the edge between *x*_{i} and *x*_{j} (*i* ≠ *j*); if there is no label then *m*_{ij} = 3. If there is no edge between *x*_{i} and *x*_{j}, then *m*_{ij} = 2.

To a given Coxeter graph *X*_{n} we associate the monoid *M*(*X*_{n}) with generators (corresponds to vertices) *x*_{1}, *x*_{2}, …, *x*_{n} and relations (corresponds to labels *m*_{ij} of the graphs) $\begin{array}{}\underset{{m}_{ij}\phantom{\rule{thinmathspace}{0ex}}\text{factors}}{\underset{\u23df}{{x}_{i}{x}_{j}{x}_{i}{x}_{j}\cdots}}=\underset{{m}_{ij}\phantom{\rule{thinmathspace}{0ex}}\text{factors}}{\underset{\u23df}{{x}_{j}{x}_{i}{x}_{j}{x}_{i}\cdots}},\end{array}$ where 1 ≤ *j* < *i* ≤ *n*.

The corresponding group *G*(*X*_{n}) associated to *X*_{n} is defined by the same presentation. From now onward we will use *X*_{n} for *G*(*X*_{n}) for simplicity.

#### Definition 1.3

If *W* is a Coxeter group with Coxeter matrix *M* = (*m*_{st})_{s,t∈S}, then the Artin group associated to *W* is defined by

$$\begin{array}{}{\displaystyle \mathcal{A}=\u3008s\in S\phantom{\rule{thinmathspace}{0ex}}|\underset{{m}_{st}\phantom{\rule{thinmathspace}{0ex}}\text{factors}}{\underset{\u23df}{sts\cdots}}=\underset{{m}_{st}\phantom{\rule{thinmathspace}{0ex}}\text{factors}}{\underset{\u23df}{tst\cdots}}\u3009.}\end{array}$$

If *W* is finite then 𝓐 is called a spherical Artin group.

#### Definition 1.4

In the spherical type Coxeter graphs, if all the labels *m*_{st} ≥ 3 are replaced by ∞ then the associated groups (monoids) are called right-angled Artin groups (*monoids*).

#### Definition 1.5

[15] Let *G* be a finitely generated group and *S* be a finite set of generators of *G*. The *word lenth* *l*_{S}(*g*) of an element *g* ∈ *G* is the smallest integer *n* for which there exists *s*_{1}, …, *s*_{n} ∈ *S* ∪ *S*^{−1} such that *g* = *s*_{1} ⋯ *s*_{n}.

#### Definition 1.6

[15] Let *G* be a finitely generated group and *S* be a finite set of generators of *G*. The *growth function* of the pair (*G*, *S*) associates to an integer *k* ≥ 0 the number *a*_{k} of elements *g* ∈ *G* such that *l*_{S}(*g*) = *k*. The corresponding *spherical growth series (also known as the generating function)* or the *Hilbert series* is given by $\begin{array}{}{\mathcal{H}}_{G}(t)=\sum _{k=0}^{\mathrm{\infty}}{a}_{k}{t}^{k}.\end{array}$

The affine (or infinite) Coxeter groups form another important series of Coxeter groups. These well-known affine Coxeter groups are *A͂*_{n}, *B͂*_{n}, *C͂*_{n}, *D͂*_{n}, *E͂*_{6}, *E͂*_{7}, *E͂*_{8}, *G͂*_{2} and *I͂*_{1} (for details, see [14]). In [7] we proved that the universal upper bound for all the spherical Artin monoids is less than 4.

In this work we discuss right-angled affine Artin monoids, specifically, we study the affine monoid $\begin{array}{}M({D}_{n}^{\mathrm{\infty}})\end{array}$ and compute its Hilbert series. We show that the growth of $\begin{array}{}M({D}_{n}^{\mathrm{\infty}})\end{array}$ series is bounded above by 4. Along with Hilbert series we also compute the recurrence relations related to $\begin{array}{}M({D}_{n}^{\mathrm{\infty}})\end{array}$. Based on some computations with mathematical softwares Derive 6 and Mathematica, we give a conjecture of the growth rate of $\begin{array}{}M({D}_{n}^{\mathrm{\infty}})\end{array}$.

The monoid $\begin{array}{}M({D}_{n}^{\mathrm{\infty}})\end{array}$ is represented by its Coxeter graph as follows:

Here *x*_{1}, *x*_{2}, …, *x*_{n} are vertices of the graph and all the labels are ∞. If all the labels in a Coxeter diagram are replaced by ∞, then there is no relation between the adjacent edges. Hence we have the associated right-angled Artin groups and the associated right-angled Artin monoids denoted by $\begin{array}{}G({D}_{n}^{\mathrm{\infty}})\end{array}$ and $\begin{array}{}M({D}_{n}^{\mathrm{\infty}})\end{array}$, respectively.

#### Definition 1.7

Let *X* be a nonempty set and *X*^{*} be the free monoid on *X*. Let *w*_{1} and *w*_{2} ∈ *X*^{*}, where *w*_{1} = *x*_{1}*x*_{2} ⋯ *x*_{r}, *w*_{2} = *y*_{1}*y*_{2} ⋯ *y*_{r} with *x*_{i}, *y*_{i} ∈ *X*. Then *w*_{1} < *w*_{2} *length*-*lexicographically* (or *quasi*-*lexicographically*) if there is a *k* ≤ *r* such that *x*_{k} < *y*_{k} and *x*_{i} = *y*_{i} for all *i* < *k*.

Let *α* = *β* be a relation in a monoid *M* and *α*_{1} = *uw* and *α*_{2} = *wv* be the words in *M*.

Then the word of the form *α*_{1} ×_{w} *α*_{2} = *uwv* is said to be an *ambiguity*. If *α*_{1}*v* = *u**α*_{2} (in the length-lexicographic order) then we say that the ambiguity *uwv* is solvable. A presentation of *M* is *complete* if and only if all the ambiguities are solvable. Corresponding to the relation *α* = *β*, the changes *γ**α**δ* ⟶ *γ**β**δ* give a rewriting system. A complete presentation is equivalent to a confluent rewriting system. In a complete presentation of a monoid a word containing *α* will be called *reducible word* and a word that does not contain *α* will be called an *irreducible word* or *canonical word*. For example *x*_{2}*x*_{1}*x*_{2} = *x*_{1}*x*_{2}*x*_{1} is a basic relation in the braid monoid *MB*_{3}. A word $\begin{array}{}v={x}_{2}^{2}{x}_{1}{x}_{2}\end{array}$ contains *α* = *x*_{2}*x*_{1}*x*_{2}. Hence *v* is a reducible word. Then $\begin{array}{}{x}_{2}^{2}{x}_{1}{x}_{2}={x}_{2}{x}_{1}{x}_{2}{x}_{1}={x}_{1}{x}_{2}{x}_{1}^{2}\end{array}$ is the canonical form of *v* [16].

In a presentation of a monoid we fix a total order *x*_{1} < *x*_{2} < ⋯ < *x*_{n} on the generators. Hence clearly we have

#### Lemma 1.8

*The presentation of the monoid* $\begin{array}{}M({D}_{n}^{\mathrm{\infty}})\end{array}$ *has generators* *x*_{1}, *x*_{2}, …, *x*_{n} *and has relations* *x*_{i}*x*_{j} = *x*_{j}*x*_{i}, 3 ≤ *j* + 2 ≤ *i* ≤ *n* − 1, *and* *x*_{n}*x*_{k} = *x*_{k}*x*_{n}, *k* ≠ *n* − 2.

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