RecurrenceRelationsandHilbertSeriesoftheMonoid ...downloads.hindawi.com/journals/jmath/2020/5215631.pdf · that the removal of the relations a2 i 1 gives Artin groups. So, Coxeter

Research ArticleRecurrence Relations and Hilbert Series of the MonoidAssociated with Star Topology

Jiang-Hua Tang1 Zaffar Iqbal2 Abdul Rauf Nizami3 Mobeen Munir 4 Faiza Azam2

and Jia-Bao Liu 5

1Department of General Education Anhui Xinhua University Hefei 230088 China2Department of Mathematics University of Gujrat Gujrat Pakistan3University of Central Punjab Department of Mathematics Lahore Pakistan4Department of Mathematics Division of Science and Technology University of Education Township Lahore Pakistan5School of Mathematics and Physics Anhui Jianzhu University Hefei 230601 China

Correspondence should be addressed to Mobeen Munir mmunirueedupk

Received 1 April 2020 Accepted 21 July 2020 Published 27 August 2020

Academic Editor Ghulam Shabbir

Copyright copy 2020 Jiang-Hua Tang et al (is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Affinemonoids are the considered as natural discrete analogues of the finitely generated cones(e interconnection between thesetwo objects has been an active area of research since last decade Star network is one of the most common in computer networktopologies In this work we study star topology Sn and associate a Coxeter structure of affine type on it We find a recurrencerelation and the Hilbert series of the associated right-angled monoid M(Sinfinn ) We observe that the growth rate of the monoidM(Sinfinn ) is unbounded

1 Introduction

Hilbert series of a graded commutative algebra is stronglyrelated with dimensions of the homogeneous components ofthe algebra (is notion has been extended to filtered al-gebras and coherent sheaves over projective schemes [1]Hilbert series is treated as a special case of HilbertndashPoincareseries of a graded vector space and is closely related with thenumber of words of an alphabet Hilbert series helpscounting words on an alphabet that do not contain a fixed setof words (is is also named as well-known forbiddensubwords problem which can be translated at the level ofmonomial algebras If we are given an alphabetX X1 X2 Xn1113864 1113865 and words W W1 W2 W d1113864 1113865then it is known that the numbers of words on X avoidingthis word W is in one-to-one correspondence with mono-mials in X having nonzero image in the ring L C[X]W IfL is a finitely generated monoid then the coefficients of itsHilbert series satisfy a recursive relation Hilbert series ofseveral popular algebras appear as power series of rational

functions [2] (ings become interesting especially when themonoid L has a rational Hilbert series (is happens exactlywhen we have a finitely presented monoid Significance ofHilbert series can be described in terms of the growth of amonoid (e growth of a monoid is said to be polynomiallybounded if the nth coefficient of its Hilbert series is 0(n d)

for some number d A monoid has exponential growth if thenth coefficient is larger than Cn for some Cgt 1 Interestinglymonoids should have polynomially bounded growths (esmallest-degree polynomial bounding the growth is one lessthan the Krull dimension In noncommutative settingstopologists used growth to study fundamental groupsMilnor proved that if a compact Riemannian manifold hasall its sectional curvatures negative then its fundamentalgroup has exponential growth and if a complete n-di-mensional Riemannian manifold has its mean curvaturetensor everywhere positive semidefinite then the finitelygenerated subgroup of its fundamental group has poly-nomially bounded growth In short a monoid has either anexponential growth or a polynomial growth (e degree of

HindawiJournal of MathematicsVolume 2020 Article ID 5215631 6 pageshttpsdoiorg10115520205215631

the growth is one less than the order of the pole at 1 of theHibert series

Coxeter groups were introduced by Canadian geometerH S M Coxeter in 1934 to solve the well-known famousword problem namely whether two words occurring ingenerators of the presentation of groups correspond to sameelement or not(ese groups have nice other properties suchas having faithful linear representations as groups of re-flections In a nontrivial way it can be proved that thesegroups are abstract analogues of the regular polytopes(esepolytopes are convex-hull of some points in Rn Coxetergroups have generators ai i isin I and have relations a2

i 1and aiajai ajaiaj with i j isin I finite and infinite groups areusually referred as spherical and affine

Star topology is one of the important topologies used innetworking and other real-world problems One way tostudy this topology is by using Dynkin diagram [3] (orCoxeter graph) and other way is by using monoids Notethat the removal of the relations a2

i 1 gives Artin groupsSo Coxeter groups are quotient groups of the Artin groupsA finite Coxeter group is a discrete acting group of reflec-tions of a sphere [3] (at is why they are known asspherical (e Artin braid group An is a spherical Coxetergroups (e Infinite Coxeter groups are generated by re-flections in affine spaces [3]

In 2009 Saito [4] found spherical growth series of Artinmonoids [5] In [6] we gave a linear system for the canonicalwords of the braid monoid MBn which lead to find Hilbertseries of MBn In [7] we computed Hilbert series of MB4 inband generators In 2006 Mairesse and Matheus [8] gavedihedral-type growth series of Artin groups In 1993 Parry[9] gave the growth series of Coxeter groups In [10] weproved that the upper bound of the growth of spherical Artinmonoids is 4 But in the affine case this result is not true In[11] we found a recurrence relation and Hilbert series of theassociated right-angled affine Artin monoid M(1113957A

infinn ) and

showed that its growth rate is unbounded In [12] we foundthe Hilbert series of M( 1113957D

infinn ) and showed that its growth rate

is also unboundedIn this paper we study the star topology Sn and find

recurrence relations and the Hilbert series of the associatedright-angled monoid M(Sinfinn ) We also compute growth rateof the monoid M(Sinfinn ) and observe that it is unbounded

2 Preliminaries

We start this section with the notion of Coxeter groups andArtin groups We study the star topology as a Dynkin di-agram and then convert it as a monoid (ese basic pre-liminary facts and notations which will be required later forformulating our main results

Definition 1 A square symmetric matrix M (mst)stisinS issaid to be a Coxeter matrix over a nonempty set S such thatall the diagonal entries are 1 and mst isin 2 3 4 infin

Definition 2 Let S be a set of vertices of a labeled graph ΓWe call Γ a Coxeter graph if any two of its vertices are

connected by an edge and if the label of each edge is greaterthan 2

By convention each edge is labeled only if the label isgreater than 3

Definition 3 A group with generator s and relations s2 1and (st)mst 1 for all s t isin S and mst neinfin is called a Coxetergroup such that (mst)stisinS is the Coxeter matrix

Definition 4 (e Artin group is

A langs isin S | sts middot middot middot1113980radicradic11139791113978radicradic1113981mstfactors

tst middot middot middot1113980radicradic11139791113978radicradic1113981mstfactorsrang

(1)

If the Coxeter group is finite thenA is called a sphericalArtin group

Definition 5 (e right-angled Artin groups or monoids areobtained if all the labels which are greater than or equal to 3of spherical Coxeter graphs are replaced with infin

(e Artin spherical groups are usually represented byCoxeter graphs (see [3 13]) these groups are An for nge 1 Bn

for nge 2 Dn for nge 4 E6 E7 E8 F4 G2 H3 H4 and I2(p)

for pge 5 and pne 6 Figure 1 contains these graphs

Definition 6 (see [14]) (e length of a word g s1 middot middot middot sn ofa finitely generated group G is the smallest nonnegativeinteger n for which s1 sn isin Scup Sminus 1 where S is the set ofgenerators of G

Definition 7 (see [14]) (e spherical growth series of a fi-nitely generated group G is HG(t) 1113936

infink0 aktk where ak is

the number of words of length kLet a b be a relation in a given monoid M (en in

length-lexicographic order a is greater then or equal to b Aword uwv has an ambiguity if uw and wv are left sides tworelations If α1v and uα2 are identical then uwv is solvable Ifα1v and uα2 differ by lexicographic order then we get a newrelation in M A presentation is said to be complete if so-lutions of all ambiguities are identical A reducible word isthe left side of a relation of a complete presentation of amonoid If w does not contain the LHS of any relation thenw is called a canonical word (e following notions are in[15ndash20] under different terminologies Grobner basescomplete presentation rewriting system and so on

3 Main Results

In this part we compute our main results

31 RecurrenceRelation of theMonoidM(Sinfinn ) In this paperwe study the star topology Sn and find recurrence relationsand the Hilbert series of the associated right-angled monoidM(Sinfinn ) We compute the growth rate of the monoidM(Sinfinn ) and using the graph we show that it is unbounded(e Coxeter graph of the star topology Sn is given by thefollowing graph (Figure 2)

2 Journal of Mathematics

We denote the right-angled monoid associated with Sn

by M(Sinfinn ) In M we fix a total order x1 ltx2 lt middot middot middot ltxn onthe generators Hence clearly we have the following lemma

Lemma 1 5e monoid M(Sinfinn ) has generators x1 x2 xn

and relations xixj xjxi for 1le j + 1le ile n minus 1 and xnxk

xkxn for 2le kle n minus 25is section covers some useful results about recursive

relations ofM(Sinfinn )Consider a system [21] of linear relations

ui(t + 1) ai1(t)u1(t) + ai2(t)u2(t) + middot middot middot + ain(t)un(t)

+ fi(t) 1le ile n

(2)

Equivalently u(t + 1) A(t)u(t) + f(t) where

u(t)

u1(t)

⋮

un(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A(t)

a11(t) middot middot middot a1n(t)

⋮ ⋱ ⋮

an1(t) middot middot middot ann(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

f(t)

f1(t)

⋮

fn(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)

5e solution of the system u(t + 1) A(t)u(t) isu(t) c1λ

t1u

1 + middot middot middot + ckλtkuk where λi and ui 1le ile k are

respectively the eigenvalues and eigenvectors of A(t)5e largest eigenvalue represents the growth rate of thesequence u1(t) u2(t) uk(t)

In the following by ck and cki we shall mean the numberof canonical words of length k and words starting with xi

Lemma 2 M(Sinfinn ) satisfies the relations c0 1 c1i 1 andck 1113936

ni1 cki (kge 1) where cki is

cki 1113936n

ji

ckminus 1j 1le ile n minus 1

ck1 i n

⎧⎪⎪⎨

⎪⎪⎩(4)

Let Sn(λ) denote the characteristic polynomial then wehave the following

Theorem 1 5e characteristic polynomial Sn(λ) of thesystem of recursive relations of M(Sinfinn ) satisfies the relation

Sn(λ) λSnminus 1(λ) minus λ(λ minus 1)nminus 2

(5)

where nge 2 and S1(λ) λ minus 1

Proof (e characteristic polynomial of the coefficientmatrix of the system of recurrence relations given in Lemma2 is

Sn(λ)

λ minus 1 minus 1 middot middot middot minus 1 minus 1 minus 1 minus 1

0 λ minus 1 middot middot middot minus 1 minus 1 minus 1 minus 1

⋮ ⋮ ⋮ ⋮ ⋮ ⋮

0 0 middot middot middot 0 λ minus 1 minus 1 minus 1

0 0 middot middot middot 0 0 λ minus 1 minus 1

minus 1 minus 1 middot middot middot minus 1 minus 1 minus 1 λ minus 1

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(6)

Adding the last row in the 2nd last row we have

(An)nge1

(Bn)nge2

(Dn)nge4

middot middot middot

4


middot middot middot(En)n=678

F4

G2

H3

H4

x1 x2 x3

x1 x2 x3

x1 x2 x3

x5 xn

xn

xn

xnndash2

xnndash1

xnndash1

middot middot middotx1 x2 x3 xnxnndash1

x4

(I2(p))pge5pne6

5x1 x2 x3 x4

x1 x2

p

4x1 x2 x3 x4

6x1 x2

5x1 x2 x3

Figure 1 Spherical Coxeter graphs

xn

xnndash1

x1

x2 x3

x4

x5

Sn

Figure 2 Sn

Journal of Mathematics 3

Sn(λ)



⋮ ⋮ ⋮ ⋮ ⋮ ⋮


minus 1 minus 1 middot middot middot minus 1 minus 1 λ minus 2 λ minus 2


1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(7)

Now subtracting the 2nd last column from the lastcolumn and simplifying we have Sn (λ) λTnminus 1(λ) where

Tnminus 1(λ)

λ minus 1 minus 1 middot middot middot minus 1 minus 1 minus 1

0 λ minus 1 middot middot middot minus 1 minus 1 minus 1

⋮ ⋮ ⋮ ⋮ ⋮

0 0 middot middot middot minus 1 λ minus 1 minus 1

minus 1 minus 1 middot middot middot minus 1 minus 1 λ minus 2

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(8)

We write Tnminus 1(λ) Unminus 1(λ) + Vnminus 1(λ) where the de-terminants Unminus 1(λ) and Vnminus 1(λ) are obtained by splittingTnminus 1(λ) such that the last row of Unminus 1(λ) is (minus 1 minus 1 λ minus

1) and the last row of Vnminus 1(λ) is (0 0 minus 1) Hence easilywe have Unminus 1(λ) Snminus 1(λ) and Vnminus 1(λ) minus (λ minus 1)nminus 2(erefore we have


(9)

Here we have an explicit formula for Sn(λ)

Lemma 3 In M(Sinfinn ) characteristic polynomial is givenexplicitly by

Sn(λ) λnminus 1(λ minus 1) minus 1113944

nminus 1

i1λi

(λ minus 1)nminus iminus 1

(10)

Proof From equation (5) we haveSnminus k(λ) λSnminus kminus 1(λ) minus λ(λ minus 1)nminus kminus 2 kge 0 Hence we have


λ2Snminus 2(λ) minus λ(λ minus 1)nminus 2

minus λ2(λ minus 1)nminus 3




λnminus 1Snminus (nminus 1) minus 1113944

nminus 1

i1λi


λnminus 1(λ minus 1) minus 1113944

nminus 1

i1λi


(11)

32 Hilbert Series of the Monoid M(Sinfinn ) From now onH

(n)M (t) 1113936kge0cktk will denote the Hilbert series of M(Sinfinn )

and H(n)Mi(t) 1113936kge1c

(n)ki tk will denote the Hilbert series of

M(Sinfinn ) of the words starting with xi

Following Lemma 2 we get the following

Theorem 2 For M(Sinfinn ) we have

(1) H(n)M (t) 1 + 1113936

ni1 H

(n)Mi(t)

(2) H(n)Mi(t) t + t 1113936

nji H

(n)Mj(t) 1le ile n minus 1

(3) H(n)M1(t) H

(n)Mn(t)

Proof

(1) Since ck 1113936ni1 cki (kge 1) Hn

M(t) 1113936kge0cktk c0 +

1113936kge1cktk 1 + 1113936kge1 1113936ni1 ckit

k 1 + 1113936ni1 1113936kge1ckit

k

1 + 1113936ni1 Hn

Mi(t)(2) Also from Lemma 2 we have cki 1113936

nji ckminus 1j

(1le ile n minus 1) Hence H(n)Mi(t) 1113936

nkge 1 cki(t)tk

c1i(t)t + 1113936nkge 2 cki(t)tk t + 1113936

nkge 2 1113936

nji ckminus 1j(t)tk

t + t 1113936nji 1113936

nkge 2 ckminus 1j(t) tkminus 1 t+ t 1113936

nji H

(n)Mj(t)

(3) It can be proved similarly

(e linear system of(eorem 2 takes the form WnX Bwhere

Wn

1 minus t minus t middot middot middot minus t minus t minus t

0 1 minus t middot middot middot minus t minus t minus t

0 0 middot middot middot minus t minus t minus t

⋮ ⋮ ⋮ ⋮ ⋮0 0 middot middot middot 0 1 minus t minus t

minus t minus t middot middot middot minus t minus t 1 minus t

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

X

H(n)M1(t)

H(n)M2(t)

⋮

H(n)Mn(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

t

t

⋮

t

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)

Lemma 4 For M(Sinfinn ) we have H(n)Mm(t) ((t(1minus

t)mminus 1)tnSn(1t)) m 1 2 n minus 1For the system of equations given in 5eorem 2 we have

the following

Lemma 5

det Wn( 1113857 tnSn

1t

1113874 1113875 (13)

Proof It is obvious to Just factor out t from each row ofdet(Wn)


Lemma 6

H(n)Mm(t)

t(1 minus t)mminus 1

tnSn(1t) m 1 2 n minus 1 (14)

Proof (e solution of the system WnX B isH

(n)Mm(t) (Tmdet(Wn)) where Tm is the determinant of

the matrix obtained by replacing 0 in the mth column of Wn

by B (at is

Tm

1 minus t minus t middot middot middot minus t t middot middot middot minus t minus t

0 1 minus t middot middot middot minus t t middot middot middot minus t minus t

0 0 middot middot middot minus t t middot middot middot minus t minus t

⋮ ⋮ ⋮ ⋮ ⋮ ⋮

0 0 middot middot middot 0 t middot middot middot 0 1 minus t

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(15)

Adding mth column of Tm to its column numbersm + 1 m + 2 n we get H

(n)Mm(t)

((t(1 minus t)mminus 1)tnSn(1t)) m 1 2 n minus 1(e following result gives the Hilbert series of

M(Sinfinn )

Theorem 3

H(n)M (t)

1tnSn(1t)

(16)

Proof Equation (5) gives tnSn(1t) tnminus 1Snminus 1(1t)minus t(1 minus t)nminus 2 Hence Lemma 6 and (eorem 2 imply that

H(n)M (t) 1 + H

(n)M1(t) + H

(n)M2(t) + middot middot middot + H

[n)Mnminus 1(t)

+ H(n)Mn(t)

1

tnSn(1t)1113888t

nSn

1t

1113874 1113875 + t + t(1 minus t)

+ t(1 minus t)2

+ middot middot middot + t(1 minus t)nminus 2

+ t1113889

1

tnSn(1t)1113888t

nminus 1Snminus 1

1t

1113874 1113875 + 2t + t(1 minus t)

+ t(1 minus t)2


1113889

tS1(1t) minus t

tnSn(1t)

1

tnSn(1t)

(17)

33 Conclusions Here we gave by constructing the affine-type Coxeter structure on star topology a recursive relationand the Hilbert series of the right-angled monoid M(Sinfinn )

associated with a star graph (e main result isH

(n)M (t) (1tnSn(1t)) We also computed the growth rate

rn of M(Sinfinn ) using Mathematica some initial values arer3 2618 r4 3147 r5 3629 r6 4079 r7 4506r8 4915 r9 5309 r10 5691 r11 6063 r12 6426

40 60 80 100 120n

rn

24

4

68

8

1012

12

1416

16

1820

20

2224262830323436

Figure 3 Sinfinn


r13 6781 r14 7130 r15 7472 r16 7809r17 8141 r18 8468 r19 8790 r20 9109 We alsocompute r40 1493 r60 2015 r80 2504 r100 297and r120 342 We have the following graph representingthe behavior of the growth rate of Sn(λ) (Figure 3)

We observe that the growth rate for M(Sinfinn ) increasesand is unbounded Hence at the end we have the followingnatural open problem emerging from our research

An open problem the growth rate of M(Sinfinn ) isunbounded

Data Availability

No such data are used in this research

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is project was supported by the Natural Science FundProject of Anhui Xinhua University ( Grant No 2017zr011)

References

[1] R P Stanley ldquoHilbert functions of graded algebrasrdquoAdvancesin Mathematics vol 28 no 1 pp 57ndash83 1978

[2] S Hal Computational Algebraic Geometry Cambridge Uni-versity Press Cambridge UK 2003

[3] N Bourbaki Groupes et algebres de Lie Chapitres 4-6 Ele-mentary Mathematics Hermann MO USA 1968

[4] K Saito ldquoGrowth functions for Artin monoidsrdquo Proceedingsof the Japan Academy Series A Mathematical Sciences vol 85no 7 pp 84ndash88 2009

[5] E Artin ldquo(eory of braidsrdquo 5e Annals of Mathematicsvol 48 no 1 pp 101ndash126 1947

[6] Z Iqbal ldquoHilbert series of positive braidsrdquo Algebra Collo-quium vol 18 pp 1017ndash1028 2011

[7] Z Iqbal and S Yousaf ldquoHilbert series of the braid monoid$MB_4$ in band generatorsrdquo Turkish Journal of Mathe-matics vol 38 pp 977ndash984 2014

[8] J Mairesse and F Matheus ldquoGrowth series for Artin groups ofdihedral typerdquo International Journal of Algebra and Com-putation vol 16 no 6 pp 1087ndash1107 2006

[9] W Parry ldquoGrowth series of Coxeter groups and salemnumbersrdquo Journal of Algebra vol 154 no 2 pp 406ndash4151993

[10] B Berceanu and Z Iqbal ldquoUniversal upper bound for thegrowth of Artin monoidsrdquo Communications in Algebravol 43 no 5 pp 1967ndash1982 2015

[11] Z Iqbal S Batool and M Akram ldquoHilbert series of right-angled affine Artin monoid M(1113957A

infinn )rdquo Kuwait Journal of

Science vol 44 no 4 pp 19ndash27 2017[12] C Young Z Iqbal A Rauf Nizami M Munir S Riaz and

M Shin ldquoSome recurrence relations and Hilbert series ofright-angled affine Artin monoid M( 1113957D

infinn )rdquo Journal of

Function Spaces vol 2018 Article ID 1901657 6 pages 2018[13] H S M Coxeter Regular Complex Polytopes Cambridge

University Press Cambridge UK 2nd edition 1991[14] P D Harpe Topics in Geometric Group5eory(eUniversity

of Chicago Press Chicago IL USA 2000

[15] D J Anick ldquoOn the homology of associative algebrasrdquoTransactions of the American Mathematical Society vol 296no 2 p 641 1986

[16] G M Bergman ldquo(e diamond lemma for ring theoryrdquoAdvances in Mathematics vol 29 no 2 pp 178ndash218 1978

[17] L A Bokut Y Fong W-F Ke and L-S Shiao ldquoGrobner-Shirshov bases for braid semigrouprdquo in Advances in Algebrapp 60ndash72 World Scientific Publishing Singapore 2003

[18] K S Brown ldquo(e geometry of rewriting systemsa proof ofAnick-Groves-Squeir theoremrdquo in Algorithms andClassification in Combinatorial Group 5eory G Baumslagand C F Miller Eds pp 137ndash164 Springer New York NYUSA 1992

[19] P M Cohn Further Algebra and Applications SpringerLondon UK 2003

[20] V A Ufnarovskij ldquoCombinatorial and asymptotic methods inalgebrardquo in Encyclopaedia of Mathematical Sciences SpringerBerlin Germany 1995

[21] W G Kelley and A C Peterson Difference equations AnIntroduction with Applications p 125 Second edition Aca-demic Press New York NY USA 2001


the growth is one less than the order of the pole at 1 of theHibert series

Coxeter groups were introduced by Canadian geometerH S M Coxeter in 1934 to solve the well-known famousword problem namely whether two words occurring ingenerators of the presentation of groups correspond to sameelement or not(ese groups have nice other properties suchas having faithful linear representations as groups of re-flections In a nontrivial way it can be proved that thesegroups are abstract analogues of the regular polytopes(esepolytopes are convex-hull of some points in Rn Coxetergroups have generators ai i isin I and have relations a2

i 1and aiajai ajaiaj with i j isin I finite and infinite groups areusually referred as spherical and affine

Star topology is one of the important topologies used innetworking and other real-world problems One way tostudy this topology is by using Dynkin diagram [3] (orCoxeter graph) and other way is by using monoids Notethat the removal of the relations a2

i 1 gives Artin groupsSo Coxeter groups are quotient groups of the Artin groupsA finite Coxeter group is a discrete acting group of reflec-tions of a sphere [3] (at is why they are known asspherical (e Artin braid group An is a spherical Coxetergroups (e Infinite Coxeter groups are generated by re-flections in affine spaces [3]

In 2009 Saito [4] found spherical growth series of Artinmonoids [5] In [6] we gave a linear system for the canonicalwords of the braid monoid MBn which lead to find Hilbertseries of MBn In [7] we computed Hilbert series of MB4 inband generators In 2006 Mairesse and Matheus [8] gavedihedral-type growth series of Artin groups In 1993 Parry[9] gave the growth series of Coxeter groups In [10] weproved that the upper bound of the growth of spherical Artinmonoids is 4 But in the affine case this result is not true In[11] we found a recurrence relation and Hilbert series of theassociated right-angled affine Artin monoid M(1113957A

infinn ) and

showed that its growth rate is unbounded In [12] we foundthe Hilbert series of M( 1113957D

infinn ) and showed that its growth rate

is also unboundedIn this paper we study the star topology Sn and find

recurrence relations and the Hilbert series of the associatedright-angled monoid M(Sinfinn ) We also compute growth rateof the monoid M(Sinfinn ) and observe that it is unbounded

2 Preliminaries

We start this section with the notion of Coxeter groups andArtin groups We study the star topology as a Dynkin di-agram and then convert it as a monoid (ese basic pre-liminary facts and notations which will be required later forformulating our main results

Definition 1 A square symmetric matrix M (mst)stisinS issaid to be a Coxeter matrix over a nonempty set S such thatall the diagonal entries are 1 and mst isin 2 3 4 infin

Definition 2 Let S be a set of vertices of a labeled graph ΓWe call Γ a Coxeter graph if any two of its vertices are

connected by an edge and if the label of each edge is greaterthan 2

By convention each edge is labeled only if the label isgreater than 3

Definition 3 A group with generator s and relations s2 1and (st)mst 1 for all s t isin S and mst neinfin is called a Coxetergroup such that (mst)stisinS is the Coxeter matrix

Definition 4 (e Artin group is

A langs isin S | sts middot middot middot1113980radicradic11139791113978radicradic1113981mstfactors

tst middot middot middot1113980radicradic11139791113978radicradic1113981mstfactorsrang

(1)

If the Coxeter group is finite thenA is called a sphericalArtin group

Definition 5 (e right-angled Artin groups or monoids areobtained if all the labels which are greater than or equal to 3of spherical Coxeter graphs are replaced with infin

(e Artin spherical groups are usually represented byCoxeter graphs (see [3 13]) these groups are An for nge 1 Bn

for nge 2 Dn for nge 4 E6 E7 E8 F4 G2 H3 H4 and I2(p)

for pge 5 and pne 6 Figure 1 contains these graphs

Definition 6 (see [14]) (e length of a word g s1 middot middot middot sn ofa finitely generated group G is the smallest nonnegativeinteger n for which s1 sn isin Scup Sminus 1 where S is the set ofgenerators of G

Definition 7 (see [14]) (e spherical growth series of a fi-nitely generated group G is HG(t) 1113936

infink0 aktk where ak is

the number of words of length kLet a b be a relation in a given monoid M (en in

length-lexicographic order a is greater then or equal to b Aword uwv has an ambiguity if uw and wv are left sides tworelations If α1v and uα2 are identical then uwv is solvable Ifα1v and uα2 differ by lexicographic order then we get a newrelation in M A presentation is said to be complete if so-lutions of all ambiguities are identical A reducible word isthe left side of a relation of a complete presentation of amonoid If w does not contain the LHS of any relation thenw is called a canonical word (e following notions are in[15ndash20] under different terminologies Grobner basescomplete presentation rewriting system and so on

3 Main Results

In this part we compute our main results

31 RecurrenceRelation of theMonoidM(Sinfinn ) In this paperwe study the star topology Sn and find recurrence relationsand the Hilbert series of the associated right-angled monoidM(Sinfinn ) We compute the growth rate of the monoidM(Sinfinn ) and using the graph we show that it is unbounded(e Coxeter graph of the star topology Sn is given by thefollowing graph (Figure 2)









+ fi(t) 1le ile n

(2)


u(t)

u1(t)

⋮

un(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A(t)


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

f(t)

f1(t)

⋮

fn(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)


t1u






cki 1113936n

ji


ck1 i n

⎧⎪⎪⎨

⎪⎪⎩(4)




(5)



Sn(λ)



⋮ ⋮ ⋮ ⋮ ⋮ ⋮




1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(6)


(An)nge1

(Bn)nge2

(Dn)nge4


4



F4

G2

H3

H4

x1 x2 x3

x1 x2 x3

x1 x2 x3

x5 xn

xn

xn

xnndash2

xnndash1

xnndash1


x4

(I2(p))pge5pne6

5x1 x2 x3 x4

x1 x2

p

4x1 x2 x3 x4

6x1 x2

5x1 x2 x3


xn

xnndash1

x1

x2 x3

x4

x5

Sn

Figure 2 Sn


Sn(λ)



⋮ ⋮ ⋮ ⋮ ⋮ ⋮




1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(7)


Tnminus 1(λ)



⋮ ⋮ ⋮ ⋮ ⋮



1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(8)




(9)




nminus 1

i1λi


(10)









nminus 1

i1λi



nminus 1

i1λi


(11)








(1) H(n)M (t) 1 + 1113936

ni1 H

(n)Mi(t)

(2) H(n)Mi(t) t + t 1113936

nji H


(3) H(n)M1(t) H

(n)Mn(t)

Proof




k 1 + 1113936ni1 1113936kge1ckit

k

1 + 1113936ni1 Hn


nji ckminus 1j


nkge 1 cki(t)tk


nkge 2 1113936

nji ckminus 1j(t)tk

t + t 1113936nji 1113936


nji H

(n)Mj(t)



Wn






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

X

H(n)M1(t)

H(n)M2(t)

⋮

H(n)Mn(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

t

t

⋮

t

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)



the following

Lemma 5


1t

1113874 1113875 (13)



Lemma 6

H(n)Mm(t)






by B (at is

Tm




⋮ ⋮ ⋮ ⋮ ⋮ ⋮


1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(15)


(n)Mm(t)


M(Sinfinn )

Theorem 3

H(n)M (t)

1tnSn(1t)

(16)


H(n)M (t) 1 + H

(n)M1(t) + H


[n)Mnminus 1(t)

+ H(n)Mn(t)

1

tnSn(1t)1113888t

nSn

1t

1113874 1113875 + t + t(1 minus t)

+ t(1 minus t)2


+ t1113889

1

tnSn(1t)1113888t

nminus 1Snminus 1

1t

1113874 1113875 + 2t + t(1 minus t)

+ t(1 minus t)2


1113889

tS1(1t) minus t

tnSn(1t)

1

tnSn(1t)

(17)





40 60 80 100 120n

rn

24

4

68

8

1012

12

1416

16

1820

20

2224262830323436

Figure 3 Sinfinn





Data Availability




Acknowledgments


References


































+ fi(t) 1le ile n

(2)


u(t)

u1(t)

⋮

un(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A(t)


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

f(t)

f1(t)

⋮

fn(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)


t1u






cki 1113936n

ji


ck1 i n

⎧⎪⎪⎨

⎪⎪⎩(4)




(5)



Sn(λ)



⋮ ⋮ ⋮ ⋮ ⋮ ⋮




1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(6)


(An)nge1

(Bn)nge2

(Dn)nge4


4



F4

G2

H3

H4

x1 x2 x3

x1 x2 x3

x1 x2 x3

x5 xn

xn

xn

xnndash2

xnndash1

xnndash1


x4

(I2(p))pge5pne6

5x1 x2 x3 x4

x1 x2

p

4x1 x2 x3 x4

6x1 x2

5x1 x2 x3


xn

xnndash1

x1

x2 x3

x4

x5

Sn

Figure 2 Sn


Sn(λ)



⋮ ⋮ ⋮ ⋮ ⋮ ⋮




1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(7)


Tnminus 1(λ)



⋮ ⋮ ⋮ ⋮ ⋮



1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(8)




(9)




nminus 1

i1λi


(10)









nminus 1

i1λi



nminus 1

i1λi


(11)








(1) H(n)M (t) 1 + 1113936

ni1 H

(n)Mi(t)

(2) H(n)Mi(t) t + t 1113936

nji H


(3) H(n)M1(t) H

(n)Mn(t)

Proof




k 1 + 1113936ni1 1113936kge1ckit

k

1 + 1113936ni1 Hn


nji ckminus 1j


nkge 1 cki(t)tk


nkge 2 1113936

nji ckminus 1j(t)tk

t + t 1113936nji 1113936


nji H

(n)Mj(t)



Wn






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

X

H(n)M1(t)

H(n)M2(t)

⋮

H(n)Mn(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

t

t

⋮

t

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)



the following

Lemma 5


1t

1113874 1113875 (13)



Lemma 6

H(n)Mm(t)






by B (at is

Tm




⋮ ⋮ ⋮ ⋮ ⋮ ⋮


1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(15)


(n)Mm(t)


M(Sinfinn )

Theorem 3

H(n)M (t)

1tnSn(1t)

(16)


H(n)M (t) 1 + H

(n)M1(t) + H


[n)Mnminus 1(t)

+ H(n)Mn(t)

1

tnSn(1t)1113888t

nSn

1t

1113874 1113875 + t + t(1 minus t)

+ t(1 minus t)2


+ t1113889

1

tnSn(1t)1113888t

nminus 1Snminus 1

1t

1113874 1113875 + 2t + t(1 minus t)

+ t(1 minus t)2


1113889

tS1(1t) minus t

tnSn(1t)

1

tnSn(1t)

(17)





40 60 80 100 120n

rn

24

4

68

8

1012

12

1416

16

1820

20

2224262830323436

Figure 3 Sinfinn





Data Availability




Acknowledgments


References



























Sn(λ)



⋮ ⋮ ⋮ ⋮ ⋮ ⋮




1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(7)


Tnminus 1(λ)



⋮ ⋮ ⋮ ⋮ ⋮



1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(8)




(9)




nminus 1

i1λi


(10)









nminus 1

i1λi



nminus 1

i1λi


(11)








(1) H(n)M (t) 1 + 1113936

ni1 H

(n)Mi(t)

(2) H(n)Mi(t) t + t 1113936

nji H


(3) H(n)M1(t) H

(n)Mn(t)

Proof




k 1 + 1113936ni1 1113936kge1ckit

k

1 + 1113936ni1 Hn


nji ckminus 1j


nkge 1 cki(t)tk


nkge 2 1113936

nji ckminus 1j(t)tk

t + t 1113936nji 1113936


nji H

(n)Mj(t)



Wn






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

X

H(n)M1(t)

H(n)M2(t)

⋮

H(n)Mn(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

t

t

⋮

t

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)



the following

Lemma 5


1t

1113874 1113875 (13)



Lemma 6

H(n)Mm(t)






by B (at is

Tm




⋮ ⋮ ⋮ ⋮ ⋮ ⋮


1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(15)


(n)Mm(t)


M(Sinfinn )

Theorem 3

H(n)M (t)

1tnSn(1t)

(16)


H(n)M (t) 1 + H

(n)M1(t) + H


[n)Mnminus 1(t)

+ H(n)Mn(t)

1

tnSn(1t)1113888t

nSn

1t

1113874 1113875 + t + t(1 minus t)

+ t(1 minus t)2


+ t1113889

1

tnSn(1t)1113888t

nminus 1Snminus 1

1t

1113874 1113875 + 2t + t(1 minus t)

+ t(1 minus t)2


1113889

tS1(1t) minus t

tnSn(1t)

1

tnSn(1t)

(17)





40 60 80 100 120n

rn

24

4

68

8

1012

12

1416

16

1820

20

2224262830323436

Figure 3 Sinfinn





Data Availability




Acknowledgments


References



























Lemma 6

H(n)Mm(t)






by B (at is

Tm




⋮ ⋮ ⋮ ⋮ ⋮ ⋮


1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(15)


(n)Mm(t)


M(Sinfinn )

Theorem 3

H(n)M (t)

1tnSn(1t)

(16)


H(n)M (t) 1 + H

(n)M1(t) + H


[n)Mnminus 1(t)

+ H(n)Mn(t)

1

tnSn(1t)1113888t

nSn

1t

1113874 1113875 + t + t(1 minus t)

+ t(1 minus t)2


+ t1113889

1

tnSn(1t)1113888t

nminus 1Snminus 1

1t

1113874 1113875 + 2t + t(1 minus t)

+ t(1 minus t)2


1113889

tS1(1t) minus t

tnSn(1t)

1

tnSn(1t)

(17)





40 60 80 100 120n

rn

24

4

68

8

1012

12

1416

16

1820

20

2224262830323436

Figure 3 Sinfinn





Data Availability




Acknowledgments


References






























Data Availability




Acknowledgments


References



























Documents

RecurrenceRelationsandHilbertSeriesoftheMonoid ...downloads.hindawi.com/journals/jmath/2020/5215631.pdf · that the removal of the relations a2 i 1 gives Artin groups. So, Coxeter