Abstract. In this paper, we consider a problem on finding the number of different single connected component containing a fixed root for a given number of vertices on semi-infinite Cayley tree. The solution of this problem is the well known Catalan numbers. The result is then extended to the complete graph. Then, we gave a suitable estimate for the given problem

Pah, Chin Hee

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BULLETIN of theMalaysian MathematicalSciences Societyhttp://math.usm.my/bulletinBull. Malays. Math. Sci. Soc. (2) 31(2) (2008), 175–183An Application of Catalan Numbers on Cayley Treeof Order 2: Single Polygon CountingC. H. PahDepartment of Computational and Theoretical Sciences,Faculty of Science, International Islamic University Malaysia,25250 Kuantan, Pahang, Malaysia.pahchinhee@hotmail.comAbstract. In this paper, we consider a problem on finding the number ofdifferent single connected component containing a fixed root for a given numberof vertices on semi-infinite Cayley tree. The solution of this problem is the wellknown Catalan numbers. The result is then extended to the complete graph.Then, we gave a suitable estimate for the given problem.2000 Mathematics Subject Classification: 82B20, 11B83Key words and phrases: Cayley tree, phase transition, contour method, Catalannumbers1. IntroductionIn statistical physics, certain algebraic problems especially on integer lattice Z2could be translated in combinatoric problems by a suitable choice of geometric rep-resentation. One of the most successful idea was suggested by Peierl i.e. the socalled “contour” method [8]. His argument is to consider the boundary of a spinconfiguration, was later given a more rigorous proof by Dobrusin and Griffith inde-pendently [3, 6] (for details, see [7, 4]). In Peierl’s study of Ising model, he proposeda geometrical object “contour” i.e. a set of connected edges which separate the dif-ferent “spin” for two nearest neighbours (sites) in a given configuration. The studyis then transformed from configuration space into geometric representation as con-tour, becoming a combinatoric problem. His investigation proceed by a fundamentalmathematical tools: counting, here counts the number of a class of contours. Thisidea turned out to be very fruitful [10, 11] in the line of research of phase transition,i.e. existence of non-uniqueness of limiting Gibbs measures.In integer lattice Z2, it is well known that the number of different non-equivalentpolygon is less than 4 · 3n−1 (the number of all polygonal paths passing througha given point), which could be found in all standard texts [7] discussed on phaseReceived: March 15, 2007; Revised: December 12, 2007.176 C. H. Pahtransition on Z2. The Gibbs measures constructed in the Ising model on Z2 (fordetails, see [7, 4]), a constant i-configuration, Λ ⊂ Z2 , is then(1.1) P i=+1Λ,β (V−) <∞∑n=4,6,...4 · 3n exp(−2βn).where n is the contour length, β is inverse temperature and V− is the event of thecenter having the negative spin value. This estimate is required in the line of theproof of existence of phase transition. The advantage of the contour method is thatno explicit calculation of the probability is required.A Cayley tree Γk = (V, L) (see [1]) of order k ≥ 1 is an infinite homogeneous tree,i.e., a graph without cycles, with exactly k + 1 edges incident to each vertex. HereV is the set of vertices and L that of edges (arcs). In Cayley tree, such an estimateas 4 · 3n−1 in Z2 is still not widely used. In [2], a similar result was given asLemma 1.1. Let G be a countable graph of maximal degree k and let Nm(x) be thenumber of connected subgraph G′ ⊂ G with x ∈ V (G′) and |E(G′)| = m. ThenNm(x) ≤ (k · e)m.Our result would be closer compare to this result only for the case k = 3. Inour investigation, we found a recurrent equation generating the number of polygonsaccording to the number of vertices. The sequence of these numbers is exactly thewell known Catalan numbers. Note that Fibonacci numbers appear naturally innature while Catalan numbers appear naturally in counting. One of the examplewhose solution is Catalan numbers is the number of orientation to write m brackets:m = 1 {()},m = 2 {()(), (())},m = 3 {()()(), ()(()), (())(), (()()), ((()))},......Catalan numbers [12, 5, 9], in term of binomial coefficient is defined as:(1.2) Cn =1n+ 1(2nn)=(2n)!(n+ 1)!n!for n ≥ 0.The main theorem is then proved based on the properties of Catalan numbers.An efficient way to calculate Cn is to write Cn in(1.3) Cn+1 =2(2n+ 1)n+ 2Cn.An Application of Catalan Numbers on Cayley Tree 177It also satisfies(1.4) C0 = 1, Cn+1 =n∑i=0CiCn−i.There are many famous counting problems in combinatorics whose solution isgiven by the Catalan numbers (see examples given in [12]), including this problem.Since the investigation on Catalan numbers is well established, we do not emphasison the properties on such sequence. This paper is mainly devoted to finding asuitable estimate for the number of polygons contain a fixed root on a Cayley treeof order 2.2. Semi-infinite Cayley TreeWe begin the investigation by the study on the semi-infinite Cayley tree (seeFigure 1). A semi-infinite Cayley tree of order 2, denoted by Γ2semi , i.e. a graphwithout cycles, with exactly 3 edges incident to each vertex, except a root x0 ∈ Vwhich only emanates 2 edges from the vertex.x0W0W1Figure 1. A semi-infinite Cayley tree J2 where each vertex except x0 exactly 3edges issue whereas x0 emanates 2 edges.We define distance d(x, y), x, y ∈ V , is the length of (i.e. the number of edges in)the shortest path connecting x with y. Two vertices x, y are called connected (ornearest neighbour) if exist single edge connect x and y, or d(x, y) = 1. For the fixedx0, we setWn = {x ∈ V |d(x, x0) = n},Vn = ∪n=0Wnwhere W0 = x0, andLn = {l =< x, y >∈ L|x, y ∈ Vn}.DenoteS(x) = {y ∈Wn+1|d(x, y) = 1, x ∈ Wn}.The defined set is called the set of direct successors. Observe that any vertex x ∈ Vnhas 2 successors in a semi-finite Cayley of order 2.Naively, we borrow the term “polygon” as in Z2.178 C. H. PahDefinition 2.1. A polygon Λ is a single connected component where Λ ⊂ V .Geometrically, we actually do not see any polygon as a subset of V .Let | · | denote the cardinality of a set.Definition 2.2. N(m) is the number of different polygons with m number of verticesthat could be constructed containing a fixed root x0.One could use a number system to represent the semi-infinite Cayley of order2. We fix root x0 = x1 as 1, then from each vertex label k, we emanate to twonumbers 2k and 2k + 1 as two successors, so that any k and 2k or k and 2k + 1are nearest neighbour. In order to generate each polygon containing x0, here we listeach sequence of combinations by following algorithm:{xi : x1 = 1 and ∀xk, exist xj , j < k, such that xk = 2xj or xk = 2xj + 1.}The first four terms can be generated as follows:N(m = 1) = 1, {(1)}N(m = 2) = 2, {(1, 2), (1, 3)}N(m = 3) = 5, {(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 6), (1, 3, 7)}N(m = 4) = 14, {(1, 2, 3, 4), (1, 2, 3, 5), (1, 2, 3, 6), (1, 2, 3, 7), (1, 2, 4, 5),(1, 2, 4, 8), (1, 2, 4, 9), (1, 2, 5, 10), (1, 2, 5, 11), (1, 3, 6, 7),(1, 3, 6, 12), (1, 3, 6, 13), (1, 3, 7, 14), (1, 3, 7, 15)}One could find N(m) easily in the next section.3. Main Result3.1. Semi-infinite Cayley tree of order 2.Lemma 3.1. The number of polygons with m number of vertices which containinga root x0, N(m) is given by recursion below:(3.1) N(m) =m−1∑r=0N(m− r − 1)N(r), N(0) = 1 and N(1) = 1.Proof. First we divide the problem of finding m number of vertices that contain aroot x0 into(i) r number of vertices which containing a root y0 which is a successor of x0 ,i.e, N(r); and(ii) m − r − 1 number of vertices which containing a root z0 which is anothersuccessor of x0, i.e. N(m− r − 1) (see Figure 2).The total combination is N(r) · N(m − r − 1). Then N(m) is directly sum over rfrom 0 to m − 1, the multiplication of all combination of N(m − r − 1)N(r). Wedefine N(0) = 1 and N(1) = 1 is simply a result from observation.An Application of Catalan Numbers on Cayley Tree 179x0z0y0Figure 2. y0 connect to r vertices and z0 connect to m− r − 1 vertices.The first 10 terms, starting from N(1), are listed as follow based on formulaabove:1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ....and this is the well-known Catalan numbers. From (1.4), with initial value N(0) = 1and N(1) = 1, evidentlyTheorem 3.1. N(m) is the Catalan numbers Cm.This is not a surprising result because the applications of a binary operator canbe represented in terms of a binary tree. It follows that Cn is the number of rootedordered binary trees with n + 1 leaves. This problem is identical to the number ofpolygons that we defined above, N(m). For any m vertices connected component,there are m + 1 leaves. Below, we shall find an estimate base on the properties ofCatalan numbers given in previous section.From equation (1.4), we have the following proposition.Proposition 3.1.(3.2)N(m)N(m− 1) < 4.Proof.Nn+1Nn=2(2n+ 1)n+ 2<2(2n+ 4)n+ 2= 4.The main theorem that we would like to prove is the estimate of N(m).Theorem 3.2. The inequality of N(m) is given as(3.3)4m−1m3/2≤ N(m) ≤ 4m−1m.Proof. We are going to prove this by induction for two cases. Form = 1, 1 = N(1) =1, the inequality (3.3) holds.Suppose that N(m)≤4m−1/m. From (1.4),N(m+ 1) =2(2m+ 1)m+ 2N(m)<2(2m+ 1)m+ 24m−1m180 C. H. Pah< 22mm+ 14m−1m=4mm+ 1,m > 1.Suppose that N(m) ≥ 4m−1m3/2and multiply both side by2(2m+ 1)m+ 2.N(m) · 2(2m+ 1)m+ 2≥ 4m−1m3/2· 2(2m+ 1)m+ 2N(m+ 1) ≥ 4m−1m3/2· 2(2m+ 1)m+ 2>4m−1m3/2· 4mm+ 2>4m(m+ 1)3/2.From (1.3), one could easily verify thatCorollary 3.1.(3.4) limm→∞N(m)N(m− 1) = 4.We are also include here a less motivated result.Corollary 3.2. The number of all possible polygons with maximally m vertices whichcontaining a root x0, is(3.5)m∑i=1N(i) <4m3.Proof. From Proposition 3.1 and geometric progression;m∑i=1N(i) <m∑i=14i−1 =4m − 13<4m3.3.2. Complete graph. Similarly, we derive the N˜(m) which carry the same mean-ing as previous section but on the complete graph of Cayley of order 2, Γ2.Definition 3.1. N˜(m) is the number different polygons Λ, where Λ ⊂ Γ2 with|Λ| = m number of vertices that could be constructed containing a root x0.Note that N(m) without tilde is merely the Catalan numbers used in previoussection.Lemma 3.2. The number of different polygons N˜(m) with m number of verticeswhich contain a root x0, is given by recursion below:(3.6) N˜(m) =m∑r=1N(m− r)N(r), N(0) = 1 and N(1) = 1.Proof. First, we decompose the problem of finding the number of polygons of mnumber of vertices containing a root x0 into counting:An Application of Catalan Numbers on Cayley Tree 181(i) r number of vertices which containing x0 , i.e, N(r) and(ii) m−r number of vertices which containing a root y which is another successorof x0, i.e. N(m− r) (see Figure 3).The former N(r) must always count x0, then it is not allow to be N(0). The totalN˜(m) is then the sum of all N(m− r)N(r) where r range from 1 to m.x0y0Figure 3. x0 connect to r vertices (top) and y0 connect to m− r vertices (bottom).The first ten terms of N˜(m) are listed:1, 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, ...In this case, N˜(0) is undefined. The recursion (3.6) allow us to state the followinglemma.Lemma 3.3.(3.7) N˜(m) = N(m+ 1)−N(m), N(0) = 1.Proof. From Lemma 3.1 and Lemma 3.2,N˜(m) =m∑r=0N(m− r)N(r) −N(m) = N(m+ 1)−N(m).From lemma above, it is not difficult to prove the result below.Lemma 3.4.(3.8) N˜(m) =3mm+ 2N(m).Proof. From Equation (1.3) and Lemma 3.3,N˜(m) = N(m+ 1)−N(m)=2(2m+ 1)m+ 2N(m)−N(m)=3mm+ 2N(m).From Theorem 3.3 and Lemma 3.4, it is not difficult to deduce the followingresult:(3.9)3(m+ 2)√m4m−1 ≤ N˜(m) ≤ 3m+ 24m−1.182 C. H. Pahandlimm→∞N˜(m)N˜(m− 1) = 4.We can see that both N(m) and N˜(m) proportional to 4m. As m increase, N(m)increase by not more than 4 times of previous N(m− 1). The exact value of N(m)would be close to the form Am4m where Am are some positive constant depend onm. Note that the Catalan numbers has the order of O(4m−1m3/2)which the lowerbound is found to be4m−1m3/2and the asymptote of the Catalan numbers is given in[14] asN(m) ∼ 4mm3/2√pi.Since N˜(m) ≤ 3m+24m−1 < 3 ·4m−1, we can choose a simple estimate i.e. 3 ·4m−1for the problem on Cayley tree of order 2 stated in introduction. Carefully identifyingeach term in the Lemma 1.1 , the estimate obtained 3 · 4m−1 is less (3 · e)m+1 fromLemma 1.1 obtained for the same problem. Somehow 3 · 4m−1 is much simple andcloser for the purpose in the study discussed in the introduction.4. ConclusionWe proved that a solution to the problem of finding number of different “polygon”containing a fixed root for a given number of vertices on semi-infinite Cayley tree oforder 2 is exactly the well known Catalan numbers. We gave two suitable estimatesasN(m) ≤ 4m−1mandN˜(m) ≤ 3 · 4m−1m+ 2< 3 · 4m−1.Acknowledgements. The author thanks the referee for the comments. This re-search is funded by the SAGA Fund P77c by the Ministry of Science, Technology andInnovation (MOSTI) through the Academy of Sciences Malaysia(ASM). The authoris grateful to the Faculty of Science of IIUM for the facilities provided to him inundertaking his doctorate programme. The author thanks his supervisor Prof. Dr.Nasir Ganikhodjaev for suggesting the problem and also his valuable discussions.References[1] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic, London, 1982.[2] C. Borgs, Statistical Physics Expansion Methods in Combinatorics and Computer Science,http://research.microsoft.com/∼borgs/CBMS.pdf, March 22, 2004.[3] R. L. Dobrushin, Existence of a phase transition in the two dimensional and three-dimensionalIsing model. Soviet Phys. Doklady 10(1965), 111–113 .[4] H.-O. Georgii, Gibbs Measures and Phase Transitions, Walte de Gruyter, Berlin, 1998.[5] R. L. Graham, D. E. Knuth and O. Patashnik, Exercise 9.8 in Concrete Mathematics: AFoundation for Computer Science, Second ed., Reading, MA: Addison-Wesley, 1994.An Application of Catalan Numbers on Cayley Tree 183[6] R. B. Griffiths, Peierls’ proof of spontaneous magnetization of a two-dimensional Ising ferro-magnet. Phys. Rev. A136(1964), 437–439.[7] R. A. Minlos, Introduction To Mathematical Statistical Physics, Amer. Math. Soc., Providence,RI, 2000.[8] R. Peierls, On Ising model of ferro magnetism. Proc. Cambridge Phil. Soc. 32(1936), 477–481.[9] S. Pemmaraju and S. Skiena, Computational Discrete Mathematics, Cambridge Univ. Press,Cambridge, 2003[10] S. A. Pirogov and Ya. G. Sinai, Phase diagrams of classical lattice systems, I. Theor. Math.Phys. 25(1975), 1185–1192 .[11] S. A. Pirogov and Ya. G. Sinai, Phase diagrams of classical lattice systems, II. Theor. Math.Phys. 26(1976), 39–49 .[12] R. P. Stanley, Catalan addendum, http://math.mit.edu/∼rstan/ec/catadd.pdf, version of6 October 2008; 72 pages.[13] J. H. van Lint and R. M. Wilson, A course in combinatorics, Second edition, Cambridge Univ.Press, Cambridge, 2001.[14] I. Vardi, Computational recreations in Mathematica, Addison-Wesley, Redwood City, CA,1991.

An application of Catalan Numbers on Cayley tree of order 2: Single Polygon Counting

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An application of Catalan Numbers on Cayley tree of order 2: Single Polygon Counting

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