DOCSLIB.ORG

Enumeration Via Ballot Numbers Martin Aigner

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Enumeration Via Ballot Numbers Martin Aigner View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 308 (2008) 2544–2563 www.elsevier.com/locate/disc Enumeration via ballot numbers Martin Aigner Mathematics Institute, Freie Universität Berlin, Arnimallee 3, D-14195 Berlin, Germany Received 27 April 2004; received in revised form 19 June 2007; accepted 21 June 2007 Available online 22 October 2007 Abstract Several interesting combinatorial coefficients such as the Catalan numbers and the Bell numbers can be described either via a 3-term recurrence or as sums of (weighted) ballot numbers. This paper gives some general results connecting 3-term recurrences with ballot sequences with several applications to the enumeration of various combinatorial instances. © 2007 Elsevier B.V. All rights reserved. Keywords: Ballot numbers; Catalan-like numbers; Weighted sums; Combinatorial instances 1. Introduction 2n Our starting point is a two-fold description of the Catalan numbers Cn = (1/(n + 1)) n . First, consider the ballot number recurrence: a0,0 = 1,a0,k = 0 (k > 0), an,k = an−1,k−1 + an−1,k +···+an−1,n−1 (n1). Then a classical result says n an,k = Cn+1,an,0 = Cn for all n. k=0 The matrix A = (an,k) indexed by N0 is lower triangular with 1’s in the main diagonal. The following table shows the first rows of A with the 0’s omitted: 1 11 221 A = 5531 14 14 9 4 1 42 42 28 14 5 1 ... 2n−k The numbers an,k =[(k + 1)/(n + 1)] n are called the (ordinary) ballot numbers. E-mail address: [email protected]. 0012-365X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2007.06.012 M. Aigner / Discrete Mathematics 308 (2008) 2544–2563 2545 The ubiquity of the Catalan numbers in enumeration problems (the book of Stanley [20] lists more than 70 instances) is partly explained by this description via the ballot number recurrence. Very often, the set Sn to be counted can be partitioned in a natural way into subsets Sn,k (0k n) such that the sizes an,k =|Sn,k| obey the ballot number recurrence, resulting in |Sn|=Cn+1. We will see several examples later on. The second description of the Catalan numbers is less well-known. Consider the 3-term recurrence: b0,0 = 1,b0,k = 0 (k > 0), bn,k = bn−1,k−1 + 2bn−1,k + bn−1,k+1 (n1). Then bn,0 = Cn+1 for all n. The matrix B = (bn,k) is again lower-triangular. The table shows the first rows: 1 21 541 B = 14 14 6 1 42 48 27 8 1 132 165 110 44 10 1 ... S = n a b We see that the sum n k=0 n,k of the ballot table equals the coefficient n,0 computed via a 3-term recurrence. It is the purpose of this paper to describe several instances where the number Sn of generalized ballot numbers is equal to bn,0 for general 3-term recurrences. In particular, this approach via ballot tables will give some indica- tion why, next to the Catalan numbers, the Motzkin numbers [1,7], Riordan numbers [7] and Fine numbers [11,12] appear so often in enumeration problems. Hence the present article may be considered as a companion paper to [7,12]. Let us consider generalized ballot numbers first. Let = (m0,m1,m2,...) be a sequence of real numbers. We associate with the lower triangular matrix A = (an,k) as follows (omitting the index ): a0,0 = 1,a0,k = 0 (k > 0), an,k = an−1,k−1 + mkan−1,k + an−1,k+1 +···+an−1,n−1 (n1). (1) n We call A the ballot table corresponding to . We are interested in the sum Sn = an,k and the element in k=0 column zero, Zn := an,0, for n0. Next we turn to 3-term recurrences. Let = (s0,s1,s2,...), = (t1,t2,...) be two sequences of real numbers, , , where ti = 0 for all i. We associate with , the matrix B = (bn,k) in the following way (again omitting the super- scripts): b0,0 = 1,b0,k = 0 (k > 0), bn,k = bn−1,k−1 + skbn−1,k + tk+1bn−1,k+1 (n1), (2) , and call Bn := bn,0 the Catalan-like numbers corresponding to and . The recurrence (2) has been considered by many authors via continued fractions, orthogonal polynomials, generating functions, etc. for special sequences and (see e.g. [2,4,13,21]). In view of later applications to enumeration problems, 2546 M. Aigner / Discrete Mathematics 308 (2008) 2544–2563 , let us note some important instances where the numbers Bn are known (see [4]): Consider = (1, 1, 1,...). The most interesting Catalan-like numbers arise in the case when is constant from the term s1 on; we then write shortly = (s0,s1): = (0, 0), B2n = Cn Catalan number,B2n+1 = 0, = (1, 1), Bn = Mn Motzkin number, = (2, 2), Bn = Cn+1, n = (1, 0), Bn = , n/2 2n + 1 = (3, 2), Bn = , n = (0, 1), Bn = Rn Riordan number, = (1, 2), Bn = Cn, = (0, 2), Bn = Fn Fine number. Other examples are 2n = (2, 2), Bn = , n = (2, 1, 1,...), = (2, 3), Bn = Schn Schröder number, = (2, 2, 2,...) and n−1 n−k = (s, s + 1), Bn = N(n, k)s , k=0 1 n n = (s,s,s,...) where N(n, k) = is the Narayana number. n k k + 1 Finally, we note that for = (1, 2, 3, 4,...), = (1, 2, 3, 4,...)we obtain the Bell number Bn = Belln, counting the number of partitions of an n-set, and for = (2, 3, 4, 5,...), = (1, 2, 3, 4,...)we obtain Bn = Belln+1 (see e.g. [3]). It was shown in [2] that the generating function , , n B (x) = Bn x for = (a,s,s,...), = (t,t,t,...) n 0 is given as follows: 2 2 , 1 − (2a − s)x − 1 − 2sx + (s − 4t)x B (x) = . (3) 2(s − a)x + 2(a2 − as + t)x2 Actually, in [2] only the case t = 1 is considered, but the more general result follows in the same way. In the next section we prove the main theorem connecting ballot numbers and Catalan-like numbers, together with some ramifications. In Section 3 we discuss several combinatorial instances, with more general weighted versions appearing in Section 4. We follow the usual terminology, for all terms not defined the reader may consult any of the standard texts [14,17,19]. M. Aigner / Discrete Mathematics 308 (2008) 2544–2563 2547 2. Ballot numbers and Catalan-like numbers S = n a A B, = b, To equate a sum n k=0 n,k of a ballot table as in (1) with a Catalan-like number n n,0 of the matrix B, as in (2), there are two approaches: , (A) Find an explicit expression of the coefficients bn,k in terms of an,k. (B) Compute the respective generating functions. We will illustrate both methods as we go along. In the sequel we use the following short notation: S , , ←→ means that Sn = Bn for all n, where Sn and Bn are defined according to (1) and (2), Z , ←→ means that Zn = Bn for all n. In this notation our starting example reads: S = (2, 2, 2,...) = (1, 1, 1,...)←→ (Sn = Bn = Cn+1), = (1, 1, 1,...) Z = (1, 2, 2,...) = (1, 1, 1,...)←→ (Zn = Bn = Cn). = (1, 1, 1,...) ←S→ (a, a, ) ←S→ (a+1,) (a, ) ←Z→ (a,) Theorem 1. If , then (1,) and (1,) . Proof. We prove the first assertion, the second is shown similarly. Consider the ballot matrix A with respect to . The columns 0, 1,...,2n determine all elements down to row 2n + 1, and hence all sums Sk (k = 0,...,2n + 1). Similarly, in the matrix B, corresponding to and , the columns 0, 1,...,n determine the Catalan-like numbers B, = b k = , ,..., n + C(x)= c xn C(x)= c xn k k,0 for 0 1 2 1. For any generating function n 0 n set n 0 n+1 , thus C(x)=(C(x)−C(0))/x. Considering the ballot recurrence for the columns 0, 1,...,2n we obtain for the generating functions Ai(x) of the ith column up to i = 2n the system, where for brevity we omit here and later often the variable x: A − 0 1 = A = m A + A +···+A x 0 0 0 1 2n A 1 = A = A + m A +···+A x 1 0 1 1 2n . A 2n = A = A + m A x 2n 2n−1 2n 2n and therefore (1 − m0x)A0 − xA1 − xA2 −···−xA2n = 1, −xA0 + (1 − m1x)A1 − xA2 −···−xA2n = 0, . −xA2n−1 + (1 − m2nx)A2n = 0. 2548 M. Aigner / Discrete Mathematics 308 (2008) 2544–2563 Denote by ⎛ ⎞ 1 − m0x −x ... −x ⎜ ⎟ ⎜ −x − m x ... −x ⎟ ⎜ 1 1 ⎟ M = ⎜ ⎟ ⎜ .. ⎟ ⎝ . ⎠ 0 −x 1 − m2nx the matrix of coefficients. By Cramer’s rule it follows that j (−1) Ej (M) Aj (x) = , det M where Ej (M) (j = 0,...,2n) is the determinant of the submatrix of M with the 0th row and jth column deleted. In particular, we obtain 2n 2n (− )j E (M) i j=0 1 j Q(x) = qix = Aj (x) = , det M i 0 j=0 and we know that qi = Si for 0i 2n + 1. Turning to the Catalan-like numbers defined by and , we find in an analogous way for the column-generating functions B0(x),...,Bn(x) up to n the coefficient matrix ⎛ ⎞ 1 − s0x −t1x 0 ...0 ⎜ ⎟ ⎜ −x − s x −t x ..
Load more

Recommended publications
  • Total Positivity of Narayana Matrices Can Also Be Obtained by a Similar Combinatorial Approach?
    Total positivity of Narayana matrices Yi Wanga, Arthur L.B. Yangb aSchool of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P.R. China bCenter for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China Abstract We prove the total positivity of the Narayana triangles of type A and type B, and thus affirmatively confirm a conjecture of Chen, Liang and Wang and a conjecture of Pan and Zeng. We also prove the strict total positivity of the Narayana squares of type A and type B. Keywords: Totally positive matrices, the Narayana triangle of type A, the Narayana triangle of type B, the Narayana square of type A, the Narayana square of type B AMS Classification 2010: 05A10, 05A20 1. Introduction Let M be a (finite or infinite) matrix of real numbers. We say that M is totally positive (TP) if all its minors are nonnegative, and we say that it is strictly totally positive (STP) if all its minors are positive. Total positivity is an important and powerful concept and arises often in analysis, algebra, statistics and probability, as well as in combinatorics. See [1, 6, 7, 9, 10, 13, 14, 18] for instance. n Let C(n, k)= k . It is well known [14, P. 137] that the Pascal triangle arXiv:1702.07822v1 [math.CO] 25 Feb 2017 1 1 1 1 2 1 P = [C(n, k)]n,k≥0 = 13 31 14641 . . .. Email addresses: [email protected] (Yi Wang), [email protected] (Arthur L.B. Yang) Preprint submitted to Elsevier April 12, 2018 is totally positive.
    [Show full text]
  • Entropoid Based Cryptography
    Entropoid Based Cryptography Danilo Gligoroski∗ April 13, 2021 Abstract The algebraic structures that are non-commutative and non-associative known as entropic groupoids that satisfy the "Palintropic" property i.e., xAB = (xA)B = (xB)A = xBA were proposed by Etherington in ’40s from the 20th century. Those relations are exactly the Diffie-Hellman key exchange protocol relations used with groups. The arithmetic for non-associative power indices known as Logarithmetic was also proposed by Etherington and later developed by others in the 50s-70s. However, as far as we know, no one has ever proposed a succinct notation for exponentially large non-associative power indices that will have the property of fast exponentiation similarly as the fast exponentiation is achieved with ordinary arithmetic via the consecutive rising to the powers of two. In this paper, we define ringoid algebraic structures (G, , ∗) where (G, ) is an Abelian group and (G, ∗) is a non-commutative and non-associative groupoid with an entropic and palintropic subgroupoid which is a quasigroup, and we name those structures as Entropoids. We further define succinct notation for non-associative bracketing patterns and propose algorithms for fast exponentiation with those patterns. Next, by an analogy with the developed cryptographic theory of discrete logarithm problems, we define several hard problems in Entropoid based cryptography, such as Discrete Entropoid Logarithm Problem (DELP), Computational Entropoid Diffie-Hellman problem (CEDHP), and Decisional Entropoid Diffie-Hellman Problem (DEDHP). We post a conjecture that DEDHP is hard in Sylow q-subquasigroups. Next, we instantiate an entropoid Diffie-Hellman key exchange protocol. Due to the non-commutativity and non-associativity, the entropoid based cryptographic primitives are supposed to be resistant to quantum algorithms.
    [Show full text]
  • Enumeration of Colored Dyck Paths Via Partial Bell Polynomials
    Enumeration of Colored Dyck Paths Via Partial Bell Polynomials Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara and Michael D. Weiner Abstract Weconsideraclassoflatticepathswithcertainrestrictionsontheirascents and down-steps and use them as building blocks to construct various families of Dyck paths. We let every building block Pj take on c j colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in a unified manner. Keywords Colored Dyck paths Colored Dyck words Colored Motzkin paths Partial Bell polynomials · · · 2010 Mathematics Subject Classification Primary: 05A15 Secondary: 05A19 · 1 Introduction A Dyck path of semilength n is a lattice path in the first quadrant, which begins at the origin (0, 0),endsat(2n, 0),andconsistsofsteps(1, 1) and (1, 1).Itiscustomary to encode an up-step (1, 1) with the letter u and a down-step (1−, 1) with the letter − D. Birmajer Department of Mathematics, Nazareth College, 4245 East Ave., Rochester, NY 14618, USA e-mail: [email protected] J. B. Gil (B) M. D. Weiner Penn State Altoona,· 3000 Ivyside Park, Altoona, PA 16601, USA e-mail: [email protected] M. D. Weiner e-mail: [email protected] P. R. W. McNamara Department of Mathematics, Bucknell University, 1 Dent Drive, Lewisburg, PA 17837, USA e-mail: [email protected] ©SpringerNatureSwitzerlandAG2019 155 G. E.
    [Show full text]
  • Combinatorial Counting
    COMBINATORIAL COUNTING MARTIN KLAZAR i Contents Preface iv 1 Catalan numbers 1 1.1 Definition, recurrence, parity, exponential growth . 1 1.2 What is an effective formula? For example, for n 7! cn? . 6 1.3 Recurrences, explicit formulae, asymptotics . 7 1.4 Dyck words, good bracketings and pattern-avoiding permutations 12 1.5 Can (cn) satisfy a linear recurrence with constant coefficients? . 18 1.6 Refining cn | the Narayana numbers . 22 1.7 Stanley's list and Valtr's theorem . 24 Comments and references . 29 2 Analytic intermezzo I. Stirling's formula for factorial 30 2.1 Approximating sums with integrals . 31 2.2 The gamma function . 32 2.3 Cauchy's formula . 34 2.4 Comments and references . 35 3 0 { 1 laws and the Blatter{Specker theorem 37 3.1 The Bollob´as{Thomasontheorem: thresholds exist . 38 3.2 The GKLT{F theorem: enumerative first-order 0 { 1 laws . 42 3.3 The Friedgut{Kalai theorem: thresholds are sharp . 49 3.4 The Shelah{Spencer theorem: irrational exponents are not first order . 49 3.5 The Blatter{Specker theorem: second-order binary structures are periodic . 49 4 Algebra of generating functions 50 4.1 The ring of formal power series . 50 4.2 Formal convergence in C[[x]] and C((x)) . 53 4.3 Differentiation, composition, exp and log . 61 4.4 Composition of GF: exponential GF and the Lagrange inversion formula . 68 4.5 Effective computation of modular reductions . 68 ii 4.6 More on algebra in C[[x]] and C((x)) . 68 4.7 Comments and references .
    [Show full text]
  • Generalized Chung-Feller Theorems for Lattice Paths
    Generalized Chung-Feller Theorems for Lattice Paths A Dissertation Presented to The Faculty of the Graduate School of Arts and Sciences Brandeis University Department of Mathematics Ira M. Gessel, Advisor In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by Aminul Huq August, 2009 This dissertation, directed and approved by Aminul Huq's committee, has been accepted and approved by the Faculty of Brandeis University in partial fulfillment of the requirements for the degree of: DOCTOR OF PHILOSOPHY Adam Jaffe, Dean of Arts and Sciences Dissertation Committee: Ira M. Gessel, Dept. of Mathematics, Chair. Susan F. Parker, Dept. of Mathematics Richard P. Stanley, Dept. of Mathematics, Massachusetts Institute of Technology c Copyright by Aminul Huq 2009 Dedication To My Parents iv Acknowledgments I wish to express my heartful gratitude to my advisor, Professor Ira M. Gessel, for his teaching, help, guidance, patience, and support. I am grateful to the members of my dissertation defense committee Professor Richard P. Stanley and Professor Susan F. Parker. Specially I'm greatly indebted to Professor Parker for her continual encouragement and mental support. I learned a great deal from her about teaching and mentoring. I owe thanks to the faculty, specially Professor Mark Adler and Professor Daniel Ruberman, to my fellow students, and to the kind and supportive staff of the Brandeis Mathematics Department. I would like to thank all my family and friends for their love and encouragement with patience and I wish to express my boundless love to my wife, Arifun Chowdhury. This thesis is dedicated to my parents, Md.
    [Show full text]
  • RIMS-1824 Rigged Configurations and Catalan, Stretched Parabolic
    RIMS-1824 Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials : Polynomiality, Unimodality and Log-concavity By Anatol N. KIRILLOV May 2015 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials : Polynomiality, Unimodality and Log-concavity anatol n. kirillov Research Institute of Mathematical Sciences ( RIMS ) Kyoto 606-8502, Japan URL: http://www.kurims.kyoto-u.ac.jp/ ~kirillov and The Kavli Institute for the Physics and Mathematics of the Universe ( IPMU ), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan Abstract We will look at the Catalan numbers from the Rigged Configurations point of view originated [9] from an combinatorial analysis of the Bethe Ansatz Equations associ- ated with the higher spin anisotropic Heisenberg models . Our strategy is to take a combinatorial interpretation of Catalan numbers Cn as the number of standard Young 2 tableaux of rectangular shape (n ), or equivalently, as the Kostka number K(n2);12n , as the starting point of research. We observe that the rectangular (or multidimen- sional) Catalan numbers C(m; n) introduced and studied by P. MacMahon [21], [30], see also [31], can be identified with the Kostka number K(nm);1mn , and therefore can be treated by Rigged Configurations technique. Based on this technique we study the stretched Kostka numbers and polynomials, and give a proof of \ a strong rationality \ of the stretched Kostka polynomials. This result implies a polynomiality property of the stretched Kostka and stretched Littlewood{Richardson coefficients [7], [26], [16]. Another application of the Rigged Configuration technique presented, is a new fam- ily of counterexamples to Okounkov's log-concavity conjecture [25].
    [Show full text]
  • Arxiv:2009.04900V2 [Math.CO] 11 Sep 2020 Counting Generalized
    Counting generalized Schr¨oder paths Xiaomei Chen∗, Yuan Xiang School of Mathematics and Computational Science Hunan University of Science and Technology Xiangtan 411201, China Mathematics Subject Classifications: 05A15 Abstract A Schr¨oder path is a lattice path from (0, 0) to (2n, 0) with steps (1, 1), (1, 1) and (2, 0) that never goes below the x axis. A small − − Schr¨oder path is a Schr¨oder path with no (2, 0) steps on the x axis. − In this paper, a 3-variable generating function RL(x,y,z) is given for Schr¨oder paths and small Schr¨oder paths respectively. As corollaries, we obtain the generating functions for several kinds of generalized Schr¨oder paths counted according to the order in a unified way. Keywords: Schr¨oder path, Narayana number, generating function 1 Introduction arXiv:2009.04900v2 [math.CO] 11 Sep 2020 In this paper, we will consider the following sets of steps for lattice paths: S = (1, 1), (1, 1) , 1 { − } S = (r, r), (r, r) r N+ , 2 { − | ∈ } S = (1, 1), (1, 1), (2, 0) , 3 { − } S = (r, r), (r, r), (2r, 0) r N+ , 4 { − | ∈ } ∗Corresponding author: [email protected] 1 S = (1, 1), (1, 1), (2r, 0) r N+ , 5 { − | ∈ } S = (r, r), (r, r), (2, 0) r N+ , 6 { − | ∈ } where (r, r), (r, r), (2r, 0) are called up steps, down steps and horizontal − steps respectively. For a given set S of steps, let LS (n) denote the set of lattice paths from (0, 0) to (2n, 0) with steps in S, and never go below the x axis.
    [Show full text]
  • Some Properties and Combinatorial Implications of Weighted Small Schr
    Some Properties and Combinatorial Implications of Weighted Small Schr¨oder Numbers Yu Hin (Gary) Au Department of Mathematics and Statistics University of Saskatchewan Saskatoon, SK Canada [email protected] Abstract th The n small Schr¨oder number is s(n) = k≥0 s(n, k), where s(n, k) denotes the number of plane rooted trees with n leaves and k internal nodes that each has at least P two children. In this manuscript, we focus on the weighted small Schr¨oder numbers k sd(n)= k≥0 s(n, k)d , where d is an arbitrary fixed real number. We provide recur- sive and asymptotic formulas for s (n), as well as some identities and combinatorial P d interpretations for these numbers. We also establish connections between sd(n) and several families of Dyck paths. 1 Introduction 1.1 Small Schr¨oder numbers The small Schr¨oder numbers, denoted s(n) for every n 1 gives the sequence ≥ n 12345 6 7 8 ··· s(n) 1 1 3 11 45 197 903 4279 ··· arXiv:1912.00555v1 [math.CO] 2 Dec 2019 (A001003 in the On-line Encyclopedia of Integer Sequences (OEIS) [13]). This sequence has been studied extensively and has many combinatorial interpretations (see, for instance, [16, Exercise 6.39]). We list two below: Let n denote the set of Schr¨oder trees with n leaves, which are plane rooted trees • whereT each internal node has at least 2 children. Then s(n) = for every n 1. |Tn| ≥ Figure 1 shows the s(4) = 11 Schr¨oder trees with 4 leaves.
    [Show full text]
  • Arxiv:Math/0611106V2 [Math.CO] 18 Oct 2007
    Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups Drew Armstrong Author address: Department of Mathematics, Cornell University, Ithaca, New York 14853 Current address: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 E-mail address: [email protected] arXiv:math/0611106v2 [math.CO] 18 Oct 2007 For Moira. Contents Abstract ix Acknowledgements xi Chapter 1. Introduction 1 1.1. Coxeter-Catalan Combinatorics 1 1.2. Noncrossing Motivation 6 1.3. Outline of the Memoir 9 Chapter 2. Coxeter Groups and Noncrossing Partitions 13 2.1. Coxeter Systems 13 2.2. Root Systems 16 2.3. Reduced Words and Weak Order 19 2.4. Absolute Order 22 2.5. Shifting and Local Self-Duality 26 2.6. Coxeter Elements and Noncrossing Partitions 29 2.7. InvariantTheoryandCatalanNumbers 35 Chapter 3. k-Divisible Noncrossing Partitions 41 3.1. Minimal Factorizations 41 3.2. Multichains and Delta Sequences 42 3.3. Definition of k-Divisible Noncrossing Partitions 46 3.4. Basic Properties of k-Divisible Noncrossing Partitions 48 3.5. Fuss-Catalan and Fuss-Narayana Numbers 59 3.6. The Iterated Construction and Chain Enumeration 66 3.7. Shellability and Euler Characteristics 72 Chapter 4. The Classical Types 81 4.1. Classical Noncrossing Partitions 81 4.2. TheClassicalKrewerasComplement 87 4.3. Classical k-Divisible Noncrossing Partitions 92 4.4. Type A 102 4.5. Type B 104 4.6. Type D 111 Chapter 5. Fuss-Catalan Combinatorics 115 5.1. Nonnesting Partitions 115 5.2. Cluster Complexes 128 5.3. Chapoton Triangles 143 5.4. Future Directions 147 vii viii CONTENTS Bibliography 155 Abstract This memoir is a refinement of the author’s PhD thesis — written at Cornell University (2006).
    [Show full text]
  • UNIVERSITY of CALIFORNIA, SAN DIEGO Flips
    UNIVERSITY OF CALIFORNIA, SAN DIEGO Flips and Juggles A Dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Jay Cummings Committee in charge: Professor Ron Graham, Chair Professor Jacques Verstra¨ete,Co-Chair Professor Fan Chung Graham Professor Shachar Lovett Professor Jeff Remmel Professor Alexander Vardy 2016 Copyright Jay Cummings, 2016 All rights reserved. The Dissertation of Jay Cummings is approved, and it is acceptable in quality and form for publication on micro- film and electronically: Co-Chair Chair University of California, San Diego 2016 iii DEDICATION To mom and dad. iv EPIGRAPH We've taught you that the Earth is round, That red and white make pink. But something else, that matters more { We've taught you how to think. | Dr. Seuss v TABLE OF CONTENTS Signature Page . iii Dedication . iv Epigraph . .v Table of Contents . vi List of Figures . viii List of Tables . ix Acknowledgements . .x Vita......................................... xii Abstract of the Dissertation . xiii Chapter 1 Juggling Cards . .1 1.1 Introduction . .1 1.1.1 Juggling card sequences . .3 1.2 Throwing one ball at a time . .6 1.2.1 The unrestricted case . .7 1.2.2 Throwing each ball at least once . .9 1.3 Throwing m ≥ 2 balls at a time . 16 1.3.1 A digraph approach . 17 1.3.2 A differential equations approach . 21 1.4 Throwing different numbers of balls at different times . 23 1.4.1 Boson normal ordering . 24 1.5 Preserving ordering while throwing . 27 1.6 Juggling card sequences with minimal crossings .
    [Show full text]
  • Enumerating a Class of Lattice Paths
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 271 (2003) 13–28 www.elsevier.com/locate/disc Enumerating a class oflattice paths Curtis Coker Department of Mathematics, Gonzaga University, Spokane, WA, USA Received 10 February 2000; received in revised form 5 November 2002; accepted 15 November 2002 Abstract Let D0(n) denote the set oflattice paths in the xy-plane that begin at (0,0), terminate at (n; n), never rise above the line y = x and have step set S = {(k; 0) : k ∈ N+}∪{(0;k):k ∈ N+}. Let E0(n) denote the set oflattice paths with step set S that begin at (0,0) and terminate at (n; n). Using primarily the symbolic method (R. Sedgewick, P. Flajolet, An Introduction to the Analysis ofAlgorithms, Addison-Wesley, Reading, MA, 1996) and the Lagrange inversion formulawe study some enumerative problems associated with D0(n) and E0(n). c 2003 Elsevier B.V. All rights reserved. Keywords: Lattice path; Lagrange inversion formula; Narayana number 1. Introduction For any positive integer n let C0(n) denote the set oflattice paths in the xy-plane that: (i) have step set S(1) = {(1; 0); (0; 1)}, (ii) begin at (0,0) and terminate at (n; n), (iii) never rise above the line y = x. 2n It is ofcourse well known that |C0(n)|=[1=(n+1)]( n ). More generally if n ¿ k ¿ 0 n+k then the ballot number [(n − k +1)=(n + 1)]( k ) counts the lattice paths that begin at (0,0), terminate at (n; k), never rise above the line y = x and have step set S(1).
    [Show full text]
  • Elements of the Sets Enumerated by Super-Catalan Numbers
    Elements of the sets enumerated by super-Catalan numbers Andrew N. Fan1, Tou¯k Mansour2, Sabrina X. M. Pang3 1;3Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P. R. China 2Department of Mathematics, University of Haifa, 31905 Haifa, Israel [email protected], 2tou¯[email protected], [email protected] 1 Introduction As we know several people tried to get many structures for ¯ne numbers (see [31, Sequence A000957]), while others on Catalan numbers (see [31, Sequence A000108]). Stanley [34,35] gave more than 130 Catalan structures while Deutsch and Shapiro [11] also discovered many settings for the Fine numbers. The structures for Fine numbers and Catalan numbers are intimately related from the list of Fine number occurences in [11], which motivated us to ¯nd out more and more super-Catalan structures by the tight link between Catalan numbers and super-Catalan numbers, whose ¯rst several terms are 1; 1; 3; 11; 45; 197; ¢ ¢ ¢ (see [31, Sequence A001003]). The purpose of this paper is to give a uni¯ed presentation of many new super-Catalan structures. We start the project with the idea of giving a restricted bi-color to the existed Catalan structures, and have included a selection of results in [18]. In the remainder of this section, we present a brief account of background for our investigation. The sequence of super-Catalan number was introcuced by Friedrich Wilhelm Karl Ernst SchrÄoderin his paper [29] during his discusses on four \bracketing problems" and the term \SchrÄodernumber" seems to have been ¯rst used by Rogers [28].
    [Show full text]