DOCSLIB.ORG

Exercises on Catalan and Related Numbers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Exercises on Catalan and Related Numbers Exercises on Catalan and Related Numb ers excerpted from Enumerative Combinatorics,vol. 2 published by Cambridge University Press 1999 by Richard P. Stanley version of 23 June 1998 2n 1 19. [1]{[3+] Show that the Catalan numb ers C = count the number of elements n n+1 n of the 66 sets S , a i nnn given b elow. We illustrate the elements of each S for i i n = 3, hoping that these illustrations will makeany unde ned terminology clear. The terms used in vv{yy are de ned in Chapter 7. Ideally S and S should b e proved i j to have the same cardinality by exhibiting a simple, elegant bijection : S ! S ij i j so 4290 bijections in all. In some cases the sets S and S will actually coincide, but i j their descriptions will di er. a triangulations of a convex n + 2-gon into n triangles by n 1 diagonals that do not intersect in their interiors b binary parenthesizations of a string of n +1 letters xx xx xxx x x xxx xx xx xx xx c binary trees with n vertices r r r r r @ @ @ @ r r r r r @r @ @ @ @ r r r r d plane binary trees with 2n +1 vertices or n +1 endp oints r r r r r H @ @ H @ @ @ H @ r @r r r r r r r r r @ @ @ @ @ @ @ @ @ r @r r r r r r r r r r r @ @ @ @ @ @ r @r r @r r r r r e plane trees with n +1 vertices r r r r r @ @ @ r r r r r r r r r @ @ @ @ r r r r r @ r f planted i.e., ro ot has degree one trivalent plane trees with 2n +2 vertices 221 r r r r r r r r r r H @ @ @ @ H @ @ @ r r r r r r r r r r H @ @ @ @ @ @ @ @ r r r r r r r r r @r r @r @ @ @ @ @ @ r r r @r r r r @r g plane trees with n +2 vertices such that the rightmost path of each subtree of the ro ot has even length r r r r r ,l @ L @ @ r, r Lrlr r @r r r r r @ @ r r r r r @ @ r r r r r h lattice paths from 0; 0 to n; n with steps 0; 1 or 1; 0, never rising ab ove the line y = x r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r i Dyck paths from 0; 0 to 2n; 0, i.e., lattice paths with steps 1; 1 and 1; 1, never falling b elow the x-axis r @ r r r r r r @ @ @ @ @ r r r @r r r r r r r r r r r @ @ @ @ @ @ @ @ @ r @r r @r r @r @r r @r @r r @r @r @r j Dyck paths as de ned in i from 0; 0 to 2n +2;0 such that any maximal sequence of consecutive steps 1; 1 ending on the x-axis has odd length r r @ @ r r r r r @ @ @ r r r r r r r r @r r @ @ @ @ @ @ @ @ r @r @r @r @r r @r @r r r r r r @ @ @ @ r r r r r @ @ r r r r r @ @ @ @ @ @ r r r r r k Dyck paths as de ned in i from 0; 0 to 2n +2;0 with no p eaks at height two. 222 r r @ @ r r r r @ @ r r r r r r r r r r @ @ @ @ @ @ @ @ r @r @r @r @r r @r @r r @r @r r @ r r r r @ @ @ @ r r r r r @ @ r r r r @ @ r @r r @r l unordered pairs of lattice paths with n +1 steps each, starting at 0; 0, us- ing steps 1; 0 or 0; 1, ending at the same p oint, and only intersecting at the b eginning and end r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r m unordered pairs of lattice paths with n 1 steps each, starting at 0; 0, using steps 1; 0 or 0; 1, ending at the same p oint, such that one path never arises ab ove the other path n n nonintersecting chords joining 2n p oints on the circumference of a circle r r r r r r r r r r @ A @ r r r r r r r @r r r @ A @ @ r r @r r @r Ar r r r r o ways of connecting 2n p oints in the plane lying on a horizontal line by n nonin- tersecting arcs, each arc connecting two of the p oints and lying ab ove the p oints r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r p ways of drawing in the plane n +1 p oints lying on a horizontal line L and n arcs connecting them such that the arcs do not pass b elow L, the graph thus formed is a tree, notwo arcs intersect in their interiors i.e., the arcs are noncrossing, and atevery vertex, all the arcs exit in the same direction left or right A A A A A @ A s s @s As s s s As s s s @As s s s @As s s s s q ways of drawing in the plane n + 1 p oints lying on a horizontal line L and n arcs connecting them such that the arcs do not pass b elow L, the graph thus 223 formed is a tree, no arc including its endp oints lies strictly b elow another arc, and atevery vertex, all the arcs exit in the same direction left or right A A A A H H s s s s @ A s sHs @As s s s As s sHs @s s s s @As r sequences of n 1's and n 1's such that every partial sum is nonnegative with 1 denoted simply as b elow 111 111 111 111 111 s sequences 1 a a of integers with a i 1 n i 111 112 113 122 123 t sequences a <a < <a of integers satisfying 1 a 2i 1 2 n1 i 12 13 14 23 24 u sequences a ;a ;:::;a of integers such that a = 0 and 0 a a +1 1 2 n 1 i+1 i 000 001 010 011 012 v sequences a ;a ;:::;a of integers such that a 1 and all partial sums are 1 2 n1 i nonnegative 0; 0 0; 1 1; 1 1; 0 1; 1 w sequences a ;a ;:::;a of integers such that a 1, all partial sums are non- 1 2 n i negative, and a + a + + a =0 1 2 n 0;0;0 0;1;1 1;0;1 1;1;0 2;1;1 x sequences a ;a ;:::;a of integers such that 0 a n i, and such that if i<j, 1 2 n i a >0, a > 0, and a = a = = a = 0, then j i>a a i j i+1 i+2 j1 i j 000 010 100 200 110 y sequences a ;a ;:::;a of integers such that i a n and such that if i j a , 1 2 n i i then a a j i 123 133 223 323 333 z sequences a ;a ;:::;a of integers such that 1 a i and such that if a = j , 1 2 n i i then a j r for 1 r j 1 ir 111 112 113 121 123 224 aa equivalence classes B of words in the alphab et [n 1] such that any three con- secutive letters of any word in B are distinct, under the equivalence relation uij v uj iv for any words u; v and any i; j 2 [n 1] satisfying ji j j 2 f;g f1g f2g f12g f21g For n =4 a representative of each class is given by ;, 1, 2, 3, 12, 21, 13, 23, 32, 123, 132, 213, 321, 2132. bb partitions = ;:::; with n 1 so the diagram of is contained in 1 n1 1 0 0 0 an n 1 n 1 square, such that if = ; ;::: denotes the conjugate 1 2 0 partition to then whenever i i i i 0; 0 1; 0 1; 1 2; 1 2; 2 2 2 2 cc p ermutations a a a of the multiset f1 ; 2 ;:::;n g such that: i the rst o c- 1 2 2n currences of 1; 2;:::;n app ear in increasing order, and ii there is no subsequence of the form 112233 112332 122331 123321 122133 dd p ermutations a a a of the set [2n] such that: i 1; 3;:::;2n 1 app ear in 1 2 2n increasing order, ii 2; 4;:::;2n app ear in increasing order, and iii 2i 1 app ears b efore 2i, 1 i n 123456 123546 132456 132546 135246 ee p ermutations a a a of [n] with longest decreasing subsequence of length at 1 2 n most two i.e., there do es not exist i<j <k, a >a >a , called 321-avoiding i j k p ermutations 123 213 132 312 231 p ermutations a a a of [n] for which there do es not exist i < j < k and 1 2 n a <a <a called 312-avoiding p ermutations j k i 123 132 213 231 321 gg p ermutations w of [2n] with n cycles of length two, such that the pro duct 1; 2;:::;2n w has n + 1 cycles 1; 2; 3; 4; 5; 61; 23; 45; 6 = 12; 4; 635 1; 2; 3; 4; 5; 61; 23; 64; 5 = 12; 63; 54 1; 2; 3; 4; 5; 61; 42; 35; 6 = 1; 324; 65 1; 2; 3; 4; 5; 61; 62; 34; 5 = 1; 3; 5246 1; 2; 3; 4; 5; 61; 62; 53; 4 = 1; 52; 436 225 hh pairs u; v of p ermutations of [n] such that u and v have a total of n +1 cycles, and uv =1;2;:::;n 123 1; 2; 3 1; 2; 3 123 1; 23 1; 32 1; 32 12; 3 12; 3 1; 23 ii p ermutations a a a of [n] that can b e put in increasing order on a single stack, 1 2 n de ned recursively as follows: If ; is the empty sequence, then let S ; = ;.
Load more

Recommended publications
  • Motzkin Paths, Motzkin Polynomials and Recurrence Relations
    Motzkin paths, Motzkin polynomials and recurrence relations Roy Oste and Joris Van der Jeugt Department of Applied Mathematics, Computer Science and Statistics Ghent University B-9000 Gent, Belgium [email protected], [email protected] Submitted: Oct 24, 2014; Accepted: Apr 4, 2015; Published: Apr 21, 2015 Mathematics Subject Classifications: 05A10, 05A15 Abstract We consider the Motzkin paths which are simple combinatorial objects appear- ing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial “counting” all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polyno- mials) have been studied before, but here we deduce some properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix en- tries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof. 1 Introduction Catalan numbers and Motzkin numbers have a long history in combinatorics [19, 20]. 1 2n A lot of enumeration problems are counted by the Catalan numbers Cn = n+1 n , see e.g. [20]. Closely related to Catalan numbers are Motzkin numbers M , n ⌊n/2⌋ n M = C n 2k k Xk=0 similarly associated to many counting problems, see e.g. [9, 1, 18]. For example, the number of different ways of drawing non-intersecting chords between n points on a circle is counted by the Motzkin numbers.
    [Show full text]
  • Motzkin Paths
    Discrete Math. 312 (2012), no. 11, 1918-1922 Noncrossing Linked Partitions and Large (3; 2)-Motzkin Paths William Y.C. Chen1, Carol J. Wang2 1Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071, P.R. China 2Department of Mathematics Beijing Technology and Business University Beijing 100048, P.R. China emails: [email protected], wang [email protected] Abstract. Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and (3; 2)-Motzkin paths, where a (3; 2)-Motzkin path can be viewed as a Motzkin path for which there are three types of horizontal steps and two types of down steps. A large (3; 2)-Motzkin path is a (3; 2)-Motzkin path for which there are only two types of horizontal steps on the x-axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of f1; : : : ; n + 1g and the set of large (3; 2)- Motzkin paths of length n, which leads to a simple explanation of the well-known relation between the large and the little Schr¨odernumbers. Keywords: Noncrossing linked partition, Schr¨oderpath, large (3; 2)-Motzkin path, Schr¨odernumber AMS Classifications: 05A15, 05A18. 1 1 Introduction The notion of noncrossing linked partitions was introduced by Dykema [5] in the study of the unsymmetrized T-transform in free probability theory. Let [n] denote f1; : : : ; ng. It has been shown that the generating function of the number of noncrossing linked partitions of [n + 1] is given by 1 p X 1 − x − 1 − 6x + x2 F (x) = f xn = : (1.1) n+1 2x n=0 This implies that the number of noncrossing linked partitions of [n + 1] is equal to the n-th large Schr¨odernumber Sn, that is, the number of large Schr¨oderpaths of length 2n.
    [Show full text]
  • Some Identities on the Catalan, Motzkin and Schr¨Oder Numbers
    Discrete Applied Mathematics 156 (2008) 2781–2789 www.elsevier.com/locate/dam Some identities on the Catalan, Motzkin and Schroder¨ numbers Eva Y.P. Denga,∗, Wei-Jun Yanb a Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China b Department of Foundation Courses, Neusoft Institute of Information, Dalian 116023, PR China Received 19 July 2006; received in revised form 1 September 2007; accepted 18 November 2007 Available online 3 January 2008 Abstract In this paper, some identities between the Catalan, Motzkin and Schroder¨ numbers are obtained by using the Riordan group. We also present two combinatorial proofs for an identity related to the Catalan numbers with the Motzkin numbers and an identity related to the Schroder¨ numbers with the Motzkin numbers, respectively. c 2007 Elsevier B.V. All rights reserved. Keywords: Catalan number; Motzkin number; Schroder¨ number; Riordan group 1. Introduction The Catalan, Motzkin and Schroder¨ numbers have been widely encountered and widely investigated. They appear in a large number of combinatorial objects, see [11] and The On-Line Encyclopedia of Integer Sequences [8] for more details. Here we describe them in terms of certain lattice paths. A Dyck path of semilength n is a lattice path from the origin (0, 0) to (2n, 0) consisting of up steps (1, 1) and down steps (1, −1) that never goes below the x-axis. The set of Dyck paths of semilength n is enumerated by the n-th Catalan number 1 2n cn = n + 1 n (sequence A000108 in [8]). The generating function for the Catalan numbers (cn)n∈N is √ 1 − 1 − 4x c(x) = .
    [Show full text]
  • Motzkin Numbers of Higher Rank: Generating Function and Explicit Expression
    1 2 Journal of Integer Sequences, Vol. 10 (2007), 3 Article 07.7.4 47 6 23 11 Motzkin Numbers of Higher Rank: Generating Function and Explicit Expression Toufik Mansour Department of Mathematics University of Haifa Haifa 31905 Israel [email protected] Matthias Schork1 Camillo-Sitte-Weg 25 60488 Frankfurt Germany [email protected] Yidong Sun Department of Mathematics Dalian Maritime University 116026 Dalian P. R. China [email protected] Abstract The generating function for the (colored) Motzkin numbers of higher rank intro- duced recently is discussed. Considering the special case of rank one yields the corre- sponding results for the conventional colored Motzkin numbers for which in addition a recursion relation is given. Some explicit expressions are given for the higher rank case in the first few instances. 1All correspondence should be directed to this author. 1 1 Introduction The classical Motzkin numbers (A001006 in [1]) count the numbers of Motzkin paths (and are also related to many other combinatorial objects, see Stanley [2]). Let us recall the definition of Motzkin paths. We consider in the Cartesian plane Z Z those lattice paths starting from (0, 0) that use the steps U, L, D , where U = (1, 1) is× an up-step, L = (1, 0) a level-step and D = (1, 1) a down-step.{ Let} M(n, k) denote the set of paths beginning in (0, 0) and ending in (n,− k) that never go below the x-axis. Paths in M(n, 0) are called Motzkin paths and m := M(n, 0) is called n-th Motzkin number.
    [Show full text]
  • Number Pattern Hui Fang Huang Su Nova Southeastern University, [email protected]
    Transformations Volume 2 Article 5 Issue 2 Winter 2016 12-27-2016 Number Pattern Hui Fang Huang Su Nova Southeastern University, [email protected] Denise Gates Janice Haramis Farrah Bell Claude Manigat See next page for additional authors Follow this and additional works at: https://nsuworks.nova.edu/transformations Part of the Curriculum and Instruction Commons, Science and Mathematics Education Commons, Special Education and Teaching Commons, and the Teacher Education and Professional Development Commons Recommended Citation Su, Hui Fang Huang; Gates, Denise; Haramis, Janice; Bell, Farrah; Manigat, Claude; Hierpe, Kristin; and Da Silva, Lourivaldo (2016) "Number Pattern," Transformations: Vol. 2 : Iss. 2 , Article 5. Available at: https://nsuworks.nova.edu/transformations/vol2/iss2/5 This Article is brought to you for free and open access by the Abraham S. Fischler College of Education at NSUWorks. It has been accepted for inclusion in Transformations by an authorized editor of NSUWorks. For more information, please contact [email protected]. Number Pattern Cover Page Footnote This article is the result of the MAT students' collaborative research work in the Pre-Algebra course. The research was under the direction of their professor, Dr. Hui Fang Su. The ap per was organized by Team Leader Denise Gates. Authors Hui Fang Huang Su, Denise Gates, Janice Haramis, Farrah Bell, Claude Manigat, Kristin Hierpe, and Lourivaldo Da Silva This article is available in Transformations: https://nsuworks.nova.edu/transformations/vol2/iss2/5 Number Patterns Abstract In this manuscript, we study the purpose of number patterns, a brief history of number patterns, and classroom uses for number patterns and magic squares.
    [Show full text]
  • Narayana Number, Chebyshev Polynomial and Motzkin Path on RNA Abstract Shapes
    Narayana number, Chebyshev polynomial and Motzkin path on RNA abstract shapes Sang Kwan Choi, Chaiho Rim and Hwajin Um Abstract We consider a certain abstract of RNA secondary structures, which is closely related to so-called RNA shapes. The generating function counting the number of the abstract structures is obtained in three different ways, namely, by means of Narayana numbers, Chebyshev polynomials and Motzkin paths. We show that a combinatorial interpretation on 2-Motzkin paths explains a relation between Motzkin paths and RNA shapes and also provides an identity related to Narayana numbers and Motzkin polynomial coefficients. 1 Introduction Ribonucleic acid (RNA) is a single stranded molecule with a backbone of nu- cleotides, each of which has one of the four bases, adenine (A), cytosine (C), gua- nine (G) and uracil (U). Base pairs are formed intra-molecularly between A-U, G-C or G-U, leading the sequence of bases to form helical regions. The primary structure of a RNA is merely the sequence of bases and its three-dimensional conformation by base pairs is called the tertiary structure. As an intermediate structure between the primary and the tertiary, the secondary structure is a planar structure allowing only nested base pairs. This is easy to see in its diagrammatic representation, see figure 1. A sequence of n bases is that of labeled vertices (1;2;··· ;n) in a horizon- tal line and base pairs are drawn as arcs in the upper half-plane. The condition of nested base pairs means non-crossing arcs: for two arcs (i,j) and (k,l) where i < j, Sang Kwan Choi Center for Theoretical Physics, College of Physical Science and Technology Sichuan University, Chengdu 6100064, China, e-mail: :[email protected] Chaiho Rim Department of Physics, Sogang University, Seoul 121-742, Korea e-mail: [email protected] Hwajin Um Department of Physics, Sogang University, Seoul 121-742, Korea e-mail: [email protected] 1 2 Sang Kwan Choi, Chaiho Rim and Hwajin Um ( ( ( ( ( .
    [Show full text]
  • Integer Sequences
    UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular
    [Show full text]
  • Numbers 1 to 100
    Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number)
    [Show full text]
  • A Bijection Between Ordered Trees and 2-Motzkin Paths and Its Many Consequences Emeric Deutscha; ∗, Louis W
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 256 (2002) 655–670 www.elsevier.com/locate/disc A bijection between ordered trees and 2-Motzkin paths and its many consequences Emeric Deutscha; ∗, Louis W. Shapirob aDepartment of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA bHoward University, Washington, DC 20059, USA Received 27 September 2000; received in revised form 4 April 2001; accepted 28 May 2001 Abstract A new bijection between ordered trees and 2-Motzkin paths is presented, together with its numerous consequences regarding ordered trees as well as other combinatorial structures such as Dyck paths, bushes, {0; 1; 2}-trees, Schr2oder paths, RNA secondary structures, noncrossing partitions, Fine paths, and Davenport–Schinzel sequences. RÃesumÃe Une nouvelle bijection entre arbres ordonnÃes et chemins de Motzkin bicolorÃes est prÃesentÃee, avec ses nombreuses consÃequences en ce qui concerne les arbres ordonnÃes ainsi que d’autres structures combinatoires telles que chemins de Dyck, buissons, arbres de type {0; 1; 2}, chemins de Schr2oder, structures secondaires de type RNA, partitions non croisÃees, chemins de Fine, et enÿn suites de Davenport–Schinzel. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Ordered trees; Motzkin paths; Binary trees; Bushes; {0,1,2}-trees; Schr2odinger paths; RNA secondary structures; Fine paths; Davenport–Schinzel sequences 1. Introduction An ordered tree is an unlabeled rooted tree where the order of the subtrees of a vertex is signiÿcant. It is well known that the number of all ordered trees with n edges 2n is the Catalan number Cn = [(1=n + 1)]( n ).
    [Show full text]
  • P R E P N E W S
    P R E P N E W S Issue 1 www.bridgehouse.org.za 15 January 2016 Grade 1H class visiting the Prep Office with Mrs Colette Hinds on their very first day of BIG SCHOOL! - 2 - Happy, smiling faces across the playgrounds! - 3 - - 4 - Wishing all our pupils a wonderful term 1, 2016! - 5 - - 6 - A warm and hearty congratulations to all our recipients at the Senior Primary Awards evening of 2015. Our Prestigious Senior Primary awards were as follows: The Good Fellowship Award went to Lloyd Phillips. He was voted as the Grade 7 pupil, who in the opinion of his Grade 6 and 7 peers had been a true role model in his demonstration of the qualities of friendship, kindness and helpfulness throughout the year. The Leadership Star was awarded to Emma Wainwright for her consistent and positive leadership throughout her year as a senior in the Prep School. The Founders’ Shield was awarded to Demeye De Leeuw. This award recognises the qualities of initiative, independence of spirit and self-motivation, and is presented to the pupil in whom we have confidence that the qualities of humanity are developing. The evening saw the introduction of a new alignment with the College in terms of awards in Grade 6 and 7: the Highly Commended Academic Award for pupils who achieved an aggregate of 75 – 79% for their year’s work. This complements the Academic Merit Award which recognises achievement from 80% and above. The introduction of the new award affords us an opportunity to acknowledge a greater number of our hard-working pupils at the Awards Ceremony.
    [Show full text]
  • Staircase Tilings and Lattice Paths
    STAIRCASE TILINGS AND LATTICE PATHS Silvia Heubach Department of Mathematics, California State University Los Angeles 5151 State University Drive, Los Angeles, CA 90032-8204 USA [email protected] Toufik Mansour Department of Mathematics, Haifa University, 31905 Haifa, Israel [email protected] Abstract We define a combinatorial structure, a tiling of the staircase in the R2 plane, that will allow us, when restricted in different ways, to create direct bijections to Dyck paths of length 2n, Motzkin paths of lengths n and n−1, as well as Schr¨oder paths and little Schr¨oder paths of length n. Keywords: Tilings, Dyck paths, Motzkin paths, Schr¨oder paths, little Schr¨oder paths. AMS Mathematical Subject Classifications: 52C20, 05A05, 05A15. 1. Introduction We define a staircase tiling which can be recognized as a visualization of a combinatorial object enumerated by the Catalan numbers and exhibit direct bijections between staircase tilings and Dyck paths, Motzkin paths, Schr¨oder paths and little Schr¨oder paths. We also enumerate staircase tilings with certain properties using these bijections. Before introducing our structure, we will give some basic definitions. A lattice path of length n is a sequence of points P1, P2,...,Pn with n ≥ 1 such that each point Pi belongs to the plane integer lattice and consecutive points Pi and Pi+1 are connected by a line segment. We will consider lattice paths in Z2 whose permitted step types are up-steps U = (1, 1), down-steps D = (1, −1), and two types of horizontal steps, h = (1, 0) and H = (2, 0). We will focus on paths that start from the origin and return to the x-axis, and never pass below the x-axis.
    [Show full text]
  • Skew-Standard Tableaux with Three Rows
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Advances in Applied Mathematics 45 (2010) 463–469 Contents lists available at ScienceDirect Advances in Applied Mathematics www.elsevier.com/locate/yaama Skew-standard tableaux with three rows Sen-Peng Eu 1 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan, ROC article info abstract Article history: Let T3 be the three-rowed strip. Recently Regev conjectured Received 10 January 2010 that the number of standard Young tableaux with n − 3entries Accepted 11 March 2010 in the “skew three-rowed strip” T3/(2, 1, 0) is mn−1 − mn−3, Available online 14 May 2010 a difference of two Motzkin numbers. This conjecture, together with hundreds of similar identities, were derived automatically MSC: 05A15 and proved rigorously by Zeilberger via his powerful program and 05A19 WZ method. It appears that each one is a linear combination of Motzkin numbers with constant coefficients. In this paper we will Keywords: introduce a simple bijection between Motzkin paths and standard Standard Young tableaux Young tableaux with at most three rows. With this bijection we Motzkin numbers answer Zeilberger’s question affirmatively that there is a uniform Motzkin paths way to construct bijective proofs for all of those identities. © 2010 Elsevier Inc. All rights reserved. 1. Introduction The enumeration of standard Young tableaux (SYTs) is a fundamental problem in combinatorics and representation theory. For example, it is known that the number of SYTs of a given shape λ n is counted by the hook-length formula [3].
    [Show full text]