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DOCUMENT RESUME ED 472 095 SE 066 382 AUTHOR Labelle, Gilbert TITLE Manipulating Combinatorial Structures. PUB DATE 2000-00-00 NOTE 18p.; In: Canadian Mathematics Education Study Group = Groupe Canadien d'Etude en Didactique des Mathematiques. Proceedings of the Annual Meeting (24th, Montreal, Quebec, Canada, May 26-30, 2000); see SE 066 379. PUB TYPE Guides Non-Classroom (055) -- Speeches/Meeting Papers (150) EDRS PRICE EDRS Price MF01/PC01 Plus Postage. DESCRIPTORS Curriculum Development; Higher Education; Interdisciplinary Approach; Mathematical Applications; *Mathematical Formulas; Mathematical Models; *Mathematics Instruction; Modern Mathematics; Teaching Methods IDENTIFIERS Combinatorics ABSTRACT This set of transparencies shows how the manipulation of combinatorial structures in the context of modern combinatorics can easily lead to interesting teaching and learning activities at every level of education from elementary school to university. The transparencies describe: (1) the importance and relations of combinatorics to science and social activities;(2) how analytic/algebraic combinatorics is similar to analytic/algebraic geometry;(3) basic combinatorial structures with terminology;(4) species of structures;(5) power series associated with any species of structures;(6) similarity of operations between power series and the corresponding species of structures;(7) examples illustrating where the structures are manipulated; and (8) general teaching/learning activities. (KHR) Reproductions supplied by EDRS are the best that can be made from the original document. PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL HAS O BEEN GRANTED BY Manipulating Combinatorial Structures Gilbert Labelle TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) Universite du Quebec a Montreal (UQAM) 1 A combinatorial structure is a finite construction made on a finite set of elements. In real-life U.S. DEPARTMENT OF EDUCATION situations, combinatorial structures often arise as "skeletons" or "schematic descriptions" Office of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION of concrete objects. For example, on a road map, the elements can be cities and the finite CENTER (ERIC) This document has been reproduced as construction can be the various roads joining these cities. Similarly, a diamond can be con- received from the person or organization originating it. sidered as a combinatorial structure: the plane facets of the diamond are connected together Minor changes have been made to according to certain rules. improve reproduction quality. Combinatorics can be defined as the mathematical analysis, classification and enu- meration of combinatorial structures. The main purpose of my presentation is to show how Points of view or opinions stated in this document do not necessarily represent the manipulation of combinatorial structures, in the context of modern combinatorics, can official OERI position cr policy. easily lead to interesting teaching/learning activities at every level of education: from el- ementary school to university. The following pages of these proceedings contain a grayscale version of my (color) transparencies (two transparencies on each page). I decided to publish directly my trans- parencies (instead of a standard typed text) for two main reasons: my talk contained a great amount of figures, which had to be reproduced anyway, and the short sentences in the transparencies are easy to read and put emphasis on the main points I wanted to stress. The first two transparencies are preliminary ones and schematically describe: a) the importance and relations of combinatorics with science/ social activities, b) how analytic/ algebraic combinatorics is similar to analytic/ algebraic geometry. The next 7 transparencies (numbered 1 to 7) contain drawings showing basic combinatorial structures together with some terminology. Collecting together similar combinatorial structures give rise to the con- cept of species of structures (transparencies 8 to 12). A power series is then associated to any species of structures enabling one to count its structures (transparencies 13 to 15). Each op- eration on power series (sum, product, substitution, derivation) is reflected by similar opera- tions on the corresponding species of structures (transparencies 16 to 18). The power of this correspondence is illustrated on explicit examples (transparencies 18 to 26) where the struc- tures are manipulated (and counted) using various combinatorial operations. Transparen- cies 27 to 30 suggest some general teaching/ learning activities (from elementary to advanced levels) that may arise from these ideas. I hope that the readers will agree that manipulating combinatorial structures constitutes a good activity to develop the mathematical mind. References The main references for the theory of combinatorial species are: Joyal, A. (1981). Une theorie combinatoire des series formelles, Advances in Mathematics, 42, 1-82. Bergeron, F., Labelle, G., & Leroux, P. (1998). Combinatorial Species and Tree-like Structures. Encyclopedia of Math. and its Applications, Vol. 67, Cambridge University Press. BEST COPYAVAILABLE 2 3 The dynamicPRELIMINARIES connections between GeometricalANALYTIC / ALGEBRAIC MathematicalGEOMETRY combinatorialwill greatlymathematics increase / discrete duringand the comingscienceactivities decades I social : figures -4 2xformulas - ey +7 = 0 Computercompression(analysis of algorithms. of data science structures, organization etc) and 0 x2+y2= 9 (classification.Chemistry symmetries of molecules, y=e2 isomers.quantum(classification,Physics etc)theory. symmetries Feynman integrals. of atoms, etc) ANALYTIC / ALGEBRAIC COMBINATORICS Combinatorics classification(analysisBiology of DNA, of species, genetic code.etc) Combinatorialstructures 4-----> Mathematicalformulas graphs,(analysis,Mathematics groups, calculus, etc) algebra, geometry, 0 Y=1"*() (publicHuman keys methods activities of encryption, y = x +xy2 encodingteachingdesignsecurebasic combinatorial oftransactions, andof plane compact learning flights, discs.stuctures.communications. shortest etc) paths. e3x 1 2 MANIPULATINGGilbert COMBINATORIAL Labelle / LaCIM - UQAM STRUCTURES Example 2.4A rooted tree on 1.7..kbc,..x,rt+3 finiteIn Combinatorics, construction amade structure,on a finites, is aset U. e 4 We say orthat : Us isIsis (built)equippedthe underlying on U with (theset of structure)s s Example 3.6An1,4 Iorienteda. 3 cyclea on IIf , 93 Exampleor 1.: s isA labelledtree on tJ by = thea, b,set c, U, + (4...;.ASP'kcL-.17 3 Variants:oddeven (oriented) (oriented) cycle cycle el" kit Example 4. (orA subsetbicoloration) on (of)tro 1%.ff., "1:\6 cif J R a y 0 _L0 Example 5.A (the) set structure on 3 Example 8.A partition= 10,*, 42,3, on 4, (7, '1'3 4 U /,0AAJ-1:1TJJ1Variants:oddeven set set Example 6. A permutation on ys..,.ITt s LY-=..egktN,Ii2/3000 /6, ai.".`t bd Example 9. An endofunction on f \--ii.el ;U ' 62-3 Se34 t a ,., ..* . 11.-1N 4 r g, ' e )% /I; . /a \--4 6 i --->.... f V241g13 e 1- 2} NN SPOf Example 7. Alinear order on U=1=d Al; .40;a3 eafj RioIf far.v.../6ri9 Vr..v,.631SyaS . H.2( er r17.-43"4iv2.,,/4itvy Iti, 'IVSoc.47,.. I 17ts 0,62, 44, F t 3 \,...0.,.........0...4".s:-",3..,..-er-'-ere...? a 3 xs44. 2 N*3 5 Example 10. ATJ=14,12,c,d,e, bicolored polygon f3on 5 Example 13. A singleton on U= I Al 6 (ThereExample is no14. singletonAn empty structurea set structure on U if (r..7(*on U=16 1.) Example 11. A graph on ET= it, .. .,ZSfq 4: /3,,-r,;--,-:---,24,..........___0#077.t.,ir 1 1 N, (There is no empty set structure if T7* +) ....,,,,,,.......4T....Ie.'4.....111-:,..... -.. 1 VAtei...---ip 23 2 99g Example 15.An Involution on U=1*-,4., A. , vS- 2..,a., qr,f5' h h A r rn s Example 12.ft' A connectedi9n graph on U= a. e 4 7 ,00.414/6 3 "111' ETC, ETC, ETC,... 7 For example, we can consider 8 CONVENTIONwill sometimes : A.beAn on represented arbitrary T...7 structureby thethethe speciesspeciesspeciesA ce.aof all evencycles,rootedtrees cycles,(arbres), trees(arbore5cences), b c thethethe speciesspeciesspecies E Ewa*vana)Eof allallall sets evensubsetsoddpermutations, (ensembles), sets, sets, (of sets), Aif LTstructure containsHere, A. U an on= even5 T..7Ad(odd)6., is c.,said ,A)number toe. be evenof elements. (odd) thethethe species speciesspeciesspecies End SL.VI'B ofof allallall edofunctions, linearbicoloredpartitions, orders, polygons) Twoif their structures underlying 4, sets and are t aredisjoint.said to be disjoint thethe species speciesspeciesempty 214.0,.species)(I of allthe0 graphs,connectedsingletons, empty(containing set, graphs, no structure), collectingStructures them can intobe studied SPECIESandOF classified STRUCTURES by : thethe species InvF ofof allallinvolutions,F- structures. 9 10 ByFLU] convention,:= the setfor anyof all finiteF- setstructures tJ we write on U PROPERTYtheFundamental proceeding 1. properties exampleFor every : satisfied finite set by t7, the set Example.thenIf F = = the species of cycles, PROPERTY 2.47E73EveryPVU bijectionIs= always rn, rijfinite. Is a 41- structure inducesCLU) another bijection,V = 14-111 ha c, 43I andCif ifrn,"if)1, 0)131= rn,n,1,, 11, then 0,1=144> 446 I411,24." "&tlivonANDirO;:74.."3Ar't 3gere't44417144,5r2 towhich form relabels a corresponding via /3 each C C-structure -structure ononV. U 0 Note. Two successive relabellingsvia A and Al11 CONVENTION. An arbitrary F -structure AL onT..712 amountseP4.)flop to a singlecmctaviou,relabelling )=CLA."3mei° via Cfpl A'o /3 : can be representedor by diagrams
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