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AUTHOR Labelle, Gilbert TITLE Manipulating Combinatorial Structures. PUB DATE 2000-00-00 NOTE 18p.; In: Canadian 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

ABSTRACT This 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 and social activities;(2) how analytic/ is similar to analytic/algebraic ;(3) basic combinatorial structures with terminology;(4) species of structures;(5) power associated with any species of structures;(6) similarity of operations between and the corresponding species of structures;(7) examples illustrating where the structures are manipulated; and (8) general teaching/learning activities. (KHR)

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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 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 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 , 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/ . 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 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 PRELIMINARIES combinatorial / discrete mathematicsThe dynamic connections between and science I social activities Geometrical ANALYTIC / figuresALGEBRAIC GEOMETRY -4 Mathematical formulas will greatly increase during the coming decades : Computer(analysis of . science organization and 2x - ey +7 = 0 x 2 + Chemistry(classification.compression of symmetries data structures, of molecules, etc) y=e2 y2= 9 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 0 4-----> Mathematical formulas (analysis, , , geometry, (publicgraphs,HumanMathematics keys groups, methods etc) activities of encryption, yY=1"*() = x + xy2 encodingteachingdesignsecurebasic of oftransactions, compact and combinatorialplane learning flights, discs. communications. shorteststuctures. paths. etc) 0 e3x 1 2 MANIPULATING COMBINATORIAL STRUCTURES Gilbert Labelle / LaCIM - UQAM Example 2. 4 A rooted tree on 1.7..kbc,..x,rt+3 finiteIn Combinatorics, construction a structure, s, is made on a finite set U. a e 1,4 4 We say that : U Is the underlying set of or : Us isis (built)equipped on U with (the structure) s s Example 3. An oriented cycle on II a. 3 a f , 93 Example 1. or : s is labelled by the set U a, b, c, , + (4...;. 6ASP'kcL-.17 3 I Variants: oddeven (oriented) (oriented) cycle cycle A tree on tJ = el" kit Example 4. (orA bicoloration) subset on (of) tro 1%.ff., J R a y "1:\6 cif 0 _L 0 Example 5. A (the) set structure on 3 Example 8. A partition on = 10,*, 42,3, 4, (7, 4 U /,0AAJ- TJJ1 Variants: oddeven set set '1'3 Example 6. ITt A on LY-= Ii2/300 /6, ai bd ys s . . \--ii.el.egktN, .".`t Example 9. An endofunctiong, on a 62-3 Se 34 f ..,.,., ..* 6 t ' i --->....11.-1N f e . ; 0 a 4 \-- )% 4 SP I ; U ' r V241g1 3 e ea1- 2} 9 NN . Of / 63 Sy H. / 1=d Al; .40;a3 fj Rio If /6 ri Vr..v,. S Example 7. A linear order on U= r17.-43"4 iv far.v... ts a ,62, 44, 2( er 2.,, /4 Iti, F itvy 'IV t 3 Soc ,.. \,...0.,...... 0...4".I 17 s:-",3..,..-er-'-ere... 1 0 a 3 1:1xs 44. .47 2 N 5 *3 ? Example 10. A bicolored polygon on TJ=14,12,c,d,e, f3 5 Example 13. A singleton on U = I Al 6 (There isExample no singleton 14. structure on U if (r..7(* 1.) a Example 11. A graph on ET= it, 0 7 . . ., ZS fq An structure on U=16 4: ....,,,,,,...... 4T /3 ,,-r,;--,-:--- ,24,...... ___ .... Ie.' 4.....1 -.. #07 .t.,ir Ate 1 ip 23 1 N, 9 Example 15. (There is no empty set structure if T7* +) An Involution on U=1*-,4., A. , q 11-: f,..... 5' vS- 2 ..,1 Va., i.. .--- 2 9 g h rn h A s Example 12. A connected graph on U= ft' r, i9 n "111' r a. e 4 7 ,00.41 4/6 3 ETC, ETC, ETC, ... 7 For example, we can consider 8 CONVENTIONwill sometimes : be represented A. on T...7An arbitrary structure by thethethe species species species a Ace. ofof all all rootedtrees evencycles, cycles,trees (arbres), (arbore5cences), b c thethe speciesthe speciesspecies species E *van Ewa a) E of ofallall all setsevensubsets oddpermutations, (ensembles), sets, sets, (of sets), A ifstructure LT contains A. onan T..7even is Here, U = 5 Ad (odd)6., number c., ,A) of elements. said to be even (odd) e. thethe speciesthespecies speciesspecies species SEnd B VI' of all L. of allof edofunctions, all linearpartitions, orders, bicolored polygons) Twoif their structures underlying 4, andsets tare are disjoint. said to be disjoint thethe speciesspecies speciesempty 214.0,. species of all 0 graphs, )( of allthe connectedsingletons, empty set, graphs, (containing no structure), collectingStructures them can intobe studied SPECIES and classified by OF STRUCTURES : the thespecies species Inv of all IF of all involutions, F - structures. 9 10 By convention, for any finite set tJ we write := the set of all structures on U PROPERTYtheFundamental proceeding 1. example properties : satisfied by For every finite set t7, the set FLU]Example. If F = = the species of cycles, F- PROPERTY 2. 47E73 Is Everyalways finite. then Is a 41- structure induces another bijection, PV V =U 14- = harn, c, 43 111 I rij and if rn, "if )1, 0)131 = rn,n,1,, 11, then CLU) 44> 446 411, 24." Cif "&tlivon AND irO;:74.." 3 Ar't gere't 3 44417144,5r 2 towhich form relabels a corresponding0,1=1 via /3 each C C-structure -structure ononV. U I 0 Note. Two successive relabellings via A and Al 11 CONVENTION. An arbitrary F -structure AL on T..7 12 amountsP4.) to a single flop ctavi ) ou, relabelling via A' mei° Cfpl o /3 : can be represented by diagrams like CD DEFINITION (Andre Joyal, e cm UCIAM,1979) =CLA."3 or A 1)combinatorial2) generates, generates, speciesfor for each each is finite abijection setanother finite set FLU), rule F which U, or ETC (3* , another bijection FE V FFLU) CountingThe properties structures of relabellingsof a given imply thatspecies 0 Any bijection F [fa] isAny a. In FLU) is called ina a coherent way : F called F- relabelling[(1° Jo via Friaj p . F-structure on U . F1(3'0 v) Al thedepends number only of onF -structures the number onofThat is, 1 FM I depends only on l tj . elements of U. t7 Hence, if we want to simply count F -structures, 13 14 it is sufficient to consider only the cases 1.7 = n =-- 1..#2. 113 n = 0, 1,2o . . Example 1. S t the species of F(x) = s xvi n DEFINITION. = IF te_11 Let F be a species and Example 2. S(x) =E pi! nvo 6:1 % the species of subsets =Ex" 0,1*0 I The generating series of the species F is = number of F -structures labelled by 00 1,2, ..., ni. Example 3. E n39 2" the species of sets eZX &I n! ECzJ= fly. ww ; = ex NOTE. We will see that generating series give rise to a dynamic connection betweenF(x) -7, = Example 4. Ewen (x) + the species+ of even sets De* + .= cosh COMBINATORICS and ANALYSIS. Example 5. (1) = Eyed + 131-t the species of+ odd sets + - - =sinlnx Example 6. L the species of linear orders 15 COMBINING and MANIPULATING SPECIES 16 111..e Ply° add,Weseries know multiply, tohow obtain to substitute, other series and differentiate: Example 7. L (x) z x" Ex" ve, C ; the species of cycles n I If F(x).. Eib. .1,,, lirk G(30 INN, p, 24+1 ExampleC(x) 8. ISMS the species of even cycles =E X" =.0/n(1% t I ak ) thenand we have the following table H (x) =Z 17., Inv , -Ei Examplec.(x)= 9. Dn.-014 m z PI even it End = the species of endofunctions it, I ir aqnft H(4= F(x) +G(x)OPERATION COEFFICIENT hn 5;, End (X) -= P1 3o n" 1-1W = F(c) - 6(%) il h== 5 + 4_101 ii. as-* Example 10. X(4) = + X : the species+ of singletonsCoif s -Fr = ti(7) =G(o)=H (x) ier = 0ti- F(x) F (6(2)) 6 h = 5+, =7" Stink , -44A 81, 3 "I. ... nit Example 11. ithm I el athe species of graphs CZ) Tc/ " combinatorialThis table suggests operations corresponding for species : DEFINITION. From given species F and G other species can be built : 17 . If F and G are species, then (F+G)(x) = F(x) + G(z), 18 The e4pecies F+G A (F+G)-structure or sEr a F-structure6-structure (41.44..tGoph.) CFo(F- 6)(21= F(GOe)), Cif F'Cx) = = F0c) GOO, Frac) . t\..) The species A ( F-G )- structure is G an cilsjoirrtofordered a F-structure pair6-structure and a 1) E that is, EXAMPLES= E.* E odd / APPLICATIONS E (X) - EevenCX) E sinb oda OC) The species F(G) = Fo G a F.-structure 2) Let D be the species of derangements,i.e., permutations without fixed points, then ez= cost, x FOG A (Fo G)-structure is disjointbuilt on aG-sfructures finite set of mss Afrls. S E D The species F' ra F-structure (here, G( 0-0) S(x)=- ECx)D(x) -riza- = ex 3) Cx.) A F'- structure on U is on U+ f4ti Dwr.: d = number of derangements of Ot 7 21 31 n obJects A. 19 20 3) End = S ° A ..,.... 4? lc" 4 41( * l mu k..),,o f) \,), ),.; e.) o 41 *a 4, .t-o /f .04.- , 14' . 4.- 4 :(4* )4 r4 LQ....." s(x)= E(c(1)) 4 , 4160= ix Eva) (m- )! _ova; ?/. C07./. 4b) Letand SO> be the species of permutations behaving the species* cycles of *-sets, 17--0 4,,04-`--/ Cick- ® . then S sEko 44, Ek Edt(x)= Enci(x)-z. / \ rns. n " 4t. S(*` )(x) absolutenumber of value permutations of Stirling of number n points of havingthe first k cycles "16 Eit(03e1) rg,, kind S(k(x)) n" PI! - 1-Ak) A,. = AM = TvNal E yin z"n = Eno,,,..tlef VII> 0 + ,41(rt,101 r 21 22 5a) WhereE+(x) E. is= the species of 33=--E0E+ BC3e)= E(E+Cx)) non-empty sets. 6) = E. E e i = = eee''e Let S be the species ofnumber2" of subsets of a setA of r, elements (known!) ON 2= (E4 ex permutations 6= number of I partitions of a set of n elementsL. n This is Dobinski's formulaIt follows that e 7) having only even cycles : ^S.-- E0C , even e 4m0. for the n th m. ec.,..(2).. eil.,Fc V7-77 5b) andLet .B E. behavingbe the the species species * classes of of *-sets, partitions z, . ., is 32. st..aft-it 0 otherwiseif h = 2k B = Elio Stirlingnumber number of partitions of the ofsecond n points kind into A classes Eli <4e> (23= [ 8) 140.4 14est - *I I YIL > 0 rht ri r) * ' (Lfr))t = (known!) <7, inn gri L'(x)= efil xl 9) C ;= L 23 12) A= X' (E°A) 24 lid(X)1=144 4.00 dx (again I ) I ./ = L odd = LI even = X(xl-E000) - eik(x) 13) Let ( a vertebrateA(30 is a pointed rooted tree ) 4.14x)be the species of vertebrates X an z n but Pb'c even f / ,i, .,0 = LOA si," 01_,A /_.: 11) B1 E B recurrence) e--.. .4 \se ...... ___\.a. a ce.....,....: ,...... , / 4, as (5-o-* fr (zee I ...... il, EA444Cx)s-.. 4...... -w, .-._.... -,.. IliB(2) 1=E(L)B0011, ex-33(z) bh. I *me (*) . ve(x).. L (A(20) 5 -1_A(.) 5 A/4 ..r. n a. r. n" number of enclofunctions on P elements a. r. n " -I (Joyal) 14) gl= the species of forests of rooted trees = 25 16) Let 84 be the species of ballots. 43.01 = L 0 Ell. 26 a' . 1/4.fe-7,1. 41' 4k. . 430 as &LIN LtE.60(7);' = L (ex- i) 4------Ter5u-kt5+6.1)11 760 claw Hence, the number of ballot outcomes for *" n candidatesinintegerY) Is Q, = number + I j ,ov-I of forests of rooted trees on nil tit n points + E,,) oi "1 0** eff "0 .11 15) Ault 17) Inv IlitheiVoirnifbitee:vvr+0:34x1n:oleutl%en4:on= Ea al)1.3.5-42*-0 n elements Is IL. t = E 18)For example : (F-0 6,Y Classical formulas of Calculus hold chain rule ] 21,14(z),.. e it'`,''` ) cs;LN., G V= v-" (Fio &)G'r [ G -Rgb N., G 7:V Zia (29 = Lt A. (x.) = L te,),T.; (r*a 11 F.6 )( G.R:: 0 -* oJC's POSSIBLE TEACHING / LEARNING ACTIVITIES ( from elementary to advanced ) 27 Tostructures find, build fromor draw real-life combinatorial situations 28 orstructuresTo fromfind, buildconcrete from or draw real-life / abstracts combinatorial situations objects. or from concrete / abstract objects.Rivers making a (plane) tree To decideToof make a givenwhether a complete species two structures areand equal non-repetitive or not. Tospecies,list makeof structures labelleda complete belongingby a and given non-repetitive to(small) a given set. Flowers Totolist elementsa findof given unlabelled combinatorial species (n small). structures on equationsn indistinguishable belonging associated 1014Vis:47 ofToto (viaspecies find series enumerative (andfrom expansions, vice-versa).their orcombinatorial structural for example). properties equations Sitting around a round table To decideof a given whether species two are structures equal or not. 29 To find combinatorial equations associated 30 s-b-c-d-e I 1 f-g-h-I-J a-g-h-l-J 1 i to species (and vice-versa). To make a complete and f-g-h-i-Jnon-repetitive = a-b-e-d-o * f-b-c-d-e, ofTo (viaspecies find series enumerative from expansion, their orcombinatorial forstructural example). properties equations species,list of structureslabelled by belonging a given (small) to a given set.The 16 trees labelled by { a, b, c, d } : a-b-d-ca-c-b-da-b-c-d b-a-c-db-d-a-cb-a-d-c b\a/db G I a cb \d/ a ab I d a-d-c-ba-c-d-ba-d-b-c c-b-a-dc-a-b-d0-c-a-d \c/ dI G I to elementslistToa given makeof unlabelled species a (n complete small), onstructures n and indistinguishable non-repetitive belonging The 2 unlabelled trees on 4 identical elements (0-4,-0-411 Y :

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