A Combinatorial Interpretation of the Generalized Fibonacci Numbers

Transcription

1 ADVANCES IN APPLIED MATHEMATICS 19, ARTICLE NO. AM A Combinatorial Interpretation of te Generalized Fibonacci Numbers Emanuele Munarini Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, Milan, Italy Received November 4, 1996; accepted December 30, 1996 In tis paper te generalized Fibonacci numbers of order k are combinatorially interpreted, in te context of te teory of linear species of Joyal, as te linear species of k-filtering partitions Academic Press 1. PARTITIONS AND FIBONACCI NUMBERS Te purpose of tis article is to give a combinatorial interpretation of te generalized Fibonacci numbers of order k, i.e., of tose numbers Fn Ž k. defined by te recurrence relation Fnk Ž k. Fnk1 Ž k. Fnk2 Ž k. Fn1 Ž k. Fn Ž k. Ž 1. and Žas in. 5, 8 by te initial conditions F0 Ž k. 0, F2 Žk. 0,..., Fk2 Žk. 0, Fk1 Žk. 1. Ž 2. As is well known, numbers can ave more tan one combinatorial interpretation and clearly tis is also true for generalized Fibonacci numbers. Our interpretation of tese numbers is te one tat naturally arises in te context of te teory of linear Joyal species Linear Species We briefly recall te concept of linear species, pointing out tose aspects tat will be used in te following and referring te reader to 4 for furter details. A linearly ordered set, oralinear order, ² L,: is a set L togeter wit a linear order relation Ži.e., a reflexive, antisymmetric, and transitive $25.00 Copyrigt 1997 by Academic Press All rigts of reproduction in any form reserved. 306

2 FIBONACCI NUMBERS 307 relation in wic all elements are comparable.. As usual, we sall write L for ² L,: and L for its cardinality. Wen L is finite we write L x, x,..., x 1 2 n. A kinteral of L is a subset of te form x; y ul: xuy4 wit exactly k elements. An order function : H L between two linearly ordered sets ² H, H : and ² L, : is a function suc tat Ž x. Ž y. L L wenever x H y. Note tat tere exists at most one order bijection between two finite linear orders H and L and tat tere exists exactly one wen H and L ave te same cardinality. Let Lin be te category of finite linearly ordered sets and order bijections, and let Set be te category of finite sets and functions. A linear species of combinatorial structures is a functor S: Lin Set. As in 4, S L is te set of all te structures of species S on te finite linear order L, and S b : S H SL is te transport of structures along te order bijection b: H L. Te cardinality of a linear species S is te formal geometric series S S n n t, n0 were 0 and n 1, 2,..., n for n 0. Two species S and T are isomorpic wen tere exists a natural equivalence between te functors S and T. In tis case S T. Te sum of two linear species S and T is te linear species S T defined by Ž S T. L SL TL, were te sum on te rigt is te disjoint union of sets. Te product of two linear species S and T is te linear species S T defined by Ž S T. L SL1 TL 2. L L L 1 2 A linear partition of a finite linear order L is a family of nonempty disjoint intervals, called blocks, wose union is L. is itself a finite linear order. Let S and T be two linear species, wit T. Te composition, or substitution, S T is te species defined in suc a manner tat giving a structure of species S T on a finite linear order L means giving a

3 308 EMANUELE MUNARINI partition of L, a structure of species T on eac block of, and a structure of species S on. Te sum, te product, and te composition of species are all preserved by passing to te cardinalities: S T S T, S T S T, ST S T. To develop our interpretation we need an operator acting on te linear species as te sift operator acts on te sequences of natural numbers. We now introduce suc an operator. Let L be a finite linear order. An augmentation of L is a linear order obtained by adding to L a new element. Clearly, we can ave more tan one augmentation of L even for te same new element. We say tat : Lin Lin is an augmentation functor if Ž L. is an augmentation of L, for every L Lin. Let be an augmentation functor and let S be a linear species. Te composition S Lin Lin Set is also a functor, i.e. a linear species. Terefore we can define te operator R by setting so tat R S S, Ž R S. L S Ž L., Ž R S. b S Ž b. for every finite linear order H and L, and for every order bijection b: H L. Te operator R does not depend, up to natural isomorpism, on te functor. In oter words, if 1 and 2 are two augmentation functors, ten R 1 S and R 2 S are naturally equivalent, for every linear species S. Because Ž L. and Ž L. 1 2 ave te same cardinality, for every L Lin, tere exists exactly one order bijection b L: 1Ž L. 2Ž L.. Terefore we can define a natural equivalence between R S and R S 1 2 by setting Sb L L, for every L Lin. From now on we sall write R witout any reference to te underlying augmentation functor. So to give a structure of species R S on a finite linear order L means to give an augmentation of L and ten a structure of species S on tat augmentation.

4 FIBONACCI NUMBERS 309 For our purposes we sall only consider te augmentation functors 1and Ž 1. tat act by adding a new left element and a new rigt element respectively. In oter words, if L x,..., x and x L, ten 1 n 1Lx, x 1,..., x n, L1x 1,..., x n, x. In general, we sall write L and L, wit N, for te linear orders obtained from L, adding new left elements and new rigt elements respectively. Tus, for te t iterate of te operator R of a linear species S, we sall ave Ž R S. L SL or Ž R S. L S L. Now we ave to see ow R acts on te formal geometric series. If Ž. n St st n is te cardinality of S, ten te cardinality of R S is te series SŽ t. SŽ 0. n RSŽ t. s n1 t. t n0 So R St Ž. is te incremental ratio Ž in 0. of te series St. Ž. For tis reason we call R S te incremental ratio of te linear species S. A linear species in -sorts, wit N, is a functor S: Lin Set, and te incremental ratio wit respect to te it sort of S is defined by Ž R S. L,...,L,...,L S L,...,Ž L.,...,L, i 1 i 1 i were is an augmentation functor. We now present some important linear species wic will play a central role in our interpretation. Te geometric species, oruniform species, G is te species defined by GL 4, for all finite linear orders L. Note tat, since G is te singleton on every L, we ave R G G. Te cardinality of G is te geometric series 1 n GŽ t. t. 1t n0 Te linear species G 1 is defined by Ž G1. L ½ 4 L L,

5 310 EMANUELE MUNARINI for every L Lin, and its cardinality is te series 1 t Ž G 1.Ž t. 1. 1t 1t Te -geometric species G is defined as te singleton on every linear order L of cardinality and as te empty set in all oter cases: 4 L GL ½ L. Ž. So R G1 G, and te cardinality of G is te series G t t Te Linear Species of Linear Partitions A linear partition of L Lin is a family of disjoint nonempty intervals wose union is L. We sall say tat B is an -block of wen B as cardinality. Let P be te linear species of te linear partitions. To give a linear partition of a finite linear order L means to give a partition of L, a structure of a nonempty linear order on eac block of, and a geometric structure on. Terefore we ave te isomorpism P GŽ G 1.. Ž 3. Ž. n Ten, passing to cardinalities and setting P Pt pt, n te relation Ž. 3 becomes 1 t 1 PŽ t.. 1t 1t t 1 1t Expanding te obtained series, we ave n1 pn Ž 2n0. 2, Ž 4. were n 0 0if n0 and n 0 1if n Te Linear Species of te k-filtering Partitions A filtering partition of order k, ora k-filtering partition, of LLin is a linear partition of L in wic eac block as at most k elements. For example, te 2-filtering partitions of L 1, 2, 3, 4 are 1,2,3,4 4, 1,2,3,4 4, 1,2,3,4 4, 1,2,3,4 4, 1,2,3,4 4.

6 FIBONACCI NUMBERS 311 Let F k be te linear species of te k-filtering partitions, and let f Ž k. n k be te cardinality of F L, wen L n. To give a k-filtering partition of a finite linear order L means to give a partition of L, a structure of linear order of cardinality at most k on eac block of, and a geometric structure on. Terefore and, passing to cardinalities, F k GŽ G1 G2 G n., Ž Žk. n 2 k f t Ž tt t.. Ž 6. 1t 1 Ž tt t. n 2 k n0 Let us now consider te species R k F k, wit k 1. For every finite linear order L, we ave, by definition, k k k R F L F Lk. k Te set F L k can be partitioned according to te fact tat te last block of a k-filtering partition of L k as cardinality 1, or 2,..., or k. For example, if L 4 and k 3, we ave te following cases: Obviously tere is a bijection between te set of all k-filtering partitions of L k wit an -block Ž 1 k. as te last block and te set of all k-filtering partitions of L Ž k.. Terefore we ave te isomorpism R k F k R k1 F k R k2 F k R F k F k Ž 7. and, passing to cardinalities, te recurrence relation f Ž k. f Ž k. f Ž k. f Ž k. f Ž k. 8 nk nk1 nk2 n1 n

7 312 EMANUELE MUNARINI wic is exactly te relation Ž. 1 by means of wic te generalized Fibonacci numbers of order k are defined. To obtain te initial conditions it suffices to observe tat a k-filtering partition of a linear-order L of cardinality less tan Ž or equal to. k is an arbitrary linear partition of L. Terefore, due to Ž. 4 in te preceeding subsection, we ave f0 Ž k. 1, f1 Žk. 1, f2 Žk. 2,..., fk2 Žk. 2 k3, fk1 Žk. 2 k2. Ž 9. Ž. So our initial conditions are different from te usual ones given by 2. Yet our Fibonacci numbers differ from te usual ones only be a sifting; more precisely, for every k 1, we ave fn Ž k. Fnk1 Ž k Te Linear Species of te k-filtering Partitions wit Blocks k Let F be te linear species of te k-filtering partitions wit blocks Ž k. k and let f be te cardinality of F L, wen L, n n. To give a structure of species F k on a finite linear order L means to give a partition of L, a structure of linear order wit at most k elements on eac block of, and a structure of a linear order wit elements on. Terefore we ave F k GŽ G1 G2 G k.. Ž 10. k Moreover, te family F 4 N being summable, we also ave F k F k Ž or GŽ G G G. G Ž G G G k 1 2 k 0 Passing to cardinalities, from 10 we obtain Ž k. n 2 k 2 k f, n t t tt t t t t, 13 n0 wereas from 11 we obtain again te series 6.

8 FIBONACCI NUMBERS 313 Expanding te series at te rigt-and side of 13, it is easy to find and ten, summing up to, ž 1 k/ Ž k. f, n r,...,r, Ž 14. r,...,r 0 1 k r r 1 k 1r kr n 1 k ž 1 k / Ž k. Žk. r1 rk fn f. n. Ž 15 r,...,r. 0 r,...,r 0 1 k 1r kr n 1 k k k k Let us consider te species R F. As te set F L k 1 1 can be partitioned according to te fact tat te last block of a k-filtering partition of L k as cardinality at most k, we ave R k F1 k R k1 F k R k2 F k R F k R F k. Ž 16. Passing to cardinalities, we obtain te recurrence relation f1, Ž k. nk f, Ž k. nk1 f, Ž k. nk2 f, Ž k. n1 f, Ž k. n Ž 17. wit te initial conditions Ž k. Žk. f0, n n0 and f,0 0, Žk. j1 f 2 j, j1,...,k1. Ž 18., j 2.1. First Application 2. APPLICATIONS Let H k be te linear species in two sorts defined, for every H, L Lin, 2 by k k H2 H, L F HL. Let us consider, for k 1, te species R k1 R k1 H k ; for H, L Lin we ave k1 k1 k k R1 R2 H2 H, L H2 H k1, k1 L k F H Ž k1. Ž k1. L. Ž 19.

9 314 EMANUELE MUNARINI Let x and y be te last element of H Ž k 1. and te first element of Ž k1. L respectively. Te set Ž 19. can be partitioned according to te fact tat in a k-filtering partition of H Ž k 1. Ž k 1. L, x and y belong eiter to different blocks or to te same block. In te first case, decomposes in an arbitrary k-filtering partition of HŽ k1. and in an arbitrary k-filtering partition of Ž k 1. L. In te second case, x and y belong to an interval wit at most k elements; so tis interval contains te last l elements of H Ž k 1. and te first r elements of Ž k 1. L, were l 1, r 1, and l r k. Tus decomposes in an arbitrary k-filtering partition of H Ž k 1 l. and in an arbitrary k-filtering partition of Ž k 1 r. L. For example, if H 5, L 3, and k 3, tere are te following cases: Tis observation immediately yields te following isomorpism: k1 k1 k k k R R H H, L F H Ž k1. F Ž k1. L k F H k1l k F k1r L. 20 l, r1 lrk Terefore, passing to cardinalities, we ave te identity f Ž k. f Ž k. f Ž k. f Ž k. f Ž k., Ž 21. mn2k2 mk1 nk1 mk1l nk1r l, r0 lrk and, for m n, te identity Ž k. Žk. 2 Žk. Žk. 2n2k2 nk1 nk1l nk1r l, r1 lrk f f f f. Ž 22.

10 FIBONACCI NUMBERS 315 Tere follow te instances of te preceding formulas for k 2, 3, 4: f f f f f mn2 m1 n1 m n f f 2 f 2 2n2 n1 n g g g g g g g g g mn4 m2 n2 m1 n1 m1 n m n1 g g 2 g 2 2g g 2n4 n2 n1 n1 n mn6 m3 n3 m2 n2 m1 n1 m2 n1 m1 n2 m2 n m n1 2n6 2 n3 2 n2 2 n12ž n1n. n2, were f f Ž2., g f Ž3., and f Ž4.. n n n n n n Let us now consider te species H k in sorts defined by k k H L,...,L F L L 1 1 for all L,...,L Lin, and, for, k 1, te species 1 We ave R k1 R 2Ž k1. R 2Ž k1. R k1 H k k1 2Žk1. 2Žk1. k1 k R R R R H L,...,L were H1 L1 Ž k 1 l 1., H H, H,...,H,H, Ž 23. k H2 Ž k1r1. L2 Ž k1l 2.,... H1 Ž k1r2. L1 Ž k1l 1., H Ž k1r1. L. Let xi and yi be te last element of Hi and te first element of Hi1 respectively, for i 1, 2,..., 1. Te set Ž 23. can be partitioned according to te fact tat xi and yi eiter belong to different blocks or belong to

11 316 EMANUELE MUNARINI te same block, for every i 1, 2,..., 1. So we ave k1 2Žk1. 2Žk1. k1 k R1 R2 R1 R H L 1,...,L :10,14 l i, riž i. lrk i i Ž. i i1,...,1 k Ł j j1 F H Ž 24. were is te function by means of wic we distinguis te case in wic x and y belong to different blocks Ž Ž i. 0.. i i from te case in wic x and y are in te same block Ž Ž i. 1.. i i. Finally, passing to cardinalities, we ave f Ž k. f Ž k. f Ž k. f Ž k. f Ž k., Ž 25. n1 n 2Ž 1.Ž k1. s1 s2 s1 s :10,14 l i, riž i. lrk i i Ž. i i1,...,1 were s1 n1 k 1 l 1, s2n22k2r1l 2,... s n 2k2r l, s n k1r. 1 Tere follow te formulas obtained from Ž 25. wen n1 n n and Ž, k. Ž 3, 2., Ž 4, 2., Ž 3, 3.: f f 3 3 f 2 f f 3 3n4 n1 n1 n n f 2f 4 4f 3 f 6f 2 f 2 f 4 4n6 n1 n1 n n1 n n g3n82gn2 3 2gn1 3 Ž gn2gn.ž 6gn2gn13gn2gn3gn Second Application km km Let us consider te species F, wit m 1. Given F L, we can partition according to te fact tat any block as cardinality k km or k. Tus we can give a structure of species F in te following way. First, we give two distinct intervals L1 and L2 wose union is L, and a structure of species F k on L 2. Ten we give a partition of L 1, for

12 FIBONACCI NUMBERS 317 eac block B of two disjoint intervals B1 and B2 wose union is B, a k structure of species F on B 1, and a structure of linear order wit cardinality between k 1 and k m on B 2. Terefore we ave te isomorpism km k k F G F Gk1 Gkm F 26 and, passing to cardinalities, km k1 km k 1 Ž t t. t t 1 Ž t t. 1 k 1 Ž t t. Ž 27. k On te oter and, F L can be partitioned according to te number of blocks of a k-filtering partition of L aving cardinality between k 1 and k m. Let Cr km te species of te k-filtering partition wit r suc blocks. km Now a partition C determines a Ž 2r 1. -tuple r Ž H, C, H, C,...,H,C,H r r r1 of disjoint intervals of L wose union is L, were te Cj are te r intervals wit k 1 C j k m, and te Hj are arbitrary intervals k, eventually empty. Let P m L be te set of suc Ž 2r 1. r -tuples. Clearly, we ave km km F L C L r0 Tus, passing to cardinalities, we obtain and r r0 k, m P L r Ž H 1,C 1,...,Hr1. r1 k Ł F Hj 28 j1 fn Ž km. f H Ž k. f H Ž k.. Ž r1 r0 k, m P L It is clear tat for every r Ž H 1,C 1,...,Hr1. Ž H 1,1, C 1,...,H 1, r,c r,h1, r1. H, C,...,H,C,H, 2,1 pž1. 2, r pžr. 2, r1

13 318 EMANUELE MUNARINI were H H and p is a permutation of r, we ave 1, j 2, j So, if we ave r were j f Žk. H f H Ž k. f H Ž k. f H Ž k.. 1,1 1, r1 2,1 2, r1 Ž kj. -blocks, wit j 1,...,m, ten Ž 29. becomes ž 1 m/ Ž km. r fn r,...,r gnž r 1,...,rm. Ž 30. r 1,...,rm0 rr1 rm gnž r 1,...,rm. f Žk. f Ž k.. 1,...,r10 1 r1nr1ž k1. rmž km. 1 r1 Obviously tis result can also be obtained by formally expanding te series at te rigt-and side of 27. REFERENCES 1. F. Dubeau, On r-generalized Fibonacci Numbers, Fibonacci Quart. 27 Ž 1989., R. L. Graam, D. E. Knut, and O. Patasnik, Concrete Matematics, AddisonWesley, Reading, MA, W.-J. Hsu, Fibonacci cubesa new interconnection topology, IEEE Trans. Parallel Distributed Systems 4 Ž 1993., A. Joyal, Une teorie combinatoire des series formelles, Ad. in Mat. 42 Ž 1981., D. E. Knut, Te Art of Computer Programming, Vol. 3, Sorting and Searcing, AddisonWesley, Reading, MA, J. Labelle, Applications diverses de la teorie combinatoire des especes ` de structurs, Ann. Sci. Mat. Quebec 7.1 Ž 1983., J. Liu, W.-J. Hsu, and M. J. Cung, Generalized Fibonacci cubes are mostly Hamiltonian, J. Grap Teory 18 Ž 1994., E. P. Miles, Jr., Generalized Fibonacci numbers and associated matrices, Amer. Mat. Montly 67 Ž 1960., M. D. Miller, On generalized Fibonacci numbers, Amer. Mat. Montly 78 Ž 1971., A. N. Pilippou, and A. A. Muwafi, Waiting for te kt consecutive success and te Fibonacci sequence of order k, Fibonacci Quart. 20 Ž 1982., M. E. Waddill, Te Tetranacci sequence and generalizations, Fibonacci Quart. 30 Ž 1992., 920.