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The Unity of Combinatorics

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Combinatorics Advances

Part of the book series: Mathematics and Its Applications ((MAIA,volume 329))

Abstract

One reason why Combinatorics has been slow to become accepted as part of mainstream Mathematics is the common belief that it consists of a bag of isolated tricks, a number of areas:

combinatorial number theory (partitions, integer sequences), combinatorial set theory, Ramsey theory, partially ordered sets, lattices (in both the poset and number-theoretic senses), error-correcting codes, combinatorial designs (latin squares and rectangles, projective and affine geometries, Steiner systems, Kirk-man’s schoolgirls problem), combinatorial games, enumerative combinatorics (recurrence relations, generating functions), 0–1 matrices, graph theory (including tournaments, topological properties, coloring problems), combinatorial geometry, packing fc covering (in number-theoretic, set-theoretic, graph-theoretic or geometric contexts)

with little or no connexion between them. We shall see that they have numerous threads weaving them together into a beautifully patterned tapestry.

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References

  1. V. E. Alekseev, On the Skolem method of constructing Steiner triple systems (Russian), Mat Zametki, 2(1967) 145–156; MR 35 #5341. [x+y = z]

    MathSciNet  Google Scholar 

  2. W. W. Rouse Ball fc H. S. M. Coxeter, Mathematical Recreations & Essays, 12th edition, Univ. of Toronto, 1974. [36-40 Nim & Wythoff’s Game; 115–116 squaring the square; 149–152 sphere packing; 189–192 latin squares; 193–221 magic squares; 222–242 map coloring; 271–311 combinatorial de-signs, an excellent survey written by J. J. Seidel]

    Google Scholar 

  3. Thøger Bang, On the sequence [nα], n = 1, 2,..., supplementary note to the previous paper by Th. Skolem, Math. Scand., 5(1957) 69–76; MR 19, 1159h. [Skolem problem]

    MathSciNet  Google Scholar 

  4. S. Beatty, Problem 3173, Amer. Math. Monthly, 33(1926) 159; 34(1927) 159. [Beatty sequences]

    Article  Google Scholar 

  5. Claude Beige, The Theory of Graphs and its Applications, Methuen, London, 1962, p. 38. [Isaacs’s Game]

    Google Scholar 

  6. E. R. Berlekamp, J. H. Conway & R. K. Guy, Winning Ways for Your Mathematical Plays, Academic Press, London, 1982. [sum of games, p. 32; nim addition, pp. 60–61; Wyt Queens, p. 61; nim-values, pp. 82–87; V-positions, p. 83; Turning Turtles, p. 429]

    MATH  Google Scholar 

  7. T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986. [a comprehensive reference for Steiner systems and difference methods]

    MATH  Google Scholar 

  8. Charles L. Bouton, Nim, a game with a complete mathematical theory, Ann. of Math., Princeton,(2) 3(1901-02) 35–39.

    MathSciNet  Google Scholar 

  9. R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte, The dissection of rectangles into squares, Duke Math. J., 7(1940) 312–340; MR 2, 153.

    Article  MathSciNet  Google Scholar 

  10. R. H. Bruck & H. J. Ryser, The non-existence of certain finite projective planes, Canad. J. Math., 1(1949) 88–93; MR 10, 319.

    Article  MathSciNet  Google Scholar 

  11. J. H. Conway, On Numbers and Games, Academic Press, London, 1976. [impartial games & Nim, chap. 11]

    MATH  Google Scholar 

  12. J. H. Conway & N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer Grundlehren der mathematischen Wissenschaften 290, 1988. [Hamming code p. 80; Golay code, p. 84; Cayley numbers, p. 122; lexi-codes, p. 327]

    MathSciNet  Google Scholar 

  13. J. H. Conway & N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Trans. Info. Theory, IT-32(1986) 337–348; MR 87f: 94049.

    Article  MathSciNet  Google Scholar 

  14. H. S. M. Coxeter, The golden section, phyllotaxis and Wythoff’s game, Scripta Math., 19(1953) 135–143; MR 15, 2

    MathSciNet  MATH  Google Scholar 

  15. Hallard T. Croft, Kenneth J. Falconer & Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991. [C2 squaring the square; D10 packing spheres; E9-E13 lattice points]

    Book  MATH  Google Scholar 

  16. R. O. Davies, On Langford’s problem, II, Math. Gaz., 43(1959) 253–255; MR 22 #5581.

    MathSciNet  MATH  Google Scholar 

  17. J. Dénes & A. D. Keedwell, Latin Squares and Their Applications, Academic Press, 1974.

    MATH  Google Scholar 

  18. J. Doyen, Recent developments in the theory of Steiner systems, Atti dei Conv. Lincei, 17(1976) Tomo I, 277–285; MR 55 #10286. [excellent bibli-ography with many early items, previously overlooked]

    MathSciNet  Google Scholar 

  19. A. J. W. Duyvestyn, Simple perfect squared square of lowest order, J. Combin. Theory Ser. B, 25(1978) 240–243; MR 80a: 05051.

    Article  MathSciNet  Google Scholar 

  20. A. J. W. Duyvestyn, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comput., 62(1994) 325–332: MR 94c:05023 (fc see 05017). [Squares with the same elements differently arranged; squares whose largest element is not on the boundary.]

    Google Scholar 

  21. G. Fano, Sui postulati fondamentali della geometria proiettiva, Giom. Mat, 30(1892) 114–124. [the ‘Fano’ configuration was anticipated by Kirk-man in 1850 ]

    Google Scholar 

  22. Branko Grunbaum & G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987, [76–81 squaring the square, 531–583 Penrose aperiodic tiles]

    Google Scholar 

  23. P. M. Grundy, Mathematics and games, Eureka, 2(1939) 6–8; [Sprague-Grundy theory of impartial games]

    Google Scholar 

  24. Richard K. Guy, A many facetted problem of Zarankiewicz, in The Many Facets of Graph Theory, Springer, New York, 1969, 129–148; MR 41 #91. [adjacency matrices, incidence matrices, packing fc covering, Turan fc Ramsey problems, tournaments, steiner triples, affine fc projective planes, difference sets]

    Article  MathSciNet  Google Scholar 

  25. Richard K. Guy, Packing [1, n] with solutions of ax + by = cz — the unity of combinatorics, Atti dei Conv. Lincei, 17(1976) Tomo II, 173–179; MR 57 #9565. [x + y = z, Langford-Skolem, Wythoff, Isaacs, Steiner, Hanani, Ringel, Zarankiewicz]

    MathSciNet  Google Scholar 

  26. Richard K. Guy, The Penrose pieces, Bull London Math. Soc., 8 (1976) 9–10.

    Google Scholar 

  27. Richard K. Guy &c Cedric A. B. Smith, The G-values for various games, Proc. Cambridge Philos. Soc., 52(1956) 514–526; MR 18, 546.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Hadamard, Resolution d’une question relative aux déterminants, Bull. Sci. Math.(2) 17(1893) 240–248. [Hadamard matrices]

    Google Scholar 

  29. R. W. Hamming, Error correcting and error detecting codes, Bell Sys. Tech. J., 29(1950) 147–160; MR 12, 35c.

    MathSciNet  Google Scholar 

  30. H. Hanani, A note on Steiner triple systems, Math. Scand., 8(1960) 154–156; MR 23 #A2330.

    MathSciNet  MATH  Google Scholar 

  31. H. Hanani, On quadruple systems, Canad. J. Math., 12(1960) 145–157; MR 22 #2558.

    Article  MathSciNet  MATH  Google Scholar 

  32. T. P. Kirkman, On a problem in combinations, Cambridge & Dublin Math. 2 (1847) 191–204.

    Google Scholar 

  33. T. P. Kirkman, Note on an unanswered prize question, Cambridge & Dublin Math. J., 5 (1850) 255–262.

    Google Scholar 

  34. T. P. Kirkman, On triads made with fifteen things, London, Edinburgh & Dublin Phil. Mag., 37 (1850) 169–171.

    Google Scholar 

  35. T. P. Kirkman, On the perfect r-partitions of N = r2 - r + 1, Trans. Historic Soc. Lanes. & Cheshire, 9(1856–57) 127–142.

    Google Scholar 

  36. T. P. Kirkman, On the puzzle of the fifteen young ladies, London, Edinburgh & Dublin Phil. Mag. (4), 23 (1862) 198–204.

    Google Scholar 

  37. Thomas Kövári, Vera Sós k Paul Turán, On a problem of K. Zarankiewicz, Colloq. Math., 3(1954) 50–57; MR 16, 456.

    MATH  Google Scholar 

  38. C. Dudley Langford, Problem, Math. Gaz., 42 (1958) 228.

    Google Scholar 

  39. John Leech, Some sphere packings in higher space, Canad. Math. J., 5(1964) 657–682; MR 29 #5166; 19(1967) 251–267; MR 35 #878.

    Article  MathSciNet  Google Scholar 

  40. Emma Lehmer, On residue difference sets, Canad. J. Math., 5(1953) 425–432; MR 15, 10.

    Article  MathSciNet  MATH  Google Scholar 

  41. Al. A. Markov, On a certain combinatorial problem (Russian), Problemy Kibernetiki, 15(1965) 263–266; MR 35 #1497. [x + y = z]

    Google Scholar 

  42. R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.Sc. thesis, The Univ. of Calgary, 1975.

    Google Scholar 

  43. R. J. Nowakowski, Zarankiewicz’s problem, Ph.D. thesis, The Univ. of Calgary, 1978.

    Google Scholar 

  44. C. J. Priday, On Langford’s problem, I, Math. Gaz. 43(1959) 250–253; MR 22 #5580.

    Article  MathSciNet  MATH  Google Scholar 

  45. D. K. Ray-Chaudhuri & R. M. Wilson, Solution of Kirkman’s schoolgirl problem, Proc. Symp. Pure Math., 19 Amer. Math. Soc. 1971, 187–203; MR 47 #3195.

    MathSciNet  Google Scholar 

  46. G. Ringel, Die toioidale Dicke des vollstandigen Graphen, Math. Zeit., 87(1965) 19–26; MR 30 #2489.

    Article  MathSciNet  Google Scholar 

  47. G. Ringel, Map Color Theorem, Grundlehren der math. Wissenschaften, 209, Springer, 1974.

    Google Scholar 

  48. G. Ringel & J. W. T. Youngs, Solution of the Heawood map-coloring problem, Proc. Nat Acad. Sci. U.S.A., 60(1968) 438–445; MR 37 #3959.

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Ringel & J. W. T. Youngs, Solution of the Heawood map-coloring problem — case 11, J. Combin. Theory, 7(1969) 71–93; MR 39 #1360; — case 2, 342–352; -case 8, 353–363; MR 41 #6723.

    Article  MathSciNet  Google Scholar 

  50. A. Rosa, A note on Steiner triple systems (Slovak), Mat.-Fyz. Časopis Sloven. Akad. Vied, 16(1966) 285–290; MR 35 #2759. [x + y = z]

    MathSciNet  MATH  Google Scholar 

  51. H. J. Ryser, Combinatorial Mathematics, Carus Math. Monograph 14, Math. Assoc. Amer., 1963.

    Google Scholar 

  52. J. Sedláček, On a set system, Ann. New York Acad. Sci., 175(1970) 329–330; MR 42 #117. [x + y = z]

    MathSciNet  MATH  Google Scholar 

  53. J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938) 377–385.

    Article  MathSciNet  Google Scholar 

  54. Th. A. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand., 5(1957) 57–68; MR 19, 1159g.

    MathSciNet  MATH  Google Scholar 

  55. Th. A. Skolem, Some remarks on the triple systems of Steiner, Math. Scand., 6(1958) 273–280; MR 21 #5582.

    MathSciNet  MATH  Google Scholar 

  56. Th. A. Skolem, Über einige Eigenschaften der Zahlenmengen [αn + ß] bei irrationalem mit einleitenden Bemerkungen uber einige kombinatorische Probleme, Norske Vid. Selsk. Forh.f Trondheim, 30(1957) 42–49; MR 19, 1159i.

    MathSciNet  Google Scholar 

  57. R. P. Sprague, Über mathematische Kampfspiele, Tôhoku Math. J., 41(1935–36) 438–444; Zbl. 13, 290. [Sprague-Grundy theory of impartial games]

    Google Scholar 

  58. R. P. Sprague, Beispiel einer Zerlegung des Quadrats in lauter verschiedene Quadrate, Math. Z., 45(1939) 607–608. [squaring the square]

    Article  MathSciNet  Google Scholar 

  59. Thomas Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.

    MATH  Google Scholar 

  60. C. M. Terry, L. R. Welch & J. W. T. Youngs, Solution of the Heawood map-coloring problem — case 4, J. Combin. Theory, 8(1970) 170–174; MR 41 #3321.

    Article  MathSciNet  MATH  Google Scholar 

  61. A. Tietäväinen, On the non-existence of perfect codes over finite fields, SIAM J. Appl. Math., 24(1973) 88–96; MR 48 #3609.

    Article  MathSciNet  MATH  Google Scholar 

  62. Thomas M. Thompson, From Error-correcting Codes through Sphere Packings to Simple Groups, Carus Math. Monograph 21, Math. Assoc. Amer., 1983.

    Google Scholar 

  63. J. A. Todd, A combinatorial problem, J. Math, and Phys., 12 (1933) 321–323.

    MATH  Google Scholar 

  64. W. S. B. Woolhouse, Prize question 1733, Lady’s and Gentleman’s Diary, 1844.

    Google Scholar 

  65. W. A. Wythoff, A modification of the game of Nim, Nieuw Arch, voor Wisk.(2), 7(1905-07) 199–202.

    Google Scholar 

  66. J. W. T. Youngs, Solution of the Heawood map-coloring problem — cases 3, 5, 6 and 9, J. Combin. Theory, 8(1970) 175–219; — cases 1, 7, and 10, 220–231; MR 41 #3322-3.

    Article  MathSciNet  MATH  Google Scholar 

  67. K. Zarankiewicz, Problem P101, Colloq. Math., 2 (1951) 301.

    Google Scholar 

  68. Š. Znám, On a combinatorial problem of K. Zarankiewicz, Colloq. Math., 11(1963) 81–84; MR 29 #37; 13(1965) 255–258; MR 32 #7434.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Calgary Calgary, Calgary, Alberta, Canada

    Richard K. Guy

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  1. Richard K. Guy
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  1. University of Waterloo, Canada

    Charles J. Colbourn

  2. Sharif University of Technology, Tehran, Iran

    Ebadollah S. Mahmoodian

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© 1995 Kluwer Academic Publishers

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Guy, R.K. (1995). The Unity of Combinatorics. In: Colbourn, C.J., Mahmoodian, E.S. (eds) Combinatorics Advances. Mathematics and Its Applications, vol 329. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3554-2_9

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  • DOI: https://doi.org/10.1007/978-1-4613-3554-2_9

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