The Ramanujan Journal

, Volume 12, Issue 1, pp 5–75 | Cite as

Partition bijections, a survey

Article

Abstract

We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.

Keywords

Integer partitions Young diagram q-series Bijection Involution Plane partitions Involution principle 

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References

  1. 1.
    Adiga, C., Berndt, B.C., Bhargava, S., Watson, G.N.: Chapter 16 of Ramanujan’s second notebook: theta-functions and q-series. Mem. Amer. Math. Soc. 53(315), v+85 (1985)Google Scholar
  2. 2.
    Ahlgren, S., Ono, K.: Addition and counting: the arithmetic of partitions. Notices Amer. Math. Soc. 48(9), 978–984 (2001)Google Scholar
  3. 3.
    Aigner, M., Ziegler, G.M.: Proofs from The Book. 2nd edition, Springer-Verlag, Berlin (2001)MATHGoogle Scholar
  4. 4.
    Alder, H.L.: Partition identities—from Euler to the present. Amer. Math. Monthly 76, 733–746 (1969)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Alladi, K.: A fundamental invariant in the theory of partitions. In Topics in Number Theory (University Park, PA, 1997), pp. 101–113. Kluwer Acad. Publ., Dordrecht (1999)Google Scholar
  6. 6.
    Alladi, K.: A variation on a theme of Sylvester—a smoother road to Göllnitz’s (big) theorem. Discrete Math. 196(1/3), 1–11 (1999)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Alladi, K., Gordon, B.: Partition identities and a continued fraction of Ramanujan. J. Combin. Theory Ser. A 63(2), 275–300 (1993)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Alladi, K., Gordon, B.: Schur’s partition theorem, companions, refinements and generalizations. Trans. Amer. Math. Soc. 347(5), 1591–1608 (1995)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Andrews, G.E.: A simple proof of Jacobi’s triple product identity. Proc. Amer. Math. Soc. 16, 333–334 (1965)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Andrews, G.E.: On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2), 17, 132–143 (1966)Google Scholar
  11. 11.
    Andrews, G.E.: On generalizations of Euler’s partition theorem. Michigan Math. J. 13, 491–498 (1966)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Andrews, G.E.: Enumerative proofs of certain q-identities. Glasgow Math. J. 8, 33–40 (1967)MATHMathSciNetGoogle Scholar
  13. 13.
    Andrews, G.E.: On a calculus of partition functions. Pacific J. Math. 31, 555–562 (1969)MATHMathSciNetGoogle Scholar
  14. 14.
    Andrews, G.E.: Two theorems of Gauss and allied identities proved arithmetically. Pacific J. Math. 41, 563–578 (1972)MATHMathSciNetGoogle Scholar
  15. 15.
    Andrews, G.E.: An extension of Carlitz’s bipartition identity. Proc. Amer. Math. Soc. 63(1), 180–184 (1977)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Andrews, G.E.: An introduction to Ramanujan’s “lost” notebook. Amer. Math. Monthly 86(2), 89–108 (1979)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Andrews, G.E.: A note on partitions and triangles with integer sides. Amer. Math. Monthly 86(6), 477–478 (1979)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Andrews, G.E.: Partitions and Durfee dissection. Amer. J. Math. 101(3), 735–742 (1979)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Andrews, G.E.: Ramanujan’s “lost” notebook. I. Partial θ-functions. Adv. in Math. 41(2), 137–172 (1981)MATHGoogle Scholar
  20. 20.
    Andrews, G.E.: On a partition theorem of N. J. Fine. J. Nat. Acad. Math. India 1(2), 105–107 (1983)MATHGoogle Scholar
  21. 21.
    Andrews, G.E.: Use and extension of Frobenius’ representation of partitions. In Enumeration and Design (Waterloo, Ont., 1982), pp. 51–65. Academic Press, Toronto, ON (1984)Google Scholar
  22. 22.
    Andrews, G.E.: Combinatorics and Ramanujan’s “lost” notebook. In Surveys in Combinatorics 1985 (Glasgow, 1985), pp. 1–23. Cambridge Univ. Press, Cambridge (1985)Google Scholar
  23. 23.
    Andrews, G.E., Sylvester, J.J.: Johns Hopkins and partitions. In A century of mathematics in America, Part I, pp. 21–40. Amer. Math. Soc., Providence, RI (1988)Google Scholar
  24. 24.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)Google Scholar
  25. 25.
    Andrews, G.E.: Some debts I owe. Sém. Lothar. Combin., 42:Art. B42a, 16 pp. (electronic) (1999)Google Scholar
  26. 26.
    Andrews, G.E.: MacMahon’s partition analysis. II. Fundamental theorems. Ann. Comb. 4(3–4):327–338, Conference on Combinatorics and Physics (Los Alamos, NM, 1998) (2000)Google Scholar
  27. 27.
    Andrews, G.E.: Schur’s theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. In q-series from a contemporary perspective (South Hadley, MA, 1998), pp. 45–56. Amer. Math. Soc., Providence, RI (2000)Google Scholar
  28. 28.
    Andrews, G.E., Ekhad, S.B., Zeilberger, D.: A short proof of Jacobi’s formula for the number of representations of an integer as a sum of four squares. Amer. Math. Monthly 100(3), 274–276 (1993)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull. Amer. Math. Soc. (N.S.) 18(2), 167–171 (1988)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Askey, R.: Ramanujan and hypergeometric and basic hypergeometric series. In Ramanujan International Symposium on Analysis (Pune, 1987), pp. 1–83. Macmillan of India, New Delhi (1989)Google Scholar
  31. 31.
    Askey, R.: The work of George Andrews: a Madison perspective. Sém. Lothar. Combin., 42:Art. B42b, 24 pp. (electronic) (1999)Google Scholar
  32. 32.
    Atkin, A.O.L., Swinnerton-Dyer, P.: Some properties of partitions. Proc. London Math. Soc. 4(3), 84–106 (1954)MATHMathSciNetGoogle Scholar
  33. 33.
    Bach, E., Shallit, J.: Algorithmic Number Theory. Vol. 1. MIT Press, Cambridge, MA (1996)MATHGoogle Scholar
  34. 34.
    Bacher, R., Manivel, L.: Hooks and powers of parts in partitions. Sém. Lothar. Combin., 47:Article B47d, 11 pp. (electronic) (2001)Google Scholar
  35. 35.
    Bell, E.T.: The form wx + xy + yz + zu. Bull. Amer. Math. Soc. 42, 377–380 (1936)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Bender, E.A., Knuth, D.E.: Enumeration of plane partitions. J. Combinatorial Theory Ser. A 13, 40–54 (1972)MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Berkovich, A., Garvan, F.G.: Some observations on Dyson’s new symmetries of partitions. J. Combin. Theory Ser. A 100(1), 61–93 (2002)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Bessenrodt, C.: A bijection for Lebesgue’s partition identity in the spirit of Sylvester. Discrete Math. 132(1–3), 1–10 (1994)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Bessenrodt, C.: On hooks of Young diagrams. Ann. Comb. 2(2), 103–110 (1998)MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Bessenrodt, C.: On pairs of partitions with steadily decreasing parts. J. Combin. Theory Ser. A 99, 162–174 (2002)MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. Ramanujan J. 1(1), 101–111 (1997)Google Scholar
  42. 42.
    Bousquet-Mélou, Eriksson, K.: A refinement of the lecture hall theorem. J. Combin. Theory Ser. A 86(1), 63–84 (1999)Google Scholar
  43. 43.
    Bressoud, D.M. 7. On a partition theorem of Göllnitz. J. Reine Angew. Math. 305, 215–217.Google Scholar
  44. 44.
    Bressoud, D.M.: A combinatorial proof of Schur’s 1926 partition theorem. Proc. Amer. Math. Soc. 79(2), 338–340 (1980)MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Bressoud, D.M., Subbarao, M.V.: On Uchimura’s connection between partitions and the number of divisors. Canad. Math. Bull 27(2), 143–145 (1984)MATHMathSciNetGoogle Scholar
  46. 46.
    Bressoud, D.M., Zeilberger, D.: A short Rogers-Ramanujan bijection. Discrete Math. 38(2–3), 313–315 (1982)MATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Bressoud, D.M., Zeilberger, D.: Bijecting Euler’s partitions-recurrence. Amer. Math. Monthly 92(1), 54–55 (1985)MATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Bressoud, D.M., Zeilberger, D.: Generalized Rogers-Ramanujan bijections. Adv. Math. 78(1), 42–75 (1989)MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Canfield, R., Corteel, S., Hitczenko, P.: Random partitions with non-negative r-th differences. Adv. in Appl. Math. 27(2–3), 298–317 (2001)MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Carlitz, L. Some generating functions. Duke Math. J. 30, 191–201 (1963)MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Carlitz, L.: Generating functions and partition problems. In Proc. Sympos. Pure Math., Vol. VIII, pp. 144–169. Amer. Math. Soc., Providence, R.I. (1965)Google Scholar
  52. 52.
    Carlitz, L., Subbarao, M.V.: A simple proof of the quintuple product identity. Proc. Amer. Math. Soc. 32, 42–44 (1972)MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Cayley, A.: A letter to Dr. Franklin (an extract). Johns Hopkins Univ. Circular 2(22), 86 (1883)Google Scholar
  54. 54.
    Chapman, R.: Franklin’s argument proves an identity of Zagier. Electron. J. Combin. 7(1), Research Paper 54, 5 pp. (electronic) (2000)Google Scholar
  55. 55.
    Cheema, M.S.: Vector partitions and combinatorial identities. Math. Comp. 18, 414–420 (1964)MATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Cohen, D.I.A.: PIE-sums: a combinatorial tool for partition theory. J. Combin. Theory Ser. A 31(3), 223–236 (1981)MATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    Corteel, S. Particle seas and basic hypergeometric series. Adv. in Appl. Math. 31(1), 199–214 (2003)Google Scholar
  58. 58.
    Corteel, S., Lovejoy, J.: Frobenius partitions and the combinatorics of Ramanujan’s 1ψ1 summation. J. Combin. Theory Ser. A 97(1), 177–183 (2002)MATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    Corteel, S., Lovejoy, J.: Overpartitions. Trans. Amer. Math. Soc. 356(4), 1623–1635 (2004)MATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Dyson, F.J.: Some Guesses in The Theory of Partitions. Eureka (Cambridge) 8, 10–15 (1944)Google Scholar
  61. 61.
    Dyson, F.J.: A new symmetry of partitions. J. Combin. Theory 7, 56–61 (1969)MATHMathSciNetGoogle Scholar
  62. 62.
    Dyson, F.J.: A walk through Ramanujan’s garden. In Ramanujan revisited (Urbana-Champaign, Ill., 1987), pp. 7–28. Academic Press, Boston, MA (1988)Google Scholar
  63. 63.
    Dyson, F.J.: Mappings and symmetries of partitions. J. Combin. Theory Ser. A 51(2), 169–180 (1989)MATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Edwards, H.M.: Fermat’s last theorem. Springer-Verlag, New York (1996)MATHGoogle Scholar
  65. 65.
    Erdös, P.: On an elementary proof of some asymptotic formulas in the theory of partitions. Ann. Math. 43(2), 437–450 (1942)MATHCrossRefGoogle Scholar
  66. 66.
    Euler, L. Introductio in analysin infinitorum. Tomus primus. Marcum-Michaelem Bousquet, Lausannae (1748)Google Scholar
  67. 67.
    Ewell, J.A.: Recurrences for the sum of divisors. Proc. Amer. Math. Soc. 64(2), 214–218 (1977)MATHMathSciNetCrossRefGoogle Scholar
  68. 68.
    Farkas, H.M., Kra, I.: Theta constants, Riemann surfaces and the modular group, vol. 37 of Graduate Studies in Mathematics. AMS, Providence, RI (2001)Google Scholar
  69. 69.
    Fine, N.J.: Some new results on partitions. Proc. Nat. Acad. Sci. USA 34, 616–618 (1948)MATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    Fine, N.J.: Basic hypergeometric series and applications. AMS, Providence, RI (1988)Google Scholar
  71. 71.
    Foata, D., Han, G.-N.: The triple, quintuple and septuple product identities revisited. Sém. Lothar. Combin., 42: Art. B42o, 12 pp. (electronic) (1999)Google Scholar
  72. 72.
    Franklin, F.: Sure le développement du produit infini (1 −x)(1 −x 2)(1 −x 3)... C. R. Acad. Paris Ser A 92:448–450 (1881)Google Scholar
  73. 73.
    Franklin, F.: On partitions. Johns Hopkins Univ. Circular 2(22), 72 (1883)Google Scholar
  74. 74.
    Garrett, K., Ismail, M.E.H., Stanton, D.: Variants of the Rogers-Ramanujan identities. Adv. in Appl. Math. 23(3), 274–299 (1999)Google Scholar
  75. 75.
    Garsia, A.M. Combinatorics Lecture Notes. (November 10, 1999), to appear.Google Scholar
  76. 76.
    Garsia, A.M., Milne, S.C.: A Rogers-Ramanujan bijection. J. Combin. Theory Ser. A 31(3), 289–339 (1981)MATHMathSciNetCrossRefGoogle Scholar
  77. 77.
    Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11. Trans. Amer. Math. Soc. 305(1), 47–77 (1988)MATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)Google Scholar
  79. 79.
    Gupta, H.: Combinatorial proof of a theorem on partitions into an even or odd number of parts. J. Combin. Theory Ser. A 21(1), 100–103 (1976)MATHCrossRefGoogle Scholar
  80. 80.
    Hardy, G.H.: Ramanujan. Twelve lectures suggested by his life and work. Cambridge University Press, Cambridge, England (1940)Google Scholar
  81. 81.
    Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. 17, 75–115 (1918)Google Scholar
  82. 82.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. 5th edn. The Clarendon Press Oxford University Press, New York, (1979)MATHGoogle Scholar
  83. 83.
    Hathaway, A.S.: A proof of a Theorem of Jacobi. Johns Hopkins Univ. Circular 2(25), 143–144 (1883)Google Scholar
  84. 84.
    Hickerson, D.R.: A partition identity of the Euler type. Amer. Math. Monthly 81, 627–629 (1974)MATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    Hirschhorn, M.D.: Simple proofs of identities of MacMahon and Jacobi. Discrete Math. 16(2), 161–162 (1976)MATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    Hirschhorn, M.D.: Polynomial identities which imply identities of Euler and Jacobi. Acta Arith. 32(1), 73–78 (1977)MATHMathSciNetGoogle Scholar
  87. 87.
    Hirschhorn, M.D.: A simple proof of Jacobi’s two-square theorem. Amer. Math. Monthly 92(8), 579–580 (1985)MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    Hirschhorn, M.D.: A simple proof of Jacobi’s four-square theorem. Proc. Amer. Math. Soc. 101(3), 436–438 (1987)MATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    Hoare, A.H.M.: An involution of blocks in the partitions of n. Amer. Math. Monthly 93(6), 475–476 (1986)MATHMathSciNetCrossRefGoogle Scholar
  90. 90.
    Ismail, M.E.H.: A simple proof of Ramanujan’s 1ψ1 sum. Proc. Amer. Math. Soc. 63(1), 185–186 (1977)MATHMathSciNetCrossRefGoogle Scholar
  91. 91.
    Joichi, J.T., Stanton, D.: An involution for Jacobi’s identity. Discrete Math. 73(3), 261–271 (1989)MATHMathSciNetCrossRefGoogle Scholar
  92. 92.
    Kanigel, R.: The Man Who Knew Infinity. Scribner, New York (1991)Google Scholar
  93. 93.
    Kim, D., Yee, A.J.: A note on partitions into distinct parts and odd parts. Ramanujan J. 3(2), 227–231 (1999)MATHMathSciNetCrossRefGoogle Scholar
  94. 94.
    Kleitman, D.J.: On the future of combinatorics. In: Essays on the future. Birkhäuser, Boston, MA (2000) pp. 123–134Google Scholar
  95. 95.
    Knuth, D.E., Paterson, M.S.: Identities from partition involutions. Fibonacci Quart. 16(3), 198–212 (1978)MATHMathSciNetGoogle Scholar
  96. 96.
    Krattenthaler, C.: Another involution principle-free bijective proof of Stanley’s hook-content formula. J. Combin. Theory Ser. A 88(1), 66–92 (1999)MATHMathSciNetCrossRefGoogle Scholar
  97. 97.
    Leibenzon, Z.L.: A simple combinatorial method for proof of the Jacobi identity and its generalizations. Funktsional. Anal. i Prilozhen. 20(1), 77–78 (1986)MATHMathSciNetGoogle Scholar
  98. 98.
    Leibenzon, Z.L.: A simple proof of the Macdonald identities for the series A. Funktsional. Anal. i Prilozhen. 25(3), 19–23 (1991)Google Scholar
  99. 99.
    Lewis, R.P.: A combinatorial proof of the triple product identity. Amer. Math. Monthly 91(7), 420–423 (1984)MATHMathSciNetCrossRefGoogle Scholar
  100. 100.
    Little, D.P.: An extension of Franklin’s bijection. Sém. Lothar. Combin., 42: Art. B42h, 10 pp. (electronic) (1999)Google Scholar
  101. 101.
    Macdonald, I.G.: Affine root systems and Dedekind’s η-function. Invent. Math. 15, 91–143 (1972)MATHMathSciNetCrossRefGoogle Scholar
  102. 102.
    MacMahon, P.A.: Combinatory analysis. Chelsea Publishing Co., New York (1960)Google Scholar
  103. 103.
    Miller, E., Pak, I. in preparation.Google Scholar
  104. 104.
    O’Hara, K.M.: Bijections for partition identities. J. Combin. Theory Ser. A 49(1), 13–25 (1988)MathSciNetCrossRefMATHGoogle Scholar
  105. 105.
    Pak, I. On Fine’s partition theorems, Dyson, Andrews, and missed opportunities. Math. Intelligencer, to appear.Google Scholar
  106. 106.
    Pak, I.: Partition identities and geometric bijections. Proc. Amer. Math. Soc. to appear.Google Scholar
  107. 107.
    Pak, I.: Hook length formula and geometric combinatorics. Sém. Lothar. Combin., 46:Art. B46f, 13 pp. (electronic) (2001)Google Scholar
  108. 108.
    Pak, I., Postnikov, A.: A generalization of Sylvester’s identity. Discrete Math. 178(1–3), 277–281 (1998)MATHMathSciNetCrossRefGoogle Scholar
  109. 109.
    Remmel, J.B.: Bijective proofs of some classical partition identities. J. Combin. Theory Ser. A 33(3), 273–286 (1982)MATHMathSciNetCrossRefGoogle Scholar
  110. 110.
    Sagan, B.E.: Bijective proofs of certain vector partition identities. Pacific J. Math. 102(1), 171–178 (1982)MATHMathSciNetGoogle Scholar
  111. 111.
    Schur, I.: Ein Beitrag zur Additiven Zahlentheorie und zur Theorie der Kettenbrüche. S.-B. Preuss. Akad. Wiss. Phys. Math. Klasse, pp. 302–321 (1917)Google Scholar
  112. 112.
    Schur, I.: Zur Additiven Zahlentheorie. S.-B. Preuss. Akad. Wiss. Phys. Math. Klasse, pp. 488–495 (1926)Google Scholar
  113. 113.
    Shanks, D.: A short proof of an identity of Euler. Proc. Amer. Math. Soc. 2, 747–749 (1951)MATHMathSciNetCrossRefGoogle Scholar
  114. 114.
    Shiu, P.: Involutions associated with sums of two squares. Publ. Inst. Math. (Beograd) (N.S.), 59(73), 18–30, avaiable at: http://www.emis.de/journals/PIMB/073/index.html. (1996)
  115. 115.
    Stanley, R.P.: Enumerative combinatorics. Vol. 1, 2. Cambrridge University Press, Cambridge (1997, 1999)Google Scholar
  116. 116.
    Stanton, D.: An elementary approach to the Macdonald identities. In q–series and partitions (Minneapolis MN, 1988), pp. 139–149. Springer, New York (1989)Google Scholar
  117. 117.
    Stockhofe, D.: Bijektive Abbildungen auf der Menge der Partitionen einer natürlichen Zahl. Bayreuth. Math. Schr. (10), 1–59 (1982)Google Scholar
  118. 118.
    Stoyanovskii, A.V., Feigin, B.L.: Functional models of the representations of current algebras, and semi-infinite Schubert cells. Funktsional. Anal. i Prilozhen. 28(1), 68–90 (1994)MATHMathSciNetGoogle Scholar
  119. 119.
    Subbarao, M.V.: Combinatorial proofs of some identities. In Proceedings of the Washington State University Conference on Number Theory (Washington State Univ., Pullman, Wash., 1971), pp. 80–91. Dept. Math., Washington State Univ., Pullman, Wash (1971)Google Scholar
  120. 120.
    Sudler, C.: Jr. Two enumerative proofs of an identity of Jacobi. Proc. Edinburgh Math. Soc. 15(2), 67–71 (1966)MATHMathSciNetGoogle Scholar
  121. 121.
    Sylvester, J.J., Franklin, F.: A constructive theory of partitions, arranged in three acts, an interact and an exodion. Amer. J. Math. 5, 251–330 (1882)MathSciNetCrossRefGoogle Scholar
  122. 122.
    Uchimura, K.: An identity for the divisor generating function arising from sorting theory. J. Combin. Theory Ser. A 31(2), 131–135 (1981)MATHMathSciNetCrossRefGoogle Scholar
  123. 123.
    Vahlen, K.T.: Beiträge zu einer additiven Zahlentheorie. J. Reine Angew. Math. 112, 1–36 (1893)MATHGoogle Scholar
  124. 124.
    Vershik, A.M.: A bijective proof of the Jacobi identity, and reshapings of the Young diagrams. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 155 (Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII), 3–6 (1986)Google Scholar
  125. 125.
    Wenkov, B.A.: Elementary Number Theory, in Russian, ONTI, Moscow, USSR (1937)Google Scholar
  126. 126.
    Wilf, H.S. Lectures on Integer Partitions. (unpublished), avaiable at: http://www.cis.upenn.edu/∼wilf
  127. 127.
    Wilf, H.S.: Identically distributed pairs of partition statistics. Sém. Lothar. Combin., 44:Art B44c, 3 pp. (electronic) (2000)Google Scholar
  128. 128.
    Wright, E.M.: An enumerative proof of an identity of Jacobi. J. London Math. Soc. 40, 55–57 (1965)MATHMathSciNetGoogle Scholar
  129. 129.
    Zagier, D.: A one-sentence proof that every prime p ≡ 1 (mod 4) is a sum of two squares. Amer. Math. Monthly 97(2), 144 (1990)Google Scholar
  130. 130.
    Zeng, J. The q-variations of Sylvester’s bijection between odd and strict partitions. Ramanujan J. 9(3), 289–303 (2005)Google Scholar
  131. 131.
    Zolnowsky, J.: A direct combinatorial proof of the Jacobi identity. Discrete Math. 9, 293–298 (1974)MATHMathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsMITCambridge

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