Partition bijections, a survey

Ramanujan Journal

Volumn 12, Issue 1, 2006, Pages 5-75

  Pak, Igor ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Bijection; Integer partitions; Involution; Involution principle; Plane partitions; Q series; Young diagram

Indexed keywords

EID: 33748704243     PISSN: 13824090     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11139-006-9576-1     Document Type: Article

References (131)

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)
2. Ahlgren, S., Ono, K.: Addition and counting: the arithmetic of partitions. Notices Amer. Math. Soc. 48(9), 978-984 (2001)
3. Aigner, M., Ziegler, G.M.: Proofs from The Book. 2nd edition, Springer-Verlag, Berlin (2001)
4. Alder, H.L.: Partition identities - from Euler to the present. Amer. Math. Monthly 76, 733-746 (1969)
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)
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)
7. Alladi, K., Gordon, B.: Partition identities and a continued fraction of Ramanujan. J. Combin. Theory Ser. A 63(2), 275-300 (1993)
8. Alladi, K., Gordon, B.: Schur's partition theorem, companions, refinements and generalizations. Trans. Amer. Math. Soc. 347(5), 1591-1608 (1995)
9. Andrews, G.E.: A simple proof of Jacobi's triple product identity. Proc. Amer. Math. Soc. 16, 333-334 (1965)
10. Andrews, G.E.: On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2), 17, 132-143 (1966)
11. Andrews, G.E.: On generalizations of Euler's partition theorem. Michigan Math. J. 13, 491-498 (1966)
12. Andrews, G.E.: Enumerative proofs of certain q-identities. Glasgow Math. J. 8, 33-40 (1967)
13. Andrews, G.E.: On a calculus of partition functions. Pacific J. Math. 31, 555-562 (1969)
14. Andrews, G.E.: Two theorems of Gauss and allied identities proved arithmetically. Pacific J. Math. 41, 563-578 (1972)
15. Andrews, G.E.: An extension of Carlitz's bipartition identity. Proc. Amer. Math. Soc. 63(1), 180-184 (1977)
16. Andrews, G.E.: An introduction to Ramanujan's "lost" notebook. Amer. Math. Monthly 86(2), 89-108 (1979)
17. Andrews, G.E.: A note on partitions and triangles with integer sides. Amer. Math. Monthly 86(6), 477-478 (1979)
18. Andrews, G.E.: Partitions and Durfee dissection. Amer. J. Math. 101(3), 735-742 (1979)
19. Andrews, G.E.: Ramanujan's "lost" notebook. I. Partial θ-functions. Adv. in Math. 41(2), 137-172 (1981)
20. Andrews, G.E.: On a partition theorem of N. J. Fine. J. Nat. Acad. Math. India 1(2), 105-107 (1983)
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)
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)
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)
24. Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)
25. Andrews, G.E.: Some debts I owe. Sém. Lothar. Combin., 42:Art. B42a, 16 pp. (electronic) (1999)
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)
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)
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)
29. Andrews, G.E., Garvan, F.G.: Dyson's crank of a partition. Bull. Amer. Math. Soc. (N.S.) 18(2), 167-171 (1988)
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)
31. Askey, R.: The work of George Andrews: a Madison perspective. Sém. Lothar. Combin., 42:Art. B42b, 24 pp. (electronic) (1999)
32. Atkin, A.O.L., Swinnerton-Dyer, P.: Some properties of partitions. Proc. London Math. Soc. 4(3), 84-106 (1954)
33. Bach, E., Shallit, J.: Algorithmic Number Theory. Vol. 1. MIT Press, Cambridge, MA (1996)
34. Bacher, R., Manivel, L.: Hooks and powers of parts in partitions. Sém. Lothar. Combin., 47:Article B47d, 11 pp. (electronic) (2001)
35. Bell, E.T.: The form wx + xy + yz + zu. Bull. Amer. Math. Soc. 42, 377-380 (1936)
36. Bender, E.A., Knuth, D.E.: Enumeration of plane partitions. J. Combinatorial Theory Ser. A13, 40-54 (1972)
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)
38. Bessenrodt, C.: A bijection for Lebesgue's partition identity in the spirit of Sylvester. Discrete Math. 132(1-3), 1-10 (1994)
39. Bessenrodt, C.: On hooks of Young diagrams. Ann. Comb. 2(2), 103-110 (1998)
40. Bessenrodt, C.: On pairs of partitions with steadily decreasing parts. J. Combin. Theory Ser. A 99, 162-174 (2002)
41. Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. Ramanujan J. 1(1), 101-111 (1997)
42. Bousquet-Mélou, Eriksson, K.: A refinement of the lecture hall theorem. J. Combin. Theory Ser. A 86(1), 63-84 (1999)
43. Bressoud, D.M. 7. On a partition theorem of Göllnitz. J. Reine Angew. Math. 305, 215-217.
44. Bressoud, D.M.: A combinatorial proof of Schur's 1926 partition theorem. Proc. Amer. Math. Soc. 79(2), 338-340 (1980)
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)
46. Bressoud, D.M., Zeilberger, D.: A short Rogers-Ramanujan bijection. Discrete Math. 38(2-3), 313315 (1982)
47. Bressoud, D.M., Zeilberger, D.: Bijecting Euler's partitions-recurrence. Amer. Math. Monthly 92(1), 54-55 (1985)
48. Bressoud, D.M., Zeilberger, D.: Generalized Rogers-Ramanujan bijections. Adv. Math. 78(1), 42-75 (1989)
49. Canfleld, R., Corteel, S., Hitczenko, P.: Random partitions with non-negative r-th differences. Adv. in Appl. Math. 27(2-3), 298-317 (2001)
50. Carlitz, L. Some generating functions. Duke Math. J. 30, 191-201 (1963)
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)
52. Carlitz, L., Subbarao, M.V.: A simple proof of the quintuple product identity. Proc. Amer. Math. Soc. 32, 42-44 (1972)
53. Cayley, A.: A letter to Dr. Franklin (an extract). Johns Hopkins Univ. Circular 2(22), 86 (1883)
54. Chapman, R.: Franklin's argument proves an identity of Zagier. Electron. J. Combin. 7(1), Research Paper 54, 5 pp. (electronic) (2000)
55. Cheema, M.S.: Vector partitions and combinatorial identities. Math. Comp. 18, 414-420 (1964)
56. Cohen, D.I.A.: PIE-sums: a combinatorial tool for partition theory. J. Combin. Theory Ser. A 31(3), 223-236 (1981)
57. Corteel, S. Particle seas and basic hypergeometric series. Adv. in Appl. Math. 31(1), 199-214 (2003)
58. 1 summation. J. Combin. Theory Ser. A 97(1), 177-183 (2002)
59. Corteel, S., Lovejoy, J.: Overpartitions. Trans. Amer. Math. Soc. 356(4), 1623-1635 (2004)
60. Dyson, F.J.: Some Guesses in The Theory of Partitions. Eureka (Cambridge) 8, 10-15 (1944)
61. Dyson, F.J.: A new symmetry of partitions. J. Combin. Theory 7, 56-61 (1969)
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)
63. Dyson, F.J.: Mappings and symmetries of partitions. J. Combin. Theory Ser. A 51(2), 169-180 (1989)
64. Edwards, H.M.: Fermat's last theorem. Springer-Verlag, New York (1996)
65. Erdös, P.: On an elementary proof of some asymptotic formulas in the theory of partitions. Ann. Math. 43(2), 437-450 (1942)
66. Euler, L. Introductio in analysin infinitorum. Tomus primus. Marcum-Michaelem Bousquet, Lausannae (1748)
67. Ewell, J.A.: Recurrences for the sum of divisors. Proc. Amer. Math. Soc. 64(2), 214-218 (1977)
68. Farkas, K.M., Kra, I.: Theta constants, Riemann surfaces and the modular group, vol. 37 of Graduate Studies in Mathematics. AMS, Providence, RI (2001)
69. Fine, N.J.: Some new results on partitions. Proc. Nat. Acad. Sci. USA 34, 616-618 (1948)
70. Fine, N.J.: Basic hypergeometric series and applications. AMS, Providence, RI (1988)
71. Foata, D., Han, G.-N.: The triple, quintuple and septuple product identities revisited. Sém. Lothar. Combin., 42: Art. B42o, 12 pp. (electronic) (1999)
72. 3)... C. R. Acad. Paris Ser A 92:448-450 (1881)
73. Franklin, F.: On partitions. Johns Hopkins Univ. Circular 2(22), 72 (1883)
74. Garrett, K., Ismail, M.E.H., Stanton, D.: Variants of the Rogers-Ramanujan identities. Adv. in Appl. Math. 23(3), 274-299 (1999)
75. Garsia, A.M. Combinatorics Lecture Notes. (November 10, 1999), to appear.
76. Garsia, A.M., Milne, S.C.: A Rogers-Ramanujan bijection. J. Combin. Theory Ser. A 31(3), 289-339 (1981)
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)
78. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)
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)
80. Hardy, G.H.: Ramanujan. Twelve lectures suggested by his life and work. Cambridge University Press, Cambridge, England (1940)
81. Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. 17, 75-115 (1918)
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)
83. Hathaway, A.S.: A proof of a Theorem of Jacobi. Johns Hopkins Univ. Circular 2(25), 143-144 (1883)
84. Hickerson, D.R.: A partition identity of the Euler type. Amer. Math. Monthly 81, 627-629 (1974)
85. Hirschhorn, M.D.: Simple proofs of identities of MacMahon and Jacobi. Discrete Math. 16(2), 161-162 (1976)
86. Hirschhorn, M.D.: Polynomial identities which imply identities of Euler and Jacobi. Acta Arith. 32(1), 73-78 (1977)
87. Hirschhorn, M.D.: A simple proof of Jacobi's two-square theorem. Amer. Math. Monthly 92(8), 579-580 (1985)
88. Hirschhorn, M.D.: A simple proof of Jacobi's four-square theorem. Proc. Amer. Math. Soc. 101(3), 436-438 (1987)
89. Hoare, A.H.M.: An involution of blocks in the partitions of n. Amer. Math. Monthly 93(6), 475-476 (1986)
90. 1 sum. Proc. Amer. Math. Soc. 63(1), 185-186 (1977)
91. Joichi, J.T., Stanton, D.: An involution for Jacobi's identity. Discrete Math. 73(3), 261-271 (1989)
92. Kanigel, R.: The Man Who Knew Infinity. Scribner, New York (1991)
93. Kim, D., Yee, A.J.: A note on partitions into distinct parts and odd parts. Ramanujan J. 3(2), 227-231 (1999)
94. Kleitman, D.J.: On the future of combinatorics. In: Essays on the future. Birkhäuser, Boston, MA (2000) pp. 123-134
95. Knuth, D.E., Paterson, M.S.: Identities from partition involutions. Fibonacci Quart. 16(3), 198-212 (1978)
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)
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)
98. Leibenzon, Z.L.: A simple proof of the Macdonald identities for the series A. Funktsional. Anal, i Prilozhen. 25(3), 19-23 (1991)
99. Lewis, R.P.: A combinatorial proof of the triple product identity. Amer. Math. Monthly 91(7), 420-423 (1984)
100. Little, D.P.: An extension of Franklin's bijection. Sém. Lothar. Combin., 42: Art. B42h, 10 pp. (electronic) (1999)
101. Macdonald, I.G.: Affine root systems and Dedekind's η-function. Invent. Math. 15, 91-143 (1972)
102. MacMahon, P.A.: Combinatory analysis. Chelsea Publishing Co., New York (1960)
103. Miller, E., Pak, I. in preparation.
104. O'Hara, K.M.: Bijections for partition identities. J. Combin. Theory Ser. A 49(1), 13-25 (1988)
105. Pak, I. On Fine's partition theorems, Dyson, Andrews, and missed opportunities. Math. Intelligencer, to appear.
106. Pak, I.: Partition identities and geometric bijections. Proc. Amer. Math. Soc. to appear.
107. Pak, I.: Hook length formula and geometric combinatorics. Sém. Lothar. Combin., 46:Art. B46f, 13 pp. (electronic) (2001)
108. Pak, I., Postnikov, A.: A generalization of Sylvester's identity. Discrete Math. 178(1-3), 277-281 (1998)
109. Remmel, J.B.: Bijective proofs of some classical partition identities. J. Combin. Theory Ser. A 33(3), 273-286 (1982)
110. Sagan, B.E.: Bijective proofs of certain vector partition identities. Pacific J. Math. 102(1), 171-178 (1982)
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)
112. Schur, I.: Zur Additiven Zahlentheorie. S.-B. Preuss. Akad. Wiss. Phys. Math. Klasse, pp. 488-495 (1926)
113. Shanks, D.: A short proof of an identity of Euler. Proc. Amer. Math. Soc. 2, 747-749 (1951)
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. Stanley, R.P.: Enumerative combinatorics. Vol. 1, 2. Cambrridge University Press, Cambridge (1997, 1999)
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)
117. Stockhofe, D.: Bijektive Abbildungen auf der Menge der Partitionen einer natürlichen Zahl. Bayreuth. Math. Sehr. (10), 1-59 (1982)
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)
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)
120. Sudler, C.: Jr. Two enumerative proofs of an identity of Jacobi. Proc. Edinburgh Math. Soc. 15(2), 67-71 (1966)
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)
122. Uchimura, K.: An identity for the divisor generating function arising from sorting theory. J. Combin. Theory Ser. A 31(2), 131-135 (1981)
123. Vahlen, K.T.: Beiträge zu einer additiven Zahlentheorie. J. Reine Angew. Math. 112, 1-36 (1893)
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)
125. Wenkov, B.A.: Elementary Number Theory, in Russian, ONTI, Moscow, USSR (1937)
126. Wilf, H.S. Lectures on Integer Partitions, (unpublished), avaiable at: http://www.cis.upenn.edu/~wilf
127. Wilf, H.S.: Identically distributed pairs of partition statistics. Sém. Lothar. Combin., 44:Art B44c, 3 pp. (electronic) (2000)
128. Wright, E.M.: An enumerative proof of an identity of Jacobi. J. London Math. Soc. 40, 55-57 (1965)
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)
130. Zeng, J. The q-variations of Sylvester's bijection between odd and strict partitions. Ramanujan J. 9(3), 289-303 (2005)
131. Zolnowsky, J.: A direct combinatorial proof of the Jacobi identity. Discrete Math. 9, 293-298 (1974)