Abstract
For primes l we obtain a simple formula for p(N) (mod l) as a weighted sum over l-square affine partitions of N. When l ϥ {3,5,7,11}, the weights are explicit divisor functions. The Ramanujan congruences modulo 5, 7, 11, 25, 49, and 121 follow immediately from these formulae.
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References
K. Alladi, “The method of weighted words and applications to partitions,” Number Theory, Paris, 1992–1993, London Math. Soc. Lecture Notes, vol. 215, 1995, pp. 1–36.
P. Fong and B. Srinivasan, “The blocks of finite general linear groups and unitary groups,” Invent. Math. 69 (1982), 109–153.
F. Garvan, D. Kim, and D. Stanton, “Cranks and t-cores,” Invent. Math. 101 (1990), 1–17.
A. Granville and K. Ono, “Defect zero p-blocks for finite simple groups,” Trans. Amer. Math. Soc. 348(1) (1996), 331–347.
M. Hirschhorn, “On the residue mod 2 and mod 4 of p(n),” Acta Arith. 38 (1980), 105–109.
M. Hirschhorn, “On the parity of p(n) II,” J. Combin. Theory (A) 62 (1993), 128–138.
M. Hirschhorn and M. Subbarao, “On the parity of p(n),” Acta Arith. 50(4) (1988), 355–356.
G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, 1979.
A. Klyachko, “Modular forms and representations of symmetric groups, integral lattices and finite linear groups,” Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 116 (1982).
O. Kolberg, “Note on the parity of the partition function,” Math. Scand. 7 (1959), 377–378.
H.H. Nakamura, “On some generating functions for McKay numbers—Prime power divisibilities of the hook products of Young diagrams,” J. Math. Sci., U. Tokyo 1 (1994), 321–337.
J. Olsson, “Remarks on symbols, hooks and degrees of unipotent characters,” J. Comb. Th. (A) 42 (1986), 223–238.
J. Olsson, “Combinatorics and representations of finite groups,” University Essen Lecture Notes, vol. 20, 1993.
K. Ono, “On the positivity of the number of partitions that are t-cores,” Acta Arith. 66(3) (1994), 221–228.
K. Ono, “Parity of the partition function in arithmetic progressions,” J. Reine Angew. Math. 472 (1996), 1–15.
K. Ono, “Odd values of the partition function,” Disc. Math. 169 (1997), 263–268.
K. Ono and L. Sze, “4-core partitions and class numbers“ Acta Arith. 65 (1997), 249–272.
T.R. Parkin and D. Shanks, “On the distribution of parity in the partition function,” Math. Comp. 21 (1967), 466–480.
J. Sturm, “On the congruence of modular forms,” Springer Lecture Notes, vol. 1240, Springer-Verlag, 1984.
M. Subbarao, “Some remarks on the partition function,” Amer. Math. Monthly 73 (1966), 851–854.
H.P.F. Swinnerton-Dyer, “On l-adic Galois representations and congruences for coefficients of modular forms,” Springer Lecture Notes, vol. 350, 1973.
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Dedicated to the memory of Paul Erdős
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Ono, K. (1998). The Residue of p(N) Modulo Small Primes. In: Alladi, K., Elliott, P.D.T.A., Granville, A., Tenebaum, G. (eds) Analytic and Elementary Number Theory. Developments in Mathematics, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4507-8_5
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DOI: https://doi.org/10.1007/978-1-4757-4507-8_5
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