The Ramanujan Journal

, Volume 46, Issue 3, pp 765–774 | Cite as

Binomial transforms and integer partitions into parts of k different magnitudes

  • Mircea Merca


A relationship between the general linear group of degree n over a finite field and the integer partitions of n into parts of k different magnitudes was investigated recently by the author. In this paper, we use a variation of the classical binomial transform to derive a new connection between partitions into parts of k different magnitudes and another finite classical group, namely the symplectic group Sp. New identities involving the number of partitions of n into parts of k different magnitudes are introduced in this context.


Binomial transform Integer partitions Symplectic group 

Mathematics Subject Classification

05E15 05A19 05A17 



The author appreciates the anonymous referees for their comments on the original version of this paper. Special thanks go to Dr. Oana Merca for the careful reading of the manuscript and helpful remarks.


  1. 1.
    Andrews, G.E.: Stacked lattice boxes. Ann. Comb. 3, 115–130 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andrews, G.E., Merca, M.: The truncated pentagonal number theorem. J. Comb. Theory Ser. A 119, 1639–1643 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Fulman, J.: A probabilistic approach toward conjugacy classes in the finite general linear and unitary groups. J. Algebra 212, 557–590 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fulman, J.: A probabilistic approach to conjugacy classes in the finite symplectic and orthogonal groups. J. Algebra 234, 207–224 (2000)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fulman, J.: Random matrix theory over finite fields. Bull. Amer. Math. Soc. 39, 51–85 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Addison-Wesley, Reading, MA (1989)MATHGoogle Scholar
  7. 7.
    Guo, V.J.W., Zeng, J.: Two truncated identities of Gauss. J. Comb. Theory Ser. A 120, 700–707 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    He, T.Y., Ji, K.Q., Zang, W.J.T.: Bilateral truncated Jacobi’s identity. Eur. J. Comb. 51, 255–267 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Knuth, D.E.: The Art of Computer Programming, vol. 3: Sorting and Searching. Addison-Wesley, Reading, MA (1973)MATHGoogle Scholar
  10. 10.
    Macdonald, I.G.: Numbers of conjugacy classes in some finite classical groups. Bull. Austral. Math. Soc 23, 23–48 (1981)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    MacMahon, P.A.: Divisors of numbers and their continuations in the theory of partitions. Proc. Lond. Math. Soc. s2–19(1), 75–113 (1921)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Malle, G.: On the distribution of class groups of number fields. Exp. Math. 19, 465–474 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mao, R.: Proofs of two conjectures on truncated series. J. Comb. Theory Ser. A 130, 15–25 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Merca, M.: A note on the partitions involving parts of \(k\) different magnitudes. J. Number Theory 162, 23–34 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Merca, M.: Lambert series and conjugacy classes in GL. Discrete Math. (2017). doi: 10.1016/j.disc.2017.04.020
  16. 16.
    Riordan, J.: Combinatorial Identities. Wiley, New York (1968)MATHGoogle Scholar
  17. 17.
    Rudvalis, A., Shinoda, K.: An enumeration in finite classical groups. U-Mass Amherst Department of Mathematics Technical, Report (1988)Google Scholar
  18. 18.
    Yee, A.J.: The truncated Jacobi triple product theorem. J. Comb. Theory Ser. A 130, 1–14 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Academy of Romanian ScientistsBucharestRomania

Personalised recommendations