Annals of Combinatorics

, Volume 17, Issue 3, pp 509–523 | Cite as

Advanced Computer Algebra for Determinants

  • Christoph Koutschan
  • Thotsaporn “Aek” Thanatipanonda


We prove three conjectures concerning the evaluation of determinants, which are related to the counting of plane partitions and rhombus tilings. One of them was posed by George Andrews in 1980, the other two were by Guoce Xin and Christian Krattenthaler. Our proofs employ computer algebra methods, namely, the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These variations make Zeilberger’s original approach even more powerful and allow for addressing a wider variety of determinants. Finally, we present, as a challenge problem, a conjecture about a closed-form evaluation of Andrews’s determinant.

Mathematics Subject Classification

33F10 15A15 05B45 


determinant computer algebra holonomic ansatz rhombus tiling 


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  1. 1.
    Amdeberhan T., Zeilberger D: Determinants through the looking glass. Adv. Appl. Math. 27(2-3), 225–230 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: Macdonald’s conjecture and descending plane partitions. In: Narayana, T.V., Mathsen, R.M., Williams, J.G. (eds.) Combinatorics, Representation Theory and Statistical Methods in Groups, pp. 91–106. Dekker, New York (1980)Google Scholar
  3. 3.
    Chyzak, F.: Fonctions holonomes en calcul formel. PhD thesis, École polytechnique (1998)Google Scholar
  4. 4.
    Ciucu M. et al.: Enumeration of lozenge tilings of hexagons with a central triangular hole. J. Combin. Theory Ser. A 95, 251–334 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Gessel, I.M., Xin, G.: The generating functions of ternary trees and continued fractions. Electron. J. Combin. 13(1), #R53 (2006)Google Scholar
  6. 6.
    Ishikawa, M., Koutschan, C.: Zeilberger’s holonomic ansatz for Pfaffians. In: van Hoeij, M., van der Hoeven, J. (eds.) ISSAC 2012: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, pp. 227–233. ACM (2012)Google Scholar
  7. 7.
    Kauers, M.: Guessing handbook. Technical Report 09-07, RISC Report Series, Johannes Kepler University Linz (2009).
  8. 8.
    Koutschan, C.: Advanced applications of the holonomic systems approach. PhD thesis, RISC, Johannes Kepler University, Linz, Austria (2009)Google Scholar
  9. 9.
    Koutschan, C.: HolonomicFunctions (User’s Guide). Technical Report 10-01, RISC Report Series, Johannes Kepler University Linz (2010). Available at:
  10. 10.
    Koutschan, C., Kauers, M., Zeilberger, D.: Proof of George Andrews’s and David Robbins’s q-TSPP conjecture. Proc. Natl. Acad. Sci. USA 108(6), 2196–2199 (2011)Google Scholar
  11. 11.
    Koutschan, C., Thanatipanonda, T.: Electronic supplementary material to the article “Advanced computer algebra for determinants” (2011).
  12. 12.
    Krattenthaler, C.: Advanced determinant calculus. Sém. Lothar. Combin. 42, Art. B42q (1999)Google Scholar
  13. 13.
    Krattenthaler C: Advanced determinant calculus: a complement. Linear Algebra Appl. 411, 68–166 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Stanley, R.: A baker’s dozen of conjectures concerning plane partitions. In: Labelle, G., Leroux, P. (eds.) Combinatoire Énumérative, pp. 285–293. Springer, Berlin (1986)Google Scholar
  15. 15.
    Zeilberger D: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32(3), 321–368 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Zeilberger D: Reverend Charles to the aid of Major Percy and Fields-medalist Enrico. Amer. Math. Monthly 103(6), 501–502 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Zeilberger, D.: The holonomic ansatz II. Automatic discovery(!) and proof(!!) of holonomic determinant evaluations. Ann. Combin. 11(2), 241–247 (2007)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Christoph Koutschan
    • 1
  • Thotsaporn “Aek” Thanatipanonda
    • 1
  1. 1.Research Institute for Symbolic Computation (RISC)Johannes Kepler UniversityLinzAustria

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