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Number Theory and Discrete Mathematics pp 13–30Cite as

Birkhäuser

A (Conjectural) 1/3-phenomenon for the Number of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus

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Abstract

We state, discuss, provide evidence for, and prove in special cases the conjecture that the prob-ability that a random tiling by rhombi of a hexagon with side lengths 2n +a,2n + b, 2n +c,2n + a, 2n + b, 2n + c contains the (horizontal) rhombus with coordinates (2n + x, 2n + y) is equal to \(\tfrac{1}{3} + {{g}_{{a,b,c,x,y}}}\left( n \right){{\left( {_{n}^{{2n}}} \right)}^{3}}/\left( {_{{3n}}^{{6n}}} \right)\),where ga b,c,x,y (n) isa rational function in n. Several specific instances of this “1/3-phenomenon” are made explicit.

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Research partially supported by the Austrian Science Foundation FWF, grant P12094-MAT and P13190-MAT.

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Author information

Authors and Affiliations

  1. Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090, Wien, Austria

    C. Krattenthalert

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  1. C. Krattenthalert
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Editor information

Editors and Affiliations

  1. Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, 160014, India

    A. K. Agarwal

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© 2002 Springer Basel AG

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Krattenthalert, C. (2002). A (Conjectural) 1/3-phenomenon for the Number of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus. In: Agarwal, A.K., Berndt, B.C., Krattenthaler, C.F., Mullen, G.L., Ramachandra, K., Waldschmidt, M. (eds) Number Theory and Discrete Mathematics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8223-1_2

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  • DOI: https://doi.org/10.1007/978-3-0348-8223-1_2

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9481-4

  • Online ISBN: 978-3-0348-8223-1

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