Automatic counting of tilings of skinny plane regions

doi:10.1017/CBO9781139506748.010

9 - Automatic counting of tilings of skinny plane regions

Published online by Cambridge University Press: 05 July 2013

Shalosh B. Ekhad and

Edited by

Stefanie Gerke and

Simon R. Blackburn: Affiliation:
Royal Holloway, University of London
Stefanie Gerke: Affiliation:
Royal Holloway, University of London
Mark Wildon: Affiliation:
Royal Holloway, University of London

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Summary

Abstract

The deductive method ruled mathematics for the last 2500 years, now it is the turn of the inductive method. We make a start by using the C-finite ansatz to enumerate tilings of skinny place regions, inspired by a Mathematics Magazine Problem proposed by Donald Knuth.

Very important

This article is accompanied by the Maple packages

http://www.math.rutgers.edu/~zeilberg/tokhniot/RITSUF,
http://www.math.rutgers.edu/~zeilberg/tokhniot/RITSUFwt,
http://www.math.rutgers.edu/~zeilberg/tokhniot/ARGF,

to be described below. In fact, more accurately, this article accompanies these packages, written by DZ and the many output files, discovering and proving deep enumeration theorems, done by SBE, that are linked to from the webpage of this article: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ritsuf.html.

How it all started: April 5, 2012

During one of the Rutgers University Experimental Mathematics Seminar dinners, the name of Don Knuth came up, and two of the participants, David Nacin, who was on sabbatical from William Patterson University, and first-year graduate student Patrick Devlin, mentioned that they recently solved a problem that Knuth proposed in Mathematics Magazine [5]. The problem was:

1868. Proposed by Donald E. Knuth, Stanford University, Stanford, California.

Let n ≥ 2 be an integer. Remove the central (n − 2)2 squares from an (n + 2) × (n + 2) array of squares. In how many ways can the remaining squares be covered with 4n dominoes?

Type: Chapter
Information: Surveys in Combinatorics 2013 , pp. 363 - 378

DOI: https://doi.org/10.1017/CBO9781139506748.010 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2013

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References

[1] Mihai, Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), 67–97.Google Scholar

[2] M., Fisher and H., Temperley, Dimer Problems in Statistical Mechanics-an exact result, Philos. Mag. 6 (1961), 1061–1063.Google Scholar

[3] P. W., Kasteleyn, The statistics of dimers on a lattice: I. The number of dimer arrangements in a quadratic lattice, Physica 27 (1961), 1209–1225.Google Scholar

[4] Manuel, Kauers and Peter, Paule, The Concrete Tetrahedron, Springer, 2011.

[5] Donald E., Knuth, Tiling a square ring with dominoes, proposed problem #1868, Math. Magazine 84 (2) (April 2011), and Solution by John Bonomo and David Offner, Math. Magazine 85(2) (April 2012), 154–155.Google Scholar

[6] Richard, Stanley, Enumerative Combinatorics, volume 1, Wadsworth and Brooks/Cole, Pacific Grove, CA, 1986, second printing, Cambridge University Press, Cambridge, 1996.

[7] Roberto, Tauraso, A New Domino Tiling Sequence, Journal of Integer Sequences, 7 (2004), Article 04.2.3. Available on-line: http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Tauraso/tauraso3.pdf.Google Scholar

[8] Doron, Zeilberger, The C-finite Ansatz, a “sanitized” version is to appear in the Ramanujan Journal, the original, uncensored version is available on-line: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/cfinite.html.

[9] Doron, Zeilberger, An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding, in C. S., Calude (editor), Randomness & Complexity, from Leibniz to Chaitin, World Scientific, Singapore, 2007. Available on-line: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/enquiry.html.

[10] Doron, Zeilberger, The Automatic Central Limit Theorems Generator (and Much More!), in Advances in Combinatorial Mathematics, Proceedings of the Waterloo Workshop in Computer Algebra 2008 in honor of Georgy P. Egorychev, chapter 8, pp. 165–174, (I., Kotsireas, E., Zima, editors), Springer Verlag, 2009. Available on-line: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/georgy.html.

[11] Doron, Zeilberger, HISTABRUT: A Maple Package for SymbolCrunching in Probability theory, Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, Aug. 25, 2010, http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/histabrut.html.

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9 - Automatic counting of tilings of skinny plane regions

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