Involve: A Journal of Mathematics

  • Involve
  • Volume 7, Number 6 (2014), 787-805.

Seating rearrangements on arbitrary graphs

Daryl DeFord

Full-text: Open access

Abstract

We exhibit a combinatorial model based on seating rearrangements, motivated by some problems proposed in the 1990s by Kennedy, Cooper, and Honsberger. We provide a simpler interpretation of their results on rectangular grids, and then generalize the model to arbitrary graphs. This generalization allows us to pose a variety of well-motivated counting problems on other frequently studied families of graphs.

Article information

Source
Involve, Volume 7, Number 6 (2014), 787-805.

Dates
Received: 4 November 2013
Revised: 3 January 2014
Accepted: 24 January 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733750

Digital Object Identifier
doi:10.2140/involve.2014.7.787

Mathematical Reviews number (MathSciNet)
MR3284885

Zentralblatt MATH identifier
1306.05098

Subjects
Primary: 05C30: Enumeration in graph theory

Keywords
matrix permanents cycle covers tilings recurrence relations

Citation

DeFord, Daryl. Seating rearrangements on arbitrary graphs. Involve 7 (2014), no. 6, 787--805. doi:10.2140/involve.2014.7.787. https://projecteuclid.org/euclid.involve/1513733750


Export citation

References

  • S. Aaronson, “A linear-optical proof that the permanent is $\#\mathrm{\bf P}$-hard”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467:2136 (2011), 3393–3405.
  • L. W. Beineke and F. Harary, “Binary matrices with equal determinant and permanent”, Studia Sci. Math. Hungar 1 (1966), 179–183.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: The art of combinatorial proof, The Dolciani Mathematical Expositions 27, Mathematical Association of America, Washington, DC, 2003.
  • G. Chartrand, L. Lesniak, and P. Zhang, Graphs & digraphs, 5th ed., CRC Press, Boca Raton, 2011.
  • D. DeTemple and W. Webb, Combinatorial reasoning: An introduction to the art of counting, Wiley, Hoboken, NJ, 2014.
  • R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics: A foundation for computer science, 2nd ed., Addison-Wesley, Reading, MA, 1994.
  • F. Harary, Graph theory and theoretical physics, Academic Press, New York, 1967.
  • F. Harary, “Determinants, permanents and bipartite graphs”, Math. Mag. 42:3 (1969), 146–148.
  • R. Honsberger, In Pólya's footsteps: Miscellaneous problems and essays, The Dolciani Mathematical Expositions 19, Mathematical Association of America, Washington, DC, 1997.
  • P. Kasteleyn, “The statistics of dimers on a lattice, I: The number of dimer arrangements on a quadratic lattice”, Physica 27:12 (1961), 1209–1225.
  • R. Kennedy and C. Cooper, “Variations on a $5 \times 5$ seating rearrangement problem”, Mathematics in College 1993 (1993), 59–67.
  • G. Kuperberg, “An exploration of the permanent-determinant method”, Electron. J. Combin. 5 (1998), R46, 1–34.
  • J. H. van Lint and R. M. Wilson, A course in combinatorics, 2nd ed., Cambridge University Press, 2001.
  • C. H. C. Little, “A characterization of convertible (0,1)-matrices”, J. Combinatorial Theory Ser. B 18:3 (1975), 187–208.
  • N. A. Loehr, Bijective combinatorics, CRC Press, Boca Raton, 2011.
  • P. Lundow, “Computation of matching polynomials and the number of 1-factors in polygraphs”, research Reports 12, Umeå University, 1996, http://www.theophys.kth.se/~phl/Text/1factors.pdf.
  • M. Marcus and H. Minc, “Permanents”, Amer. Math. Monthly 72 (1965), 577–591.
  • OEIS, “The On–Line Encyclopedia of Integer Sequences”, 2012, http://oeis.org.
  • T. Otake, R. Kennedy, and C. Cooper, “On a seating rearrangement problem”, Mathematics and Informatics Quarterly 52 (1996), 63–71.
  • G. Pólya, “Aufgabe 424”, Arch. Math. Phys. 20:3 (1913), 271.
  • N. Robertson, P. D. Seymour, and R. Thomas, “Permanents, Pfaffian orientations, and even directed circuits”, Ann. of Math. $(2)$ 150:3 (1999), 929–975.
  • G. E. Shilov, Linear algebra, Dover, New York, 1977.
  • H. N. V. Temperley and M. E. Fisher, “Dimer problem in statistical mechanics–-an exact result”, Philos. Mag. $(8)$ 6:68 (1961), 1061–1063.
  • L. G. Valiant, “The complexity of computing the permanent”, Theoret. Comput. Sci. 8:2 (1979), 189–201.
  • V. V. Vazirani and M. Yannakakis, “Pfaffian orientations, $0/1$ permanents, and even cycles in directed graphs”, pp. 667–681 in Automata, languages and programming (Tampere, 1988), edited by T. Lepist ö and A. Salomaa, Lecture Notes in Comput. Sci. 317, Springer, Berlin, 1988.