Abstract
It is known that, in general, the coboundary polynomial and the Möbius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will investigate if it is possible that the Möbius polynomial of a matroid, together with the Möbius polynomial of the dual matroid, define the coboundary polynomial of the matroid. In some cases, the answer is affirmative, and we will give two constructions to determine the coboundary polynomial in these cases.
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References
Aigner M.: Combinatorial Theory. Springer, New York (1979)
Brylawski, T.H.: The Tutte polynomial. I. General theory. In C.I.M.E. Summer Schools (1980)
Crapo H.: The Tutte polynomial. Aequat. Math. 3, 211–229 (1969)
de Boer M.A.: Almost MDS codes. Des. Codes Cryptogr. 9, 143–155 (1996)
Duursma I.M.: Weight distributions of geometric Goppa codes. Trans. Am. Math. Soc. 351, 3609–3639 (1999)
Duursma I.M.: From weight enumerators to zeta functions. Discrete Appl. Math. 111, 55–73 (2001)
Duursma, I.M.: Combinatorics of the two-variable zeta function. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds.) International Conference on Finite Fields and Applications. In: Lecture Notes in Computer Science, vol. 2948, pp. 109–136. Springer, Berlin (2003)
Duursma I.M.: Extremal weight enumerators and ultraspherical polynomials. Discrete Math. 268, 103–127 (2003)
Faldum A., Willems W.: Codes of small defect. Des. Codes Cryptogr. 10(3), 341–350 (1997)
Greene C.: Weight enumeration and the geometry of linear codes. Stud. Appl. Math. 55, 119–128 (1976)
Jurrius R.P.M.J., Pellikaan R.: Codes, arrangements and matroids. In: Series on Coding Theory and Cryptology. World Scientific Publishing, Singapore (2011)
Kløve T.: The weight distribution of linear codes over GF(q l) having generator matrix over GF(q). Discrete Math. 23, 159–168 (1978)
MacWilliams F.J., Sloane N.J.A.: The theory of error-correcting codes. North-Holland Mathematical Library, Amsterdam (1977)
Oxley J.G.: Matroid Theory, 2nd edn. Oxford University Press, Oxford (2011)
Stanley, R.P.: An introduction to hyperplane arrangements. In: Geometric Combinatorics, IAS/Park City Math. Ser., vol. 13, pp. 389–496. American Mathematical Society, Providence, (2007)
Welsh D.J.A.: Matroid Theory. Academic Press, London (1976)
White N.: Theory of matroids. In: Encyclopedia of Mathmatics and its Applications, vol. 26. Cambridge University Press, Cambridge (1986)
Zaslavsky, T.: Facing up to arrangements: face-count fomulas for partitions of space by hyperplanes. Memoirs of American Mathematical Society, vol. 1, No. 154. American Mathematical Society, Providence (1975)
Zaslavsky T.: Signed graph colouring. Discrete Math. 39, 215–228 (1982)
Acknowledgments
The author would like to thank Ruud Pellikaan for stating the Main Question, and for valuable comments on this paper. The author is also indebted to an anonymous referee for the various comments on the paper and for the material in Sect. 6.
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Jurrius, R. Relations Between Möbius and Coboundary Polynomials. Math.Comput.Sci. 6, 109–120 (2012). https://doi.org/10.1007/s11786-012-0117-6
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DOI: https://doi.org/10.1007/s11786-012-0117-6