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Tutte-Whitney polynomials: some history and generalizations. (English) Zbl 1124.05020
Grimmett, Geoffrey (ed.) et al., Combinatorics, complexity, and chance. A tribute to Dominic Welsh. Oxford: Oxford University Press (ISBN 0-19-857127-5/hbk). Oxford Lecture Series in Mathematics and its Applications 34, 28-52 (2007).
This article provides a fairly brisk introduction to the Tutte-Whitney polynomial, i.e., the two-variable generating function for the coranks and nullities of subsets of a matroid. (The specific form often referred to as the “Tutte polynomial,” defined using activities, does not make an appearance.) The author begins with a survey of the polynomial’s properties, with special attention to those discovered by Welsh, in whose honor the volume containing this article was published; there are too many to list here, but the fundamentally important [F. Jaeger, D. L. Vertigan and D. J. A. Welsh, Math. Proc. Camb. Philos. Soc. 108, 35–53 (1990; Zbl 0747.57006)] should certainly be mentioned. The history of the polynomial’s introduction into the literature is told next, including W. T. Tutte’s assessment [Adv. Appl. Math. 32, No. 1–2, 5–9 (2004; Zbl 1041.05001)] that H. Whitney [Ann. Math. (2) 33, 688–718 (1932; Zbl 0005.31301)] “used analogous coefficients without bothering to affix them to two variables.” A useful survey of many recent generalizations of the Tutte-Whitney polynomial follows; again, there are too many to list here. The paper closes with a detailed summary of the author’s own work generalizing Tutte-Whitney polynomials to invariants of binary functions [G. Farr, Adv. Appl. Math. 32, No. 1–2, 239–262 (2004; Zbl 1041.05016)].
MSC:
05B35Matroids, geometric lattices (combinatorics)