Umbral calculus, binomial enumeration and chromatic polynomials
Author:
Nigel Ray
Journal:
Trans. Amer. Math. Soc. 309 (1988), 191-213
MSC:
Primary 05A40; Secondary 05C15
DOI:
https://doi.org/10.1090/S0002-9947-1988-0957067-0
MathSciNet review:
957067
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We develop the concept of partition categories, in order to extend the Mullin-Rota theory of binomial enumeration, and simultaneously to provide a natural setting for recent applications of the Roman-Rota umbral calculus to computations in algebraic topology. As a further application, we describe a generalisation of the chromatic polynomial of a graph.
- [1] M. Aigner, Combinatorial theory, Springer-Verlag, 1979. MR 542445 (80h:05002)
- [2] C. Berge, Principles of combinatorics, Academic Press, 1971. MR 0270922 (42:5805)
- [3] J. Blissard, Theory of generic equations, Quart. J. Pure Appl. Math. 4 (1861), 279-305.
- [4] L. Comtet, Advanced combinatorics, Reidel, 1974. MR 0460128 (57:124)
- [5] M. Content, F. Lemay and P. Leroux, Catégories de Möbius et fonctorialités; un cadre général pour l'inversion de Möbius, J. Combin. Theory A Ser. 28 (1980), 169-190. MR 563554 (82f:18002)
- [6] P. Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. 36 (1932), 39-95.
- [7] M. Henle, Binomial enumeration on dissects, Trans. Amer. Math. Soc. 202 (1975), 1-39. MR 0357133 (50:9601)
- [8] S. A. Joni & G.-C. Rota, Coalgebras and algebras in combinatorics, Stud. Appl. Math. 61 (1979), 93-139. MR 544721 (81c:05002)
- [9] A. Joyal, Une théorie combinatoire des séries formelles, Adv. in Math. 42 (1981), 1-82. MR 633783 (84d:05025)
- [10] G. Labelle, Une nouvelle démonstration combinatoire des formules d'inversion de Lagrange, Adv. in Math. 42 (1981), 217-247. MR 642392 (83e:05016)
- [11] E. K. Lloyd, Enumeration, Handbook of Applicable Mathematics 5, Wiley, 1985, pp. 531-621.
- [12] S. Mac Lane, Categories for the working mathematician, Springer-Verlag, 1971. MR 1712872 (2001j:18001)
- [13] R. Mullin and G.-C. Rota, On the foundations of combinatorial theory III: Theory of binomial enumeration, Graph Theory and its Applications, Academic Press, 1971, pp. 168-213. MR 0274300 (43:65)
- [14] O. Nava & G.-C. Rota, Plethysm, categories and combinatorics, Adv. in Math. 58 (1985), 61-88. MR 812935 (87e:05010)
- [15] N. Ray, Extensions of umbral calculus: penumbral coalgebras and generalised Bernoulli numbers, Adv. in Math. 61 (1986), 49-100. MR 847728 (88b:05019)
- [16]
-, Symbolic calculus: a 19th century approach to
and
, Proc. 1985 Durham Sympos. on Homotopy Theory, London Math. Soc. Lecture Note Ser. 117, Cambridge Univ. Press, 1987, pp. 195-238.
- [17] N. Ray and C. Wright, Umbral calculus and a new chromatic polynomial, Ars Combinatoria (to appear).
- [18] J. Riordan, An introduction to combinatorial analysis, Wiley, 1958. MR 0096594 (20:3077)
- [19] S. Roman, The umbral calculus, Academic Press, 1984. MR 741185 (87c:05015)
- [20] G.-C. Rota, On the foundations of combinatorial theory. I: Theory of Möbius functions, Z. Wahrsch. Verw. Gebiete 2 (1964), 340-368. MR 0174487 (30:4688)
- [21] P. J. Webb, A local method in group cohomology, Comment. Math. Helv. 62 (1987), 135-167. MR 882969 (88h:20065)
- [22] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press, 1927. MR 1424469 (97k:01072)
Retrieve articles in Transactions of the American Mathematical Society with MSC: 05A40, 05C15
Retrieve articles in all journals with MSC: 05A40, 05C15
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1988-0957067-0
Article copyright:
© Copyright 1988
American Mathematical Society