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TWO APPROACHES TO MÖBIUS INVERSION

Part of: Commutative algebra: Homological methods Combinatorics

Published online by Cambridge University Press:  15 August 2011

I-CHIAU HUANG
I-CHIAU HUANG*
Affiliation:
Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, ROC (email: ichuang@math.sinica.edu.tw)
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Abstract

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The Möbius inversion formula for a locally finite partially ordered set is realized as a Lagrange inversion formula. Schauder bases are introduced to interpret Möbius inversion.

Keywords

inverse relationLagrange inversionMöbius inversionSchauder basis

MSC classification

Secondary: 05A19: Combinatorial identities, bijective combinatorics 13D45: Local cohomology
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 1 , February 2012 , pp. 68 - 78
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Bailey, W. N., Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, 32 (Stechert-Hafner, New York, 1964).Google Scholar
[2]Haiman, M. and Schmitt, W., ‘Incidence algebra antipodes and Lagrange inversion in one and several variables’, J. Combin. Theory Ser. A 50(2) (1989), 172–185.CrossRefGoogle Scholar
[3]Huang, I-C., ‘Pseudofunctors on modules with zero dimensional support’, Mem. Amer. Math. Soc. 114(548) (1995), xii+53.Google Scholar
[4]Huang, I-C., ‘Applications of residues to combinatorial identities’, Proc. Amer. Math. Soc. 125(4) (1997), 1011–1017.CrossRefGoogle Scholar
[5]Huang, I-C., ‘Reversion of power series by residues’, Comm. Algebra 26(3) (1998), 803–812.CrossRefGoogle Scholar
[6]Huang, I-C., ‘Residue methods in combinatorial analysis’, in: Local Cohomology and its Applications, Lecture Notes in Pure and Applied Mathematics, 226 (Marcel Dekker, New York, NY, 2001), pp. 255–342.Google Scholar
[7]Huang, I-C., ‘Inverse relations and Schauder bases’, J. Combin. Theory Ser. A 97(2) (2002), 203–224.CrossRefGoogle Scholar
[8]Huang, I-C., ‘Method of generating differentials’, in: Advances in Combinatorial Mathematics: Proceedings of the Waterloo Workshop in Computer Algebra 2008 (Springer, Berlin, 2009), pp. 125–152.Google Scholar
[9]Huang, I-C., ‘Changes of parameters for generalized power series’, Comm. Algebra 38 (2010), 2480–2498.CrossRefGoogle Scholar
[10]Jacobi, C. G. I., ‘De resolutione aequationum per series infinitas’, J. reine angew. Math. 6 (1830), 257–286.Google Scholar
[11]Lindström, B., ‘Determinants on semilattices’, Proc. Amer. Math. Soc. 20 (1969), 207–208.CrossRefGoogle Scholar
[12]MacMahon, P. A., Combinatory Analysis (Chelsea, New York, 1960), two volumes (bound as one).Google Scholar
[13]Matsumura, H., Commutative Ring Theory (Cambridge University Press, Cambridge, 1986).Google Scholar
[14]Redheffer, R., ‘Eine explizit lösbare Optimierungsaufgabe’, in: Numerische Methoden bei Optimierungsaufgaben, Band 3 (Tagung, Math. Forschungsinst., Oberwolfach, 1976), International Series of Numerical Mathematics, 36 (Birkhäuser, Basel, 1977), pp. 213–216.CrossRefGoogle Scholar
[15]Rota, G.-C., ‘On the foundations of combinatorial theory. I. Theory of Möbius functions’, Z. Wahrscheinlichkeitstheor. Verw. Geb. 2 (1964), 340–368.CrossRefGoogle Scholar
[16]Spiegel, E. and O’Donnell, C. J., Incidence Algebras, Monographs and Textbooks in Pure and Applied Mathematics, 206 (Marcel Dekker, New York, NY, 1997).Google Scholar
[17]Stanley, R. P., ‘Structure of incidence algebras and their automorphism groups’, Bull. Amer. Math. Soc. 76 (1970), 1236–1239.CrossRefGoogle Scholar
[18]Wilf, H. S., ‘Hadamard determinants, Möbius functions, and the chromatic number of a graph’, Bull. Amer. Math. Soc. 74 (1968), 960–964.CrossRefGoogle Scholar
[19]Wilf, H. S., ‘The Redheffer matrix of a partially ordered set’, Electron. J. Combin. 11(2) (2004/06), Research Paper 10, 5 pp (electronic).CrossRefGoogle Scholar