The Möbius Function of a Partially Ordered Set

The history of the Mobius function has many threads, involving aspects of number theory, algebra, geometry, topology, and combinatorics. The subject received considerable focus from Rota’s by now classic paper in which the Mobius function of a partially ordered set emerged in clear view as an important object of study. On the one hand, it can be viewed as an enumerative tool, defined implicitly by the relations $$f(x) = \sum\limits_{yx} {g(y)} {\text{ and }}g(x) = \sum\limits_{yx} {\mu (y,x)f(y)} $$ where f and g are arbitrary functions on a poset P. On the other hand, one can study μ for its own sake as a combinatorial invariant giving important and useful information about the structure of P.

[1]  C. Greene.  Weight Enumeration and the Geometry of Linear Codes , 1976 .

[2]  H. Crapo,et al.  The Möbius function of a lattice , 1966 .

[3]  R. P. Dilworth.  Proof of a Conjecture on Finite Modular Lattices , 1954 .

[4]  David A. Smith.  Incidence functions as generalized arithmetic functions, II. , 1969 .

[5]  Thomas A. Dowling,et al.  Complementing permutations in finite lattices , 1977, J. Comb. Theory, Ser. B.

[6]  R. Stanley.  Ordered Structures And Partitions , 1972 .

[7]  Anders Björner,et al.  A Homotopy Complementation Formula for Partially Ordered Sets , 1983, Eur. J. Comb..

[8]  H. Whitney.  A logical expansion in mathematics , 1932 .

[9]  T. Zaslavsky.  Facing Up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes , 1975 .

[10]  R. Stanley,et al.  On the foundations of combinatorial theory. VI. The idea of generating function , 1972 .

[11]  G. Rota,et al.  Finite operator calculus , 1975 .

[12]  P. Orlik,et al.  Combinatorics and topology of complements of hyperplanes , 1980 .

[13]  Thomas A. Dowling,et al.  A class of geometric lattices based on finite groups , 1973 .

[14]  Kenneth Baclawski,et al.  Fixed points in partially ordered sets , 1979 .

[15]  Richard P. Stanley,et al.  Some Aspects of Groups Acting on Finite Posets , 1982, J. Comb. Theory, Ser. A.

[16]  Gian-Carlo Rota,et al.  Enumeration under group action , 1977 .

[17]  Curtis Greenl,et al.  On the Möbius algebra of a partially ordered set , 1973 .

[18]  P. Orlik,et al.  Unitary reflection groups and cohomology , 1980 .

[19]  Kenneth Baclawski,et al.  Whitney Numbers of Geometric Lattices , 1975 .

[20]  Richard P. Stanley,et al.  Modular elements of geometric lattices , 1971 .

[21]  Anders Björner,et al.  Homotopy Type of Posets and Lattice Complementation , 1981, J. Comb. Theory, Ser. A.

[22]  Harry Lakser,et al.  The homology of a lattice , 1971, Discret. Math..

[23]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[24]  Thomas A. Dowling,et al.  Whitney Number Inequalities for Geometric Lattices , 1975 .

[25]  Gian-Carlo Rota,et al.  Baxter algebras and combinatorial identities. II , 1969 .

[26]  E. Bender,et al.  On the Applications of Möbius Inversion in Combinatorial Analysis , 1975 .

[27]  A. Björner.  Shellable and Cohen-Macaulay partially ordered sets , 1980 .

[28]  Thomas Zaslavsky,et al.  Signed graph coloring , 1982, Discret. Math..

[29]  Richard P. Stanley,et al.  An Introduction to Cohen-Macaulay Partially Ordered Sets , 1982 .

[30]  Thomas Zaslavsky,et al.  The Geometry of Root Systems and Signed Graphs , 1981 .

[31]  Richard P. Stanley,et al.  Finite lattices and Jordan-Hölder sets , 1974 .

[32]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[33]  Kenneth Baclawski,et al.  Cohen-Macaulay Connectivity and Geometric Lattices , 1982, Eur. J. Comb..

[34]  R. Stanley.  Structure of incidence algebras and their automorphism groups , 1970 .

[35]  Ladnor Geissinger,et al.  Valuations on distributive lattices I , 1973 .

[36]  T. Zaslavsky.  A combinatorial analysis of topological dissections , 1977 .

[37]  R. Stanley,et al.  Combinatorial reciprocity theorems , 1974 .

[38]  B. Lindström.  Determinants on semilattices , 1969 .

[39]  P. Erdos,et al.  Studies in Pure Mathematics , 1983 .

[40]  T. A. Dowling,et al.  The slimmest geometric lattices , 1974 .

[41]  Richard P. Stanley,et al.  Binomial Posets, Möbius Inversion, and Permutation Enumeration , 1976, J. Comb. Theory A.

[42]  Victor Klee,et al.  The Euler Characteristic in Combinatorial Geometry , 1963 .

[43]  J. L. Freeman.  Sixth Berkeley Symposium on Mathematical Statistics and Probability , 1973 .

[44]  Louis Solomon,et al.  The Burnside algebra of a finite group , 1967 .

[45]  James W. Walker,et al.  Homotopy Type and Euler Characteristic of Partially Ordered Sets , 1981, Eur. J. Comb..

[46]  P. Hall,et al.  THE EULERIAN FUNCTIONS OF A GROUP , 1936 .

[47]  Kenneth Baclawski,et al.  Fixed Points and Complements in Finite Lattices , 1981, J. Comb. Theory, Ser. A.

[48]  Thomas Zaslavsky,et al.  ON THE INTERPRETATION OF WHITNEY NUMBERS THROUGH ARRANGEMENTS OF HYPERPLANES, ZONOTOPES, NON-RADON PARTITIONS, AND ORIENTATIONS OF GRAPHS , 1983 .

[49]  L. Weisner,et al.  Abstract theory of inversion of finite series , 1935 .

[50]  Henry H. Crapo.  Möbius Inversion in Lattices , 1969 .

[51]  T. Zaslavsky.  The slimmest arrangements of hyperplanes: II. Basepointed geometric lattices and Euclidean arrangements , 1981 .

[52]  Curtis Greene,et al.  A rank inequality for finite geometric lattices , 1970 .

[53]  Pierre Leroux,et al.  Catégories de Möbius et fonctorialités: un cadre général pour I'inversioin de Möbius , 1980, J. Comb. Theory, Ser. A.

[54]  K. Baclawski.  The Möbius algebra as a Grothendieck ring , 1979 .

[55]  Kenneth Baclawski,et al.  Cohen-Macaulay ordered sets , 1980 .

[56]  Tom Brylawski,et al.  A decomposition for combinatorial geometries , 1972 .

[57]  P. Doubilet,et al.  On the Foundations of Combinatorial Theory. VII: Symmetric Functions through the Theory of Distribution and Occupancy , 1972 .

[58]  Anders Björner,et al.  On the homology of geometric lattices , 1982 .

[59]  D. Quillen,et al.  Homotopy properties of the poset of nontrivial p-subgroups of a group , 1978 .

[60]  J. Mather,et al.  INVARIANCE OF THE HOMOLOGY OF A LATTICE , 1966 .

[61]  Curtis Greene,et al.  An Inequality for the Mobius Function of a Geometric Lattice , 1975 .

[62]  Richard P. Stanley.  Acyclic orientations of graphs , 1973, Discret. Math..

[63]  Jon Folkman,et al.  THE HOMOLOGY GROUPS OF A LATTICE , 1964 .

[64]  Kenneth Baclawski,et al.  Galois connections and the leray spectral sequence , 1977 .

[65]  P. Hall,et al.  A Contribution to the Theory of Groups of Prime‐Power Order , 1934 .

[66]  Pierre Cartier,et al.  Problemes combinatoires de commutation et rearrangements , 1969 .

[67]  H. Wilf.  Hadamard determinants Möbius functions, and the chromatic number of a graph , 1968 .