Abstract
The papers reprinted in this chapter are concerned with enumeration within a geometric lattice. They are closely related to papers on the Tutte decomposition reprinted in the next chapter.
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References
Baclawski, K.: The Möbius algebra as a Grothendieck ring, J. Algebra 57(1979), 167–179.
Edelman, P.H.: Zeta polynomials and the Möbius function, Europ. J. Combin. 1(1980), 335–340.
Greene, C.: On the Möbius algebra of a partially ordered set, Advances in Math. 10(1973), 177–187.
Rota, G.-C. and Smith, D.: Enumeration under group action, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(1977), 637–646.
Solomon, L.: The Burnside algebra of a finite group, J. Combin. Theory 2(1967), 603–615.
Baclawski, K.: Whitney numbers of geometric lattices, Advances in Math. 16(1975), 125–138.
Baclawski, K.: Galois connections and the Leray spectral sequence, Advances in Math. 25(1977), 191–215.
Baclawski, K.: Cohen-Macaulay ordered sets, J. Algebra 63(1980), 226–258.
Baclawski, K.: Cohen-Macaulay connectivity and geometric lattices, Europ. J. Combin. 3(1982), 293–305.
Baclawski, K.: Nonpositive Cohen-Macaulay connectivity, preprint.
Baclawski, K. and Björner, A.: Fixed points in partially ordered sets, Advances in Math. 31(1979), 263–287.
Baclawski, K. and Björner, A.: Fixed points and complements in finite lattices, J. Combin. Theory Ser. A 30(1981), 335–338.
Björner, A.: Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260(1980), 159–183.
Björner, A.: Homotopy type of posets and lattice complementation, J. Combin. Theory Ser. A 30(1981), 90–100.
Björner, A.: On the homology of geometric lattices, Algebra Universalis 14(1982), 107–128.
Björner, A., Garsia, A.M. and Stanley, R.P.: An introduction to Cohen-Macaulay partially ordered sets, Ordered Sets (I. Rival, ed.), pp. 583–615, Reidel, Dordrecht and Boston, 1982.
Björner, A. and Walker, J.W.: A homotopy complementation formula for partially ordered sets, Europ. J. Combin. 4(1983), 11–19.
Brini, A.: Some homological properties of partially ordered sets, Advances in Math. 43(1982), 197–201.
Farmer, F.D.: Homology of reflexive relations, Math. Japon. 20(1975), 303–310.
Farmer, F.D.: Homology of products and joins of reflexive relations, Discrete Math. 11(1975), 23–27.
Farmer, F.D.: Characteristic for reflexive relations, Aequationes Math. 15(1977), 195–199.
Farmer, F.D.: Cellular homology for posets, Math. Japon. 23(1978/79), 607–613.
Griffiths, H.B.: The homology groups of some ordered systems, Acta Math. 129(1972), 195–235.
Griffiths, H.B.: An exact homology sequence induced by a Galois connection, Proc. London Math. Soc. (3) 32(1976), 101–116.
Lakser, H.: The homology of a lattice, Discrete Math. 1(1971), 187–192.
Mather, J.: Invariance of the homology of a lattice, Proc. Amer. Math. Soc. 17(1966), 1120–1124.
Orlik, P. and Solomon, L.: Combinatorics and topology of complements of hyperplanes, Invent. Math. 56(1980), 167–189.
Quillen, D.: Homotopy properties of the poset of non-trivial p-subgroups of a group, Advances in Math. 28(1978), 101–128.
Rota, G.-C: On the combinatorics of the Euler characteristic, Studies in Pure Mathematics (Presented to Richard Rado), pp. 221–233, Academic Press, London, 1971.
Stanley, R.P.: Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc. 249(1979), 139–157.
Walker, J.W.: Homotopy type and Euler characteristic, Europ. J. Combin. 2(1981), 373–384.
Brylawski, T.: Modular constructions for combinatorial geometries, Trans. Amer. Math. Soc. 203(1975), 1–44.
Brylawski, T. and Oxley, J.G.: Several identities for the characteristic polynomial of a combinatorial geometry, Discrete Math. 31(1980), 161–170.
Brylawski, T. and Oxley, J.G.: The broken-circuit complex: its structure and factorization, Europ. J. Combin. 2(1981), 107–121.
Dowling, T.A.: A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14(1973), 61–86.
Greene, C.: On the Möbius algebra of a partially ordered set, Advances in Math. 10(1973), 177–187.
Stanley, R.P.: Supersolvable lattices, Algebra Universalis 2(1972), 197–217.
Stanley, R.P.: Finite lattices and Jordan-Hölder sets, Algebra Universalis 4(1974), 361–371.
Basterfield, J.G. and Kelly, L.M.: A characterization of sets of n points which determine n hyperplanes, Proc. Cambridge Philos. Soc. 64(1968), 585–588.
Björner, A.: Some matroid inequalities, Discrete Math. 31(1980), 101–103.
Björner, A.: The unimodality conjecture for convex polytopes, Bull. Amer. Math. Soc. (New Series) 4(1981), 187–188.
Brylawski, T.: Connected matroids with the smallest Whitney numbers, Discrete Math. 18(1977), 243–252.
Dowling, T.A.: Complementing permutations in finite lattices, J. Combin. Theory Ser. B 23(1977), 223–226.
Dowling, T.A.: On the independent set numbers of a finite matroid, Combinatorics 79, Part I, Ann. Discrete Math. 8(1980), 21–28.
Dowling, T.A.: On the orbit numbers of a finite geometric lattice, preprint, 1981.
Dowling, T.A. and Wilson, R.M.: The slimmest geometric lattices, Trans. Amer. Math. Soc. 196(1974), 203–215.
Greene, C.: A rank inequality for finite geometric lattices, J. Combin. Theory 9(1970), 357–364.
Greene, C.: A inequality for the Möbius function of a geometric lattice, Stud. Appl. Math. 54(1975), 71–74.
Harper, L.H.: Stirling behavior is asymptotically normal, Ann. Math. Statist. 38(1967), 410–414.
Heron, A.P.: A property of the hyperplanes of a matroid and an extension of Dilworth’s theorem, J. Math. Anal. Appl. 42(1973), 119–131.
Kung, J.P.S.: The Radon transforms of a combinatorial geometry, I, J. Combin. Theory Ser. A 26(1979), 97–102.
Kurtz, D.C.: A note on concavity properties of triangular arrays of numbers, J. Combin. Theory Ser. A 13(1972), 135–139.
Lieb, E.H.: Concavity properties and a generating function for Stirling numbers, J. Combin. Theory 5(1968), 203–206.
Mahoney, C.R.: On unimodality of the independent set numbers of a class of matroids, J. Combin. Theory Ser. B 39(1985), 77–85.
Mason, J.H.: Matroids: unimodal conjectures and Motzkin’s theorem, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), pp. 207–220, Inst. Math. Appl., Southend-on-Sea, 1972.
Motzkin, T.S.: The lines and planes connecting the points of a finite set, Trans. Amer. Math. Soc. 70(1951), 451–464.
Seymour, P.D.: On the points-lines-planes conjecture, J. Combin. Theory Ser. B 33(1982), 17–26.
Stanley, R.P.: Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combin. Theory Ser. A 31(1981), 56–65.
Stonesifer, J.R.: Logarithmic concavity for edge lattices of graphs, J. Combin. Theory Ser. A 18(1975), 36–46.
Stonesifer, J.R.: Logarithmic concavity for a class of geometric lattices, J. Combin. Theory Ser. A 18(1975), 216–218.
Woodall, D.R.: The inequality b ⩾ v, Proc. Fifth British Combinatorial Conf. (Univ. Aberdeen, Aberdeen, 1975), pp. 661–664, Congressus Numerantium, No. 15, Utilitas Math., Winnipeg, Man,. 1976.
Zaslavsky, T.: The slimmest arrangements of hyperplanes I: Geometric lattices and projective arrangements, Geometriae Dedicata 14(1983), 243–259.
Zaslavsky, T.: The slimmest arrangements of hyperplanes. II: Basepointed geometric lattices and Euclidean arrangements, Mathematika 28(1981), 169–190.
Baker, K.A.: A generalization of Sperner’s lemma, J. Combin. Theory 6(1969), 224–225.
Canfield, E.R.: On a problem of Rota, Advances in Math. 29(1978), 1–10.
Greene, C.: A rank inequality for finite geometric lattices, J. Combin. Theory 9(1970), 357–364.
Greene, C. and Kleitman, D.J.: The structure of Sperner k-families, J. Combin. Theory Ser. A 20(1976), 41–68.
Greene, C. and Kleitman, D.J.: Proof techniques in the theory of finite sets, Studies in Combinatorics (G.-C. Rota, ed.) pp. 22–79, Math. Assoc. Amer., Washington, D.C., 1978.
Harper, L.H.: The morphology of partially ordered sets, J. Combin. Theory Ser. A 17(1974), 44–58.
Harper, L.H. and Rota, G.-C.: Matching theory, an introduction, Advances in Probability, Vol. 1(P. Ney, ed.), pp. 169–215, Marcel Dekker, New York, 1971.
Kahn, J.: Some non-Sperner paving matroids, Bull. London Math. Soc. 12(1980), 268.
Peck, G.W.: On Canfield type antichains of partitions, preprint.
Shearer, J.B.: A simple counterexample to a conjecture of Rota, Discrete Math. 28(1979), 327–330.
Aslander, L., and H. M. Trent: Incidence matrices and linear graphs. J. Math. Mech. 8, 827–835 (1959).
Bell, E. T.: Algebraic Arithmetic. New York: Amer. Math. Soc. (1927)
Bell, E. T.: Exponential polynomials. Ann. of Math., 11. Ser. 35. 258–277 (1934).
Berge, C.: Théorie des graphes et ses applications. Paris: Dounod 1958.
Birkhoff, Garrett: Lattice Theory, third preliminary edition. Harvard University. 1963.
Birkhoff, Garrett: Lattice Theory, revised edition. American Mathematical Society, 1948.
Birkhoff, G. D.: A determinant formula for the number of ways of coloring a map. Ann. of Math., II. Ser. 14, 42–46 (1913).
Birkhoff, G. D., and D. C. Lewis: Chromatic polynomials. Trans. Amer. math. Soc. 60, 355–451 (1946).
Bleicher, M. N., and G. B. Preston: Abstract linear dependence relations. Publ. Math., Debrecen 8, 55–63 (1961).
Bougayev, N. V.: Theory of numerical derivatives. Moscow, 1870–1873, pp. 1–222.
Bruijn, N. G. de: Generalization of Polya’s fundamental theorem in enumerative combinatorial analysis. Indagationes math. 21, 59–69 (1959).
Chung, K.-L., and L. T. C. Hsu: A combinatorial formula with its application to the theory of probability of arbitrary events. Ann. math. Statistics 16, 91–95 (1945).
Dedekind, R.: Gesammelte Mathematische Werke, volls. I-II-III. Hamburg: Deutsche Math. Verein. (1930).
Delsarte, S.: Fonctions de Möbius sur les groupes abéliens finis. Ann. of Math., II. Ser. 49, 600–609 (1948).
Dilworth, R. P.: Proof of a conjecture on finite modular lattices. Ann. of Math., II. Ser. 60, 359–304 (1954).
Dirac, G. A.: On the four-color conjecture. Proc. London math. Society, III. Ser. 13, 193 to 218 (1963).
Dowker, C. H.: Homology groups of relations. Ann. of Math., II. Ser. 56, 84–95 (1952).
Dubreil-Jacotin, M.-L., L. Lesieur et R. Croisot: Lecons sur la théorie des treilles des structures algebriques ordonnées et des treilles géometriques. Paris: Gauthier-Villars 1953.
Eilenberg, S., and N. Steenrod: Foundations of algebraic topology. Princeton: University Press 1952.
Fary, I.: On straight-line representation of planar graphs. Acta Sci. math. Szeged 11. 229–233 (1948).
Feller, W.: An introduction to probability theory and its applications, second edition. New York: Wiley 1960.
Franklin, P.: The four-color problem. Amer. J. Math. 44, 225–236 (1922).
Fréchet, M.: Les probabilités associées à un système d’évenements compatibles et dépendants. Actualitées scientifiques et industrielles, nos. 859 et 942. Paris: Hermann 1940 et 1943.
Frontera Marqués, B.: Una función numérica en los retículos finitos que se anula para los retículos reducibles. Actas de la 2a, Reunión de matemáticos españoles. Zaragoza 103–111 1962.
Frucht, R., and G.-C. Rota: La función de Möbius para el retículo di particiones de un conjunto finito. To appear in Scientia (Chile).
Goldberg, K., M.S. Green and R. E. Nettleton: Dense subgraphs and connectivity. Canadian J. Math. 11 (1959).
Golomb, S. W.: A mathematical theory of discrete classification. Fourth Symposium in Information Theory, London, 1961.
Green, M.S., and R. E. Nettleton: Möbius function on the lattice of dense subgraphs. J. Res. nat. Bur. Standards 64B, 41–47 (1962).
Green, M.S., and R. E. Nettleton: Expression in terms of modular distribution functions for the entropy density in an infinite system. J. Chemical Physisc 29, 1365–1370 (1958).
Hadwiger, H.: Eulers Charakteristik und kombinatorische Geometrie. J. reine angew. Math. 194, 101–110 (1955).
Hall, Philip: A contribution to the theory of groups of prime power order. Proc. London math. Soc., II. Ser. 36, 39–95 (1932).
Hall, Philip: The Eulerian functions of a group. Quart. J. Math. Oxford Ser. 134–151, 1936.
Harary, V.: Unsolved problems in the enumeration of graphs. Publ. math. Inst. Hungar. Acad. Sci. 5, 63–95 (1960).
Hardy, G. H.: Ramanujan. Cambridge: University Press 1940.
Hardy, G. H., and E. M. Wright: An introduction to the theorv of numbers. Oxford: University Press 1954.
Hartmanis, J.: Lattice theory of generalized partitions. Canadian J. Math. 11, 97–106 (1959).
Hille, E.: The inversion problems of Möbius. Duke math. J. 3, 549–568 (1937).
Hsu, L. T. C.: Abstract theory of inversion of iterated summation. Duke math. J. 14, 465 to 473 (1947).
Hsu, L. T. C.: On Romanov’s device of orthogonalization. Sci. Rep. Nat. Tsing Hua Univ. 5, 1–12 (1948).
Hsu, L. T. C.: Note on an abstract inversion principle. Proc. Edinburgh math. Soc (2) 9, 71–73 (1954).
Jackson, F. H.: Series connected with the enumeration of partitions. Proc. London math. Soc., II. Ser. 1, 63–88 (1904).
Jackson, F. H.: The q-form of Taylor’s theorem. Messenger of Mathematics 38, 57–61 (1909).
Jónsson, B.: Lattice-theoretic approach to projective and affine geometry. Symposium on the Axiomatic Method. Amsterdam, North-Holland Publishing Company, 1959, 188–205.
Jónsson, B., and A. Tarski: Direct decomposition of finite algebraic systems. Notre Dame Mathematical lectures, no. 5. Indiana: Notre Dame 1947.
Kac, M., and J. C. Ward: A combinatorial solution of the two-dimensional Ising model. Phys. Review 88, 1332–1337 (1952).
Kaplanski, I., and J. Riordan: The problème des ménages. Scripta math. 12, 113–124 (1946).
Klee, V.: The Euler characteristic in combinatorial geometry. Amer. math. Monthly 70, 119–127 (1963).
Lazarson, T.: The representation problem for independence functions. J. London math. Soc. 33, 21–25 (1958).
Maclane, S.: A lattice formulation of transcendence degrees and p-bases. Duke math. J. 4, 455–468 (1938).
MacMillan, B.: Absolutely monotone functions. Ann. of Math., II. Ser. 60, 467–501 (1954).
Möbius, A. F.: Über eine besondere Art von Umkehrung der Reihen. J. reine angew. Math. 9, 105–123 (1832).
Ore, O.: Theory of graphs. Providence: American Mathematical Society 1962.
Polya, G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Vcrbindungen. Acta math. 68, 145–253 (1937).
Rado, R.: Note on independence functions. Proc. London math. Soc., III. Ser. 7, 300–320 (1957).
Read, R. C.: The enumeration of locally restricted graphs, I. J. London math. Soc. 34, 417 to 436 (1959).
Redfield, J. H.: The theory of group-reduced distributions. Amer. J. Math. 49, 433–455 (1927).
Revuz, André: Fonctions croissantes et mesures sur les espaces topologiques ordonnés. Ann. Inst. Fourier 6 187–268 (1955).
Riordan, J.: An introduction to combinatorial analysis. New York: Wiley 1958.
Romanov, N. P.: On a special orthonormal system and its connection with the theory of primes. Math. Sbornik, N. S. 16, 353–364 (1945).
Rota, G.-C.: Combinatorial theory and Möbius functions. To appear in Amer. math. Monthly.
Rota, G.-C.: The number of partitions of a set. To appear in Amer. math. Monthly.
Ryser, H. J.: Combinatorial Mathematics. Buffalo: Mathematical Association of America 1963.
Schützenberger, M. P.: Contribution aux applications statistiques de la théorie de l’information. Publ. Inst. Stat. Univ. Paris, 3, 5–117 (1954).
Tarski, A.: Ordinal algebras. Amsterdam: North-Holland Publishing Company 1956.
Touchard, J.: Sur un problème de permutations. C. r. Acad. Sci., Paris, 198, 631–633 (1934).
Tutte, W. T.: A contribution to the theory of chromatic polynomials. Canadian J. Math. 6, 80–91 (1953).
Tutte, W. T.: A class of Abelian group. Canadian J. Math. 8, 13–28 (1950).
Tutte, W. T.: A homotopy theorem for matroids. I. and II. Trans. Amer. math. Soc. 88, 144–140 (1958).
Tutte, W. T.: Matroids and graphs. Trans. Amer. math. Soc. 90, 527–552 (1959).
Ward, M.: The algebra of lattice functions. Duke math. J. 5, 357–371 (1939).
Weisser, L.: Abstract theory of inversion of finite series. Trans. Amer. math. Soc. 38. 474–484 (1935).
Weisser, L.: Some properties of prime-power groups. Trans. Amer. math. Soc. 38, 485–492 (1935).
Whitney, H.: A logical expansion in mathematics. Bull. Amer. math. Soc. 38, 572–579 (1932).
Whitney, H.: Characteristic functions and the algebra of logic. Ann. of Math., II. Ser. 34, 405–414 (1933).
Whitney, H.: The abstract properties of linear dependence. Amer. J. Math. 57, 507–533 (1935).
Wielandt, H.: Beziehungen zwischen den Fixpunktzahlen von Automorphismengruppen eincr endlichen Gruppe. Math. Z. 73, 140–158 (1960).
Wintner, A.: Eratosthenian Averages. Baltimore (privately printed) 1943.
Eilenberg, S., & N. E. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, New Jersey, 1952.
Rota, Gian-Carlo, On the Foundations of Combinatorial Theory, I. Theory of Möbius Functions, Zeitschrift für Wahrsheinlichkeitstheorie und Verwandte Gebiete, 2(1964) 340–368.
Garrett Birkhoff, Lattice Theory, Third Edition (American Mathematical Society, 1967).
Henry Crapo, The Möbius Function of a Lattice, J. Combinatorial Theory 1 (1966), 126–131.
Henry Crapo, and Gian-Carlo Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, M.I.T. Press.
Gian-Carlo Rota, On the Foundations of Combinatorial Theory, I: Theory of Möbius Functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340–368.
E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1928), 544–548.
L. H. Harper, The Morphology of Geometric Lattices (to appear).
K. A. Baker, A Generalization of Sperner’s Lemma (to appear).
J. E. McLaughlin, Structure Theorems for Relatively Complemented Lattices, Pacific J. Math. 3 (1953), 197–208.
J. G. Basterfield and L. M. Kelly, A characterization of sets of n points which determine n hyperplanes, Proc. Cambridge Philos. Soc. 64(1968), 585–588. MR 38 #2040.
J. E. Blackburn, H. H. Crapo and D. A. Higgs, A catalogue of combinatorial geometries, University of Waterloo, Waterloo, Ontario, 1969.
G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR 37 #2638.
H. H. Crapo and G.-C. Rota, On the foundations of combinatorial theory: Combinatorial geometries, M. I. T. Press, Cambridge, Mass., 1970. (preliminary edition). MR 45 #74.
R. P. Dilworth, Proof of a conjecture on finite modular lattices, Ann. of Math. (2) 60 (1954), 359–364. MR 16, 106.
T. A. Dowling and R. M. Wilson, The slimmest geometric lattices, Trans. Amer. Math. Soc. 196(1974), 203–215.
C. Greene, A rank inequality for finite geometric lattices, J. Combinatorial Theory 9 (1970), 357–364. MR 42 #1727.
C. Greene, Inequalities for geometric lattices, Proc. Conf. on Möbius Algebras (H. Crapo and G. Roulet, editors), University of Waterloo, Waterloo, Ont., 1971.
L. H. Harper, Stirling behaviour is asymptotically normal, Ann. Math. Statist. 38 (1967), 410–414. MR 35 #2312.
D. G. Kelly, Disjoining permutations in finite boolean algebras, Utilitas Mathematica 3 (1973), 65–74.
E. Lieb, Concavity properties and a generating function for Stirling numbers, J. Combinatorial Theory 5 (1968), 203–206. MR 37 #6195.
G.-C. Rota, On the foundations of combinatorial theory. I: Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368. MR 30 #4688.
P. Young, U. S. R. Murty and J. Edmonds, Equicardinal matroids and matroid designs, Proc. Second Chapel Hill Conference on Combinatorial Mathematics and its Applications, University of North Carolina, Chapel Hill, N. C., 1970.
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Rota, GC. et al. (1986). Enumeration in geometric lattices. In: A Source Book in Matroid Theory. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-9199-9_3
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