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  • Print publication year: 2011
  • Online publication date: June 2012

3 - Partially Ordered Sets

Bibliography
[1] C. A., Athanasiadis, Algebraic combinatorics of graph spectra, subspace arrangements, and Tutte polynomials, Ph.D. thesis, M.I.T., 1996.
[2] C. A., Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields, Advances in Math. 122 (1996), 193–233.
[3] K., Baclawski, Cohen-Macaulay ordered sets, J. Algebra 63 (1980), 226–258.
[4] K., Baclawski, Cohen-Macaulay connectivity and geometric lattices, Europ. J. Combinatorics 3 (1984), 293–305.
[5] M. M., Bayer and L. J., Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), 143–157.
[6] M. M., Bayer and A., Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), 33–47.
[7] E. A., Bender and J. R., Goldman, Enumerative uses of generating functions, Indiana Univ. Math. J. 20 (1971), 753–765.
[8] F., Bergeron, G., Labelle, and P., Leroux, Combinatorial Species and Tree-like Structures, Encyclopedia of Mathematics and Its Applications 67, Cambridge University Press, Cambridge, 1998.
[9] G., Benkart and T., Roby, Down-up algebras, J. Algebra 209 (1998), 305–344.
[10] L. J., Billera and R., Ehrenborg, Monotonicity properties of the cd-index for polytopes, Math. Z. 233 (2000), 421–441.
[11] G., Birkhoff, On the combination of subalgebras, Proc. Camb. Phil. Soc. 29 (1933), 441–464.
[12] G., Birkhoff, Von Neumann and latttice theory, Bull. Amer. Math. 64 (1958), 50–56.
[13] G., Birkhoff, Lattice Theory, 3rd ed., American Mathematical Society, Providence, R.I., 1967.
[14] A., Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159–183.
[15] A., Björner, Posets, regular CW complexes and Bruhat order, Europ. J. Combinatorics 5 (1984), 7–16.
[16] A., Björner, A., Garsia, and R., Stanley, An introduction to the theory of Cohen-Macaulay partially ordered sets, in Ordered Sets (I., Rival, ed.), Reidel, Dordrecht/Boston/-London, 1982, pp. 583–615.
[17] A., Björner and M. L., Wachs, Bruhat order of Coxeter groups and shellability, Advances in Math. 43 (1982), 87–100.
[18] A., Björner, Topological methods, in Handbook of Combinatorics (R., Graham, M., Grötschel, and L., Lovász, eds.), North-Holland, Amsterdam, 1995, pp. 1819–1872.
[19] A., Björner and M. L., Wachs, On lexicographically shellable posets, Trans. Amer. Math. Soc. 277 (1983), 323–341.
[20] A., Björner and J. W., Walker, A homotopy complementation formula for partially ordered sets, Europ. J. Combinatorics 4 (1983), 11–19.
[21] A., Cayley, On the theory of the analytical forms called trees, Phil. Mag. 18 (1859), 374–378.
[22] H., Crapo and G.-C., Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, preliminary edition, M.I.T. Press, Cambridge, Mass., 1970.
[23] R. J., Daverman, Decompositions of Manifolds, American Mathematical Society, Providence, R.I. 2007.
[24] B. A., Davey and H. A., Priestly, Distributive lattices and duality, Appendix B of [3.33], pp. 499–517.
[25] P., Doubilet, G.-C., Rota, and R., Stanley, On the foundations of combinatorial theory (VI): The idea of generating function, in Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory, University of California, Berkeley and Los Angeles, 1972, pp. 267–318.
[26] P. H., Edelman, Zeta polynomials and the Möbius function, Europ. J. Combinatorics 1 (1980), 335–340.
[27] F. D., Farmer, Cellular homology for posets, Math. Japonica 23 (1979), 607–613.
[28] S. V., Fomin, Two-dimensional growth in Dedekind lattices, M.S. thesis, Leningrad State University, 1979.
[29] S., Fomin, Duality of graded graphs, J. Algebraic Combinatorics 3 (1994), 357–404.
[30] S., Fomin, Schensted algorithms for dual graded graphs, J. Algebraic Combinatorics 4 (1995), 5–45.
[31] I. M., Gessel, Generating functions and enumeration of sequences, thesis, M.I.T., 1977.
[32] I. P., Goulden and D. M., Jackson, Combinatorial Enumeration, John Wiley, New York, 1983; reissued by Dover, New York, 2004.
[33] G., Grätzer, General Lattice Theory: Vol. 1: The Foundation, 2nd ed., Birkhäuser, Basel/Boston/Berlin, 2003.
[34] C., Greene, On the Möbius algebra of a partially ordered set, Advances in Math. 10 (1973), 177–187.
[35] C., Greene, The Möbius function of a partially ordered set, in [3.57], pp. 555–581.
[36] O. A., Gross, Preferential arrangements, Amer. Math. Monthly 69 (1962), 4–8.
[37] B., Grünbaum, Convex Polytopes, 2nd ed., Springer-Verlag, New York, 2003.
[38] M. D., Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math. 99 (1992), 79–113.
[39] P., Hall, The Eulerian functions of a group, Quart. J. Math. 7 (1936), 134–151.
[40] I., Halperin, A survey of John von Neumann's books on continuous geometry, Order 1 (1985), 301–305.
[41] P., Headley, On a family of hyperplane arrangements related to affine Weyl groups, J. Algebraic Combinatorics 6 (1997), 331–338.
[42] M., Henle, Dissection of generating functions, Studies in Applied Math. 51 (1972), 397–410.
[43] M. E., Hoffman, Updown categories, preprint; math.CO/0402450.
[44] A., Joyal, Une théorie combinatoire des séries formelles, Advances in Math. 42 (1981), 1–82.
[45] V., Klee, A combinatorial analogue of Poincaré's duality theorem, Canad. J. Math. 16 (1964), 517–531.
[46] D. E., Knuth, A note on solid partitions, Math. Comp. 24 (1970), 955–962.
[47] D. N., Kozlov, General lexicographic shellability and orbit arrangements, Ann. Comb. 1 (1997), 67–90.
[48] D. N., Kozlov, Combinatorial Algebraic Topology, Springer, Berlin, 2008.
[49] T., Lam, Signed differential posets and sign-imbalance, J. Combinatorial Theory Ser. A 115 (2008), 466–484.
[50] T., Lam, Quantized dual graded graphs, Electronic J. Combinatorics 17(1) (2010), R88.
[51] C., Malvenuto and C., Reutenauer, Evacuation of labelled graphs, Discrete Math. 132 (1994), 137–143.
[52] E., Miller and B., Sturmfels, Combinatorial Commutative Algebra, Springer, NewYork, 2005.
[53] P., Orlik and H., Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin/Heidelberg, 1992.
[54] J. G., Oxley, Matroid Theory, Oxford University Press, New York, 1992.
[55] H. A., Priestley, Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186–190.
[56] H. A., Priestley, Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. (3) 24 (1972), 507–530.
[57] I., Rival (ed.), Ordered Sets, Reidel, Dordrecht/Boston, 1982.
[58] G.-C., Rota, On the foundations of combinatorial theory I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340–368.
[59] L., Schläfli, Theorie der vielfachen Kontinuität, in Neue Denkschrifter den allgemeinen schweizerischen Gesellschaft für die gesamten Naturwissenschaften, vol. 38, IV, Zürich, 1901; Ges. Math. Abh., vol. 1, Birkhaäuser, Basel, 1950, p. 209.
[60] J.-Y., Shi, The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups, Lecture Notes in Mathematics, no. 1179, Springer-Verlag, Berlin/Heidelberg/New York, 1986.
[61] J.-Y., Shi, Sign types corresponding to an affine Weyl group, J. London Math. Soc. 35 (1987), 56–74.
[62] M.-P., Schützenberger, Quelques remarques sur une construction de Schensted, Canad. J. Math. 13 (1961), 117–128.
[63] M.-P., Schützenberger, Promotions des morphismes d'ensembles ordonnés, Discrete Math. 2 (1972), 73–94.
[64] M.-P., Schützenberger, Evacuations, in Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo I, Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Rome, 1976, pp. 257–264.
[65] L., Solomon, The Burnside algebra of a finite group, J. Combinatorial Theory 2 (1967), 603–615.
[66] E., Spiegel and C., O'Donnell, Incidence Algebras, Marcel Dekker, New York, 1997.
[67] R., Stanley, Ordered structures and partitions, thesis, Harvard Univ., 1971.
[68] R., Stanley, Ordered structures and partitions, Memoirs Amer. Math. Soc., no. 119 (1972).
[69] R., Stanley, Supersolvable lattices, Alg. Univ. 2 (1972), 197–217.
[70] R., Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171–178.
[71] R., Stanley, Finite lattices and Jordan-Hölder sets, Alg. Univ. 4 (1974), 361–371.
[72] R., Stanley, Combinatorial reciprocity theorems, Advances in Math. 14 (1974), 194–253.
[73] R., Stanley, The Fibonacci lattice, Fib. Quart. 13 (1975), 215–232.
[74] R., Stanley, Binomial posets, Möbius inversion, and permutation enumeration, J. Combinatorial Theory Ser. A 20 (1976), 336–356.
[75] R., Stanley, Cohen-Macaulay complexes, in Higher Combinatorics (M., Aigner, ed.), Reidel, Dordrecht/Boston, 1977, pp. 51–62.
[76] R., Stanley, Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc. 249 (1979), 139–157.
[77] R., Stanley, Some aspects of groups acting on finite posets, J. Combinatorial Theory Ser. A 32 (1982), 132–161.
[78] R., Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, vol. 41, Birkhäuser, Boston/Basel/Stuttgart, 1983; 2nd ed., Boston/Basel/Berlin, 1996.
[79] R., Stanley, Generalized h-vectors, intersection cohomology of toric varieties, and related results, in Commutative Algebra and Combinatorics (M., Nagata and H., Matsumura, eds.), Advanced Studies in Pure Mathematics 11, Kinokuniya, Tokyo, and North-Holland, Amsterdam/New York, 1987, pp. 187–213.
[80] R., Stanley, Differential posets, J. Amer. Math. Soc. 1 (1988), 919–961.
[81] R., Stanley, Variations on differential posets, in Invariant Theory and Tableaux (D., Stanton, ed.), The IMA Volumes in Mathematics and Its Applications, vol. 19, Springer-Verlag, New York, 1990, pp. 145–165.
[82] R., Stanley, A survey of Eulerian posets, in Polytopes: Abstract, Convex, and Computational (T., Bisztriczky, P., McMullen, R., Schneider, A. I., Weiss, eds.), NATO ASI Series C, vol. 440, Kluwer Academic Publishers, Dordrecht/Boston/London, 1994, pp. 301–333.
[83] R., Stanley, An introduction to hyperplane arrangements, in Geometric Combinatorics (E., Miller, V., Reiner, and B., Sturmfels, eds.), IAS/Park City Mathematics Series, vol. 13, American Mathematical Society, Providence, R.I., 2007, pp. 389–496.
[84] R., Stanley, Promotion and evacuation, Electronic J. Combinatorics 15(2) (20082009), R9.
[85] B. S., Stečkin, Imbedding theorems for Möbius functions, Soviet Math. Dokl. 24 (1981), 232–235 (translated from 260 (1981)).
[86] M. H., Stone, Applications of the theory of boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375–481.
[87] J., Walker, Homotopy type and Euler characteristic of partially ordered sets, European J. Combinatorics 2 (1981), 373–384.
[88] J., Walker, Topology and combinatorics of ordered sets, Ph.D. thesis, M.I.T., 1981.
[89] D. J. A., Welsh, Matroid Theory, Academic Press, London, 1976; reprinted by Dover, New York, 2010.
[90] N., White (ed.), Theory of Matroids, Cambridge University Press, Cambridge, 1986.
[91] N., White (ed.), Combinatorial Geometries, Cambridge University Press, Cambridge, 1987.
[92] N., White (ed.), Matroid Applications, Cambridge University Press, Cambridge, 1992.
[93] H., Whitney, A logical expansion in mathematics, Bull. Amer. Math. Soc. 38 (1932), 572–579.
[94] H., Whitney, The coloring of graphs, Ann. Math. 33(2) (1932), 688–718.
[95] T., Zaslavsky, Counting the faces of cut-up spaces, Bull. Amer. Math. Soc. 81 (1975), 916–918.
[96] T., Zaslavsky, Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. 154, vii+102 pages.
[97] G. M., Ziegler, Lectures on Polytopes, Springer, New York, 1995.