Sperner理论中的几个问题
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摘要
Sperner理论是组合数学的一个分支，其研究对象是偏序集，主要考虑偏序集上满足某些条件的极值问题，它的起源可以追朔到1928年Sperner的一个定理：在子集格中，最大秩集构成基数最大的反链．经过近一个世纪的发展，Sperner定理已经发展成为一门系统的理论。
本文第一章是Sperner理论的一个简单的综述，包括相关的记号、术语以及在后面需要用到的主要方法．
第二章给出q-阶对数凹性的定义，并把一系列保持对数凹性的线性变换推广到q-阶对数凹性。
第三章给出加权偏序集q-直积的定义，并证明关于偏序集的正规匹配性的q直积定理，即：若两个偏序集都是q-阶对数凹的并且具有正规匹配性，那么它们的每一个q-直积都是都是q-阶对数凹的并且具有正规匹配性。利用这个定理得到了子空间格的几个子偏序集是q-阶对数凹的并且具有正规匹配性。
第四章讨论子集格的四个与Lih猜想相关的子偏序集，然后通过子集格的对称链分解导出它们的套链分解。
第五章考虑置换偏序集B(n，n)的LYM性质和局部EKR性质。
Sperner Theory is one of research branches in combinatorics, whose reserch object is posets, whose main content is to investgate the extremal problems on posets. Its source came from a theorem of Sperner in 1928: in a subset lattice, a size maximal rank set forms a size maximal antichain. After about one century's development Sperner's theorem has become into a systematic theory.
The first chapter is a simple survey on this theory, including the relative notations, basic terminology and the main methods used later.
In the second chapter, we introduce the definition of the q-degree log-concavity of a sequence, and give a series of linear transformations which preserve the q-degree log-concavity of a sequence.
In the third chapter, we first give the definition of q-direct product, and then deduce the q-direct product theorem from product theorem: if (Q, v) and (P, w) are both q-degree log-concave and have the normalized matching property, then each q-direct product of them is q-degree log-concave and has the normalized matching property. By this theorem we prove that some subposets of L(V) are q-degree log-concave and have the normalized matching property.
In the fourth chapter, we construct the nested chain decompositions for four subposets of the Boolean lattice.
In the last chapter, we consider the LYM property and the local EKR property of the partial permutation poset B(n,n).
本文第一章是Sperner理论的一个简单的综述，包括相关的记号、术语以及在后面需要用到的主要方法．
第二章给出q-阶对数凹性的定义，并把一系列保持对数凹性的线性变换推广到q-阶对数凹性。
第三章给出加权偏序集q-直积的定义，并证明关于偏序集的正规匹配性的q直积定理，即：若两个偏序集都是q-阶对数凹的并且具有正规匹配性，那么它们的每一个q-直积都是都是q-阶对数凹的并且具有正规匹配性。利用这个定理得到了子空间格的几个子偏序集是q-阶对数凹的并且具有正规匹配性。
第四章讨论子集格的四个与Lih猜想相关的子偏序集，然后通过子集格的对称链分解导出它们的套链分解。
第五章考虑置换偏序集B(n，n)的LYM性质和局部EKR性质。
Sperner Theory is one of research branches in combinatorics, whose reserch object is posets, whose main content is to investgate the extremal problems on posets. Its source came from a theorem of Sperner in 1928: in a subset lattice, a size maximal rank set forms a size maximal antichain. After about one century's development Sperner's theorem has become into a systematic theory.
The first chapter is a simple survey on this theory, including the relative notations, basic terminology and the main methods used later.
In the second chapter, we introduce the definition of the q-degree log-concavity of a sequence, and give a series of linear transformations which preserve the q-degree log-concavity of a sequence.
In the third chapter, we first give the definition of q-direct product, and then deduce the q-direct product theorem from product theorem: if (Q, v) and (P, w) are both q-degree log-concave and have the normalized matching property, then each q-direct product of them is q-degree log-concave and has the normalized matching property. By this theorem we prove that some subposets of L(V) are q-degree log-concave and have the normalized matching property.
In the fourth chapter, we construct the nested chain decompositions for four subposets of the Boolean lattice.
In the last chapter, we consider the LYM property and the local EKR property of the partial permutation poset B(n,n).
引文
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[13] E.R. Canfield, On a problem of Rota, Adv. Math., 29(1978), 1-10.
[14] William Y. C. Chen and G. C. Rota, q-analogs of the inclusion-exclision principle and permutations with restricted position, Discrete Math., 104 (1992), 7-22.
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[16] M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A, 22 (1977), 352-362.
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[38] J.R. Griggs, Sufficient conditions for a symmetric chain order. SIAM J. Appl. Math.. 32 (1977), 807-809.
[39] J.R. Griggs, On chains and Sperner κ-families in ranked posers, J. Combin. Theory Ser. A, 28 (1980), 156-168.
[40] J.R. Griggs, Collections of subsets with the sperner property, Trans. Amer. Math. Soc., 269 (1982), 575-591.
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[63] D. Lubell, A short proof of Sperner's theorem, J. Combin. Theory Ser., 1 (1966), 299.
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[2] I. Anderson, Some problems in combinatorial number theory, Ph.D. Thesis, University of Nottingham.
[3] K. Baker, A generalization of Sperner's lemma, J. Combin. Theory, 6 (1969), 224-5.
[4] B. Bollobás, Sperner systems consisting of pairs of complementary subsets, J. Combin. Theory Ser. A, 15 (1973), 363-6.
[5] B. Bollobás and I. Leader, An ErdSs-Ko-Rado theorem for signed sets, Comput. Math. Applic., 34 (1997), 9-13.
[6] A. Brace and D.E. Daykin, Sperner type theorems for finite sets, Proc. Br. Combin. Conf., Oxford, 1972, pp. 18-37.
[7] F. Brenti, Log concave and unimodal sequences in algebra, combinatorics and geometry: an update, Contemp. Math., 178 (1994), 71-89.
[8] F. Brenti, Log concave and PSlya sequences in combinatorics, Mere. Amer. Math. Soc., 81 (1989), No.413, pp.106.
[9] N.D. Bruijin, C.A. Van, E. Tengbergen and D. R. Kruyswijk, On the set of divisiors of a number, Nieuw Arch. Wisk, 23 (1952), 191-193.
[10] L.M. Butler, A unimodality result in the enumeration of subgroups of a finite abelian group, Proc. Amer. Math. Soc., 101 (1987), 771-775.
[11] L.M. Butler, The q-log-concavity of q-binomial coefficients, J. Combin. Theory Ser. A, 54 (1990), 54-63.
[12] P.J. Cameron and C.Y. Ku, Intersecting families of permutations, European J. Combin., 24 (2003), 881-890.
[13] E.R. Canfield, On a problem of Rota, Adv. Math., 29(1978), 1-10.
[14] William Y. C. Chen and G. C. Rota, q-analogs of the inclusion-exclision principle and permutations with restricted position, Discrete Math., 104 (1992), 7-22.
[15] S.E. Cheng and K.W. Lih, An improvement on a spernerity proof of Horrocks, Theoret. Comput. Sci., 263 (2001), 355-377.
[16] M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A, 22 (1977), 352-362.
[17] M. Deza and P. Frankl, ErdSs-Ko-Rado theorem-22 years later, SIAM J. Algebraic Discrete Methods, 4 (1983), 419-431.
[18] R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math., 51 (1950), 161-165.
[19] R.G. Donnelly, Symplectic Analogs of the Distributive Lattices L(m, n), J. Combin. Theory Ser. A, 88 (1999), 217-234.
[20] R.G. Donnelly, Explicit constructions of the fundamental representations of the symplectic Lie algebras, J. Algebra, 233 (2000), 37-64.
[21] R.G. Donnelly, S.J. Lewis and R. Pervine, Constructions of representations of O(2n + 1; C) that imply Molev and Reiner-Stanton lattices are strongly Sperner, Discrete Math., 263 (2003), 61-79.
[22] K. Engel, Sperner theory, Cambridge University Press, Cambridge, (1997), P. 160.
[23] K. Engel, An Erd(?)s-Ko-Rado theorem for the subcubes of cube, Combinatorica, 4 (1984), 133-140.
[24] P. Erd(?)s, On a Lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51(1945), 898-902
[25] P. Erd(?)s, C. Ko and R. Rado. Intersection theorems for systems of finite sets, Q. J. Math. Oxf., 2(1961)12, 313-320.
[26] P. Frankl and N. Tokushige, Random walks and multiply intersecting families, J. Combin. Theory Ser. A, 109 (2005), 121-134.
[27] P. Frankl and N. Tokushige, Weighted multiply intersecting families, Studia Sci. Math. Hungar., 40 (2003), 287-291.
[28] P. Frankl and N. Tokushige, Weighted 3-wise 2-intersecting families, J. Combin. Theory Ser. A, 100 (2002), 94-115.
[29] P. Frankl and N. Tokushige, Weighted non-trivial multiply intersecting families, Combina-torica, 26 (2006) 37-46.
[30] R. Freese, An application of Dilworth's lattice of maximal antichains, Discrete Math., 7 (1974), 107-109.
[31] E.R. Gansner, On the of order ideals of an up-down poset, Discrete Math., 39(1982), 113-132.
[32] R.L. Graham and L.H. Harper, Some results on matchings in bipartite graphs, SIAM J. Appl. Math., 17 (1969), 1017-22.
[33] C. Greene and D.J, Kleitman, Strong vertions of sperner's theorem, J. Combin. Theory Ser. A, 20 (1976), 80-88.
[34] C. Greene and D.J, Kleitman, The structure of Sperner k-families, J. Combin. Theory Ser. A, 20 (1976), 41-68.
[35] C. Greene, G.O.H. Katona and D.J. Kleitman, Extensions of the Erd(?)s-Ko-Rado theorem, Studies in Appl. Math., 55 (1976), 1-8.
[36] C. Greene and D. J. Kleitman, Proof techniques in the theory of finite sets, In G. -C. Rota, editor, Studies in Combinatorics, Vol.17 of MAA Studies in Math., pp.22-79, Math. Assoc. America, Washington, DC (1978).
[37] J.R. Griggs, Symmetric chain orders, Sperner theorems, and loop matchings, Ph.D. Thesis (1977).
[38] J.R. Griggs, Sufficient conditions for a symmetric chain order. SIAM J. Appl. Math.. 32 (1977), 807-809.
[39] J.R. Griggs, On chains and Sperner κ-families in ranked posers, J. Combin. Theory Ser. A, 28 (1980), 156-168.
[40] J.R. Griggs, Collections of subsets with the sperner property, Trans. Amer. Math. Soc., 269 (1982), 575-591.
[41] G.R. Grimmett, A generalization of a theorem of Kleitman and Milner, Bull. London Math. Soc., 5 (1973), 157-158.
[42] H-D.O.F. Gronau, On Sperner families in which no 3-sets have an empty intersection, Acta Cybernet., 4 (1978), 213-220.
[43] H-D.O.F. Gronau, On Sperner families in which no k-sets have an empty intersection, J. Combin. Theory Ser. A, 28 (1980), 54-63.
[44] H-D.O.F. Gronau, On Sperner families in which no k-sets have an empty intersection Ⅱ, J. Combin. Theory Ser. A, 30 (1981), 298-316.
[45] H-D.O.F. Gronau, On Sperner families in which no k-sets have an empty intersection Ⅲ, Combinatorica, 2(1982), 25-36.
[46] L.H. Harper, The morphology of partially ordered sets, J. Combin. Theory Ser. A, 17 (1974), 44-58.
[47] L. H. Harper, The global theory of flow in networks, Adv. Appl. Math., 1 (1980), 158-181.
[48] D.G.C. Horrocks, Nested chain partitions of hamiltonian filters, J. Combin. Theory Ser. A, 81 (1998), 176-189.
[49] D.G.C. Horrocks, On Lih's Conjecture concerning Spernerity, European J, Combin.. 20(1999), 131-148.
[50] D.G.C. Horrocks, Filters of the Boolean Algebra having the LYM property, Technical Report CORR 96-09, University of Waterloo, 1996.
[51] W.N. Hsieh and D.J. Kleitman, Normalized matching in direct products of partial orders, Studies in Appl. Math., 52 (1973), 285-9.
[52] S. Karlin, Total Positivity, Vol.1, Stanford University Press, 1968.
[53] G.O.H. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hungar., 15 (1964), 329-337.
[54] G.O.H. Katona, Extremal problems for finite sets and convex hulls-a survey, Discrete Math.. 164 (1997), 175-185.
[55] G.O.H. Katona, A simple proof of a theorem of Milner, J. Combin. Theory Ser. A. 83 (1998), 138-140.
[56] D. Kleitman, M. Edelberg and D. Lubell. Maximal-sized antichains in partial orders. Discrete Math., 1 (1971), 47-53.
[57] D. Kleitman. On an extremal property of antichains in partial orders, the LYM property and some of its implications and application, Combinatorics, (Proc.NATO Advanced Study Inst.. Breukelen, 1974), Part 2: Graph theory: foundations, partitions and combinatorial geometry, pp. 77-90. Math. Centre Tracts, No. 56, Math. Centrum, Amsterdam, 1974.
[58] C.Y. Ku and I. Leader, An Erd'os-Ko-Rado theorem for partial permutations, Discrete Math, 306 (2006), 74-86.
[59] B. Larose and C. Malvenuto, Stable sets of maximal size in Kneser-type graphs, European J. Combin., 25 (2004), 657-673.
[60] Y.S. Li and J. Wang, Erd(?)s-Ko-Rado-type theorem for colored sets, preprint.
[61] K.W. Lih, Sperner families over a subset, J. Combin. Theory Ser. A, 29 (1980), 182-185.
[62] M.J. Logan, Sperner theory in a difference of boolean lattices, Discrete Math., 257 (2002), 501-512.
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