C. Greene, D.J. Kleitman, The structure of Sperner k-families, J. Combinatorial Theory, Ser.A 20 (1976),41-68.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... (d 1 ; d 2 d 1 ; d 3 d 2 ; d t d t 1 ) where t is maximal subject to the condition that d t 1 d t (meaning that d t = n) It may not be obvious from the de nition that (w) is a partition (i.e. d 1 d 2 d 1 d t d t 1 ) but this follows by a result of Greene and Kleitman [11]. Recall that if and are two partitions of n, we say that D ( dominates ) if for all k we have i i : This de nes a partial order on the set of partitions of n. As mentioned before, there is also a natural partial order LR on the two sided Kazhdan Lusztig cells of a ....

C. Greene and D.J. Kleitman, The structure of Sperner k-families, J. Combin. Theory Ser. A 20 (1976), 41-68.

....then we may drop the G from the notation for simplicity. If and are partitions (i.e. 1 2 3 and 1 2 3 ) and furthermore are conjugates of each other, then we say that G satis es conjugacy . It is a famous theorem, due to Greene and Kleitman [9, 10], that comparability graphs of nite posets satisfy conjugacy. A clique cover of G is a vertex disjoint union of complete subgraphs whose union covers all vertices of G. If is a clique cover, then we abuse notation and also let denote the integer partition consisting of the sizes of the ....

C. Greene and D. J. Kleitman. The structure of Sperner k-families. J. Combin. Theory Ser. A, 20:41-68, 1976.

....shape, tableau. The second author was supported in part by NSF grant #DMS 9700927. 1. Introduction This survey, written at the suggestion of the late G. C. Rota, focuses on the fundamental correspondence originally discovered by C. Greene [13] following his joint work with D. J. Kleitman [14] that associates a Ferrers shape (P ) to every finite poset P . The number of boxes in the first k rows (resp. columns) of (P ) equals the maximal number of elements in a union of k chains (resp. antichains) in P . The correspondence P 7 (P ) is intimately related to at least three areas of ....

....interpretation mentioned above; and a proof based on Frank s approach, which as a byproduct yields a Ford Fulkerson type result for maximal chain families (Theorem 9.5) The latter proof, as well as our proofs of Theorems 2.3 and 2. 4, are new, although some of the ingredients were recycled from [4, 5, 6, 10, 14]. 2. Main Theorems Let P be a finite partially ordered set of cardinality n. A chain is a totally ordered subset of P . An antichain is a subset of P in which no two elements are comparable. The famous theorem of Dilworth [1] states that the maximal size of an antichain in P is equal to the ....

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C. Greene and D. J. Kleitman, The structure of Sperner k-families, J. Comb. Theory, Ser. A 20 (1976), 41--68.

.... G, with the ith term equal to the maximum number of vertices that can be additionally colored by using i instead of i 1 colors, see Albertson and Berman, 1, 2] The rst appearance of this idea is Greene s and Kleitman s proof that comparability graphs have monotonically decreasing sequences, see [10, 11]; proofs from other perspectives and related works appear in [8, 9, 21, 23, 24] In another direction, Stanley has developed a symmetric function generalization of the chromatic polynomial which contains the chromatic di erence sequence, see [26, 27] 1991 Mathematics Subject Classi cation. 05C. ....

C. Greene and D. G. Kleitman, The structure of Sperner k-families, JCT(A), 20 (1976), pp. 41-68.

....hand, an algorithm to compute a longest increasing subsequence in a permutation in time O(n log n) pertains to mathematicians folklore. A careful treatment of the algorithm can be found in [3] older sources are e.g. 1] or [8] Interest in k chains of orders goes back to Greene and Kleitman [7, 6] who discovered a rich duality between maximum k chains and maximum antichains. From this theory we quote a theorem relating maximum k chains to maximum antichains. Theorem 1 For an order P with n elements there exists a partition ff of n, such that the Ferrers diagram F ff of ff has the ....

C. Greene and D. J. Kleitman. The structure of sperner k-families. J. Comb. Th. (A), 20:41--68, 1976. 19

....partitioning P . To derive Dilworth s theorem two observations suffice: 1) For any 1 weighting Gamma1 (1) is an antichain. 2) For 1 weightings of maximal value Gamma1 ( Gamma1) Thus a 1 weighting of maximal value is the characteristic function of an antichain. Greene and Kleitman [9] found a nice generalization of Dilworth s result. Define a k antichain family to be a family of k pairwise disjoint antichains. Theorem 1 For any partially ordered set P and any positive integer k max X A2A jAj = min X C2C min(jCj; k) partially supported by Deutsche ....

....supported by Deutsche Forschungsgesellschaft 1 where the maximum is taken over all k antichain families A and the minimum over all chain partitions C of P . A chain partition C which minimizes the right hand side is called k saturated. In fact a somewhat stronger result was obtained in [9]. Theorem 2 For any k 1 there is a chain partition which is simultaneously k saturated and (k 1) saturated. Greene [8] stated the duals of these theorems. Let a chain family be a family of pairwise disjoint chains. Theorem 3 For any partially ordered set P and any positive integer ....

C. Greene and D.J. Kleitman, The structure of Sperner k-families. J. Comb. Th. (A) 20(1976), 41-68.

....: be the maximal number of vertices in a union of k cliques in G. Dually, let Q k be the maximal number of vertices in a union of k independent subsets of G. Denote x k = R k Gamma R k Gamma1 and y k = Q k Gamma Q k Gamma1 . The following result was first proved by C. Greene and D. Kleitman [5, 6] (see also [2, 4, 7, 8] Theorem. Suppose that either G or its complement is a comparability graph of some partially ordered set. Then (x 1 ; x 2 ; and (y 1 ; y 2 ; are conjugate partitions. Problem. Find other classes of graphs for which this result is true. Generalize the ....

C. Greene, D. Kleitman, The structure of Sperner k-families, J. Comb. Theory, Ser. A 20 (1976), 41-68.

....0 so l l i . 3. The Reduction Theorem The main result is that the k walks of G are P colorable if and only if the (k 1) walks of G are A(P) colorable, where A(P) is a poset on the antichains of P (actually A(P) is a distributive lattice) given by Birkhoff [1,3] see also Greene and Kleitman [10]. Recall that an antichain of P is a subset of P in which every pair of distinct elements are incomparable. We will use the fact that A(P) is order isomorphic to the collection of the order ideals of P. Definition. Let A(P) denote the collection of all antichains on P. For X,Y A(P) let X Y if ....

C. Greene and D. J. Kleitman, The structure of Sperner k-families, J. Combinatorial Theory, 20 (1976), pp. 41-68.

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C. Greene, D.J. Kleitman, The Structure of Sperner k-Families, J. Combinatorial Theory, Ser.A 20 (1976),41-68.

....grateful to Curtis Greene and Andrei Zelevinsky for a number of valuable suggestions and corrections. 25 Notes Theorem 1.1 was proved by C. Greene [5] generalizing a result of C. Schensted [18] Corollary 1.2 can be extended to arbitrary finite posets, as shown by C. Greene and D. J. Kleitman [6, 7]. Knuth equivalence and Theorems 1.4 and 1.6 are due to D. E. Knuth [10] who studied this equivalence in a more general setting, with permutations replaced by arbitrary words in the alphabet f1; ng. It is often useful to work in the plactic monoid [11] which is the quotient of the free ....

C. Greene and D. J. Kleitman, The structure of Sperner k-families, J.Combin.Theory, Ser.A 20 (1976), 41-68.

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C. Greene, D.J. Kleitman, The structure of Sperner k-families, J. Combinatorial Theory, Ser.A 20 (1976),41-68.

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C. Greene, D.J. Kleitman, The structure of Sperner k-families, J. Combinatorial Theory, Ser.A 20 (1976),41-68.

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C. Greene, D.J. Kleitman, The structure of Sperner k-families, J. Combin. Theory Ser. A 20 (1976) 41--68.

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