C. Bennett and B. E. Sagan, A generalization of semimodular supersolvable lattices, J. Algebraic Combin. 72 (1995), 209--231.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....A is free with exponents e 1 , e 2 , e # then #(A, q) # Y i=1 (q e i ) Deformations of Coxeter arrangements 3 Theorem 2. 4 is one of a number of results which explain factorization phenomena for #(A, q) Other approaches include supersolvability [58] and its generalizations [14, 19] inductive freeness [63] recursive freeness [75] factorization of rooted complexes [18] factorization [26, 67] and inductive factorization [34] For background we refer to these sources, 42, Chapter 4] and the survey article [50] A purely algebraic combinatorial proof of Theorem 2.4 ....

....in F ; see, e.g. 44, 45] Deletion Restriction: this powerful technique in the theory of arrangements yields the formula ( 42, Theorem 2.56] #(A, q) #(A # , q) #(A ## , q) where A # and A ## are obtained from A by deleting or restricting on a hyperplane H # A [42, p. 14] see, e.g. 23, 24, 6] Chromatic Polynomials: the signed chromatic polynomial interpretation of Zaslavsky [70] if A consists of some of the reflecting hyperplanes of Coxeter type B) and its generalization to gain graph coloring [73, 4] see, e.g. 70, 71, 20] 42, 2.4] and [29, ....

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C. Bennett and B.E. Sagan, A generalization of semimodular supersolvable lattices, J. Combin. Theory Ser. A. 72 (1995), 209--231.

.... of an arrangement of hyperplanes or a matroid that guarantee integral roots: we mention freeness [T81] inductive freeness [T80] and recursive freeness [Z87] rooted complex factorization [BZ] factorization [FJ, T92] and inductive factorization [FJ, JP] and existence of an atom decision tree [BS]. Why, then, characterize a comparatively weak property The best reason is simply that it can be done in considerable generality. There is no known general characterization of any other property aside from two special cases: graphic arrangements (subarrangements of A n Gamma1 = fx i = x j : ....

C. Bennett and B. Sagan, A generalization of semimodular supersolvable lattices, submitted.

....6.2 would imply that some element x # L(D n ) of rank two would have to cover at least 3 A 1 =4 atoms. It is easy to verify that there is no such element. It is frustrating that L(D n ) is not supersolvable. To get around this problem, Bennett and I have introduced a more general concept [5]. Looking at the previous proof, the reader will note that it would still go through if every NBC base could be obtained in the following manner. First pick an atom from a set A 1 = a 1 ,a 1 ,a 1 , Then pick the second atom from one of a family of sets A 2 ,A ,A , according to ....

C. Bennett and B. E. Sagan, A generalization of semimodular supersolvable lattices, J. Algebraic Combin. 72 (1995), 209--231.

....6.2 would imply that some element x 2 L(D n ) of rank two would have to cover at least 3 jA 1 j = 4 atoms. It is easy to verify that there is no such element. It is frustrating that L(D n ) is not supersolvable. To get around this problem, Bennett and I have introduced a more general concept [5]. Looking at the previous proof, the reader will note that it would still go through if every NBC base could be obtained in the following manner. First pick an atom from a set A 1 = fa 1 ; a 0 1 ; a 00 1 ; g. Then pick the second atom from one of a family of sets A 2 ; A 0 2 ; A 00 2 ....

C. Bennett and B. E. Sagan, A generalization of semimodular supersolvable lattices, J. Algebraic Combin. 72 (1995), 209--231.

....4.2 would imply that some element x 2 L(D n ) of rank two would have to cover at least 3 jA 1 j = 4 atoms. It is easy to verify that there is no such element. It is frustrating that L(D n ) is not supersolvable. To get around this problem, Bennett and I have introduced a more general concept [2]. Looking at the previous proof, the reader will note that it would still go trough if every NBC base could be obtained in the following manner. First pick an atom from a set A 1 = fa 1 ; a 0 1 ; a 00 1 ; g. Then pick the second atom from one of a family of sets A 2 ; A 0 2 ; A 00 2 ....

C. Bennett and B. E. Sagan, A generalization of semimodular supersolvable lattices, J. Algebraic Combin., to appear.

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C. Bennett and B.E. Sagan, A generalization of semimodular supersolvable lattices. J. Combin. Theory Ser. A 72 (1995), 209-231. MR 96i:05180.

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