S. Sundaram, Applications of the Hopf trace formula to computing homology representations, Jerusalem combinatorics '93, 277--309, Contemp. Math., 178 Amer. Math. Soc., Providence, RI, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....note that S k Gamma1 (n; j) 0 if j dn= k Gamma 1)e because j sets of at most k Gamma 1 objects can partition a set of size of at most n = j(k Gamma 1) Plugging this into (4) finishes the proof. We should note that expansion (4) was derived by Bjorner and Lov asz [3] and by Sundaram [19] using formal power series techniques. Analogs of this expansion for type B and D can be found in a paper of Bjorner and Sagan [5] while applications to the Boolean algebra are in Zhang s thesis [26] Corollary 3.2 Let A be a subspace arrangement. a) If A is embedded in A n and we write a ....

S. Sundaram, Applications of the Hopf trace formula to computing homology representations, Contemp. Math. 178 (1994), 277--309.

....to be able to talk about the Mobius function for lower intervals of the poset. If there is no unique largest element 1 in a poset P then we will use the convention that (P ) 0. The posets Pi n;T were introduced in [BL] and have been studied by several different authors recently, see [B] BW] L][Su]. The more general class of posets Pi n;T;V appeared in [BS] The posets Pi n;T and Pi n;T;V will not be intersection lattices of subspace arrangements in general. A natural question is: when do they correspond to a subspace arrangement embedded in A n ,B n or D n An answer is given in ....

....had if n had belonged to V . Hence if n = 2 V one can replace T;V (n) by Gammas T;V (n) in Lemma 4.2. In the case of Pi n;T the exponential generating function F T (x) was described in [BL] when 1 2 T , and a complete description for general T was given in [L] and independently by Sundaram in [Su]. Define p T (x) s T (n) Theorem 4.3. The exponential generating function for Pi n;T is given by: F T (x) ln(1 p T (x) if 1 2 T ; F T (x) Gamma ln(e Gamma p T (x) if 1 = 2 T: There is a very pleasing duality between the cases 1 2 T and 1 = 2 T . This will continue ....

S. Sundaram, Applications of the Hopf trace formula to computing homology representations, in "Proceedings of Jerusalem Combinatorics Conference 1993" (H. Barcelo and G. Kalai eds.), Contemporary Math. 178, AMS(1994), 277--309.

....In [BW2] the question of whether shellability is preserved by taking the direct product of pure posets which have bottom elements but no top elements, was left open. This is equivalent to asking whether shellability is preserved by taking the reduced product of two bounded posets (as defined in [Su]) The upper reduced product of bounded posets P 1 and P 2 is defined to be the poset obtained by attaching a top 1 to (P 1 Gamma f 1 1 g) Theta (P 2 Gamma f 1 2 g) Similarly the lower reduced product is defined to be the poset obtained by attaching a bottom 0 to (P 1 Gammaf 0 1 ....

....is essentially the same with coatoms playing the role of atoms. We remark that the following immediate consequence of Theorems 5. 9(i) and 10.17 is known to be valid for all bounded posets with torsion free homology by means of the Kunneth formula and results of Quillen [Q] and Walker [Wa] see [Su]) 10.19. Corollary. Suppose P 1 and P 2 are CL shellable posets. Then H i ( Delta(P 1 Theta P 2 ) H i Gamma1 ( Delta(P 1 ThetaP 2 ) H i Gamma1 ( Delta(P 1 ThetaP 2 ) j k=i Gamma2 H j ( Delta(P 1 ) Omega H k ( Delta(P 2 ) Products of CL shellable posets can ....

S. Sundaram, Applications of the Hopf trace formula to computing homology representations, Proceedings of Jerusalem Combinatorics Conference,

....is that the two posets are pure shellable which was proved respectively by Wachs and Bjorner (cf. CHR] and [Sa] For general k, the k mod d partition poset is not pure and that is where the difficulty lies in computing the homology representations of the general k mod d partition posets. In [Su2], Sundaram gives a formula for the virtual representation of the symmetric group on the alternating sum of homology of the k mod d partition poset. When the poset is pure and shellable, as in the case k = 0; 1, Sundaram s formula gives the nonvirtual representation on the top dimensional homology ....

....to derive (4 8) and (4 9) directly from these formulas (cf. Proposition 6.1) Corollary 4.7. For k 1, 4 10) nk H( Pi r ) 4 ik 5 : Proof. Set d = 1 in (4 6) or use Corollary 3.28. Remark. By setting u = v = 1 in (4 6) resp. 4 10) we obtain Sundaram s formula [Su2] for the alternating sum of homology of the k mod d partition poset (resp. at least k partition lattice) By also setting k = d in Corollary 4.5, we obtain the original formula of Calderbank, Hanlon and Robinson [CHR] for the homology of the d divisible partition lattice. One can obtain formulas ....

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S. Sundaram, Applications of the Hopf trace formula to computing homology representations, Proceedings of Jerusalem Combinatorics Conference,

....simple combinatorial interpretation. In particular, let S k (n, j) denote the number of partitions of an n element set into j subsets each of which is of size at most k. Thus these are generalizations of the Stirling numbers of the second kind. We now have the expansion, first derived by Sundaram [47] #(A n,k ,t) j S k 1 (n, j)#t# j . 10) To see why this is true, consider an arbitrary point c # [ s, s] A n,k .Soc can have at most k 1 of its coordinates equal. Consider the c s with exactly j di#erent coordinates. Then there are S k 1 (n, j) ways to distribute the j values among the ....

S. Sundaram, Applications of the Hopf trace formula to computing homology representations, Contemp. Math. 178 (1994), 277--309. 27

....combinatorial interpretation. In particular, let S k (n; j) denote the number of partitions of an n element set into j subsets each of which is of size at most k. Thus these are generalizations of the Stirling numbers of the second kind. We now have the expansion, first derived by Sundaram [47] (A n;k ; t) X j S k Gamma1 (n; j)hti j : 10) To see why this is true, consider an arbitrary point c 2 [ Gammas; s] n n S A n;k . So c can have at most k Gamma 1 of its coordinates equal. Consider the c s with exactly j different coordinates. Then there are S k Gamma1 (n; j) ways to ....

S. Sundaram, Applications of the Hopf trace formula to computing homology representations, Contemp. Math. 178 (1994), 277--309.

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S. Sundaram, Applications of the Hopf trace formula to computing homology representations, Jerusalem combinatorics '93, 277--309, Contemp. Math., 178 Amer. Math. Soc., Providence, RI, 1994.

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