A comparison theorem for ff-vectors of simplicial polytopes

Let $f_i(P)$ denote the number of $i$-dimensional faces of a convex polytope $P$. Furthermore, let $S(n,d)$ and $C(n,d)$ denote, respectively, the stacked and the cyclic $d$-dimensional polytopes on $n$ vertices. Our main result is that for every simplicial $d$-polytope $P$, if $$f_r(S(n_1,d))\le f_r(P) \le f_r(C(n_2,d))$$ for some integers $n_1, n_2$ and $r$, then $$f_s(S(n_1,d))\le f_s(P) \le f_s(C(n_2,d))$$ for all $s$ such that $r<s$. For $r=0$ these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain ``comparison theorem'' for $f$-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.Comment: 8 pages. Revised and corrected version. To appear in "Pure and Applied Mathematics Quarterly

A cell complex in number theory

Let De_n be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic (the Mertens function) is closely related to deep properties of the prime number system. In this paper we study the asymptotic behavior of the individual Betti numbers and of their sum. We show that De_n has the homotopy type of a wedge of spheres, and that as n tends to infinity: \sum \be_k(\De_n) = \frac{2n}{\pi^2} + O(n^{\theta}),\;\; \mbox{for all} \theta > \frac{17}{54}. We also study a CW complex tDe_n that extends the previous simplicial complex. In tDe_n all numbers up to n correspond to cells and its Euler characteristic is the summatory Liouville function. This cell complex is shown to be homotopy equivalent to a wedge of spheres, and as n tends to infinity: \sum \be_k(\tDe_n) = \frac{n}{3} + O(n^{\theta}),\;\; \mbox{for all} \theta > \frac{22}{27}.Comment: 16 page

A note on blockers in posets

The blocker $A^{*}$ of an antichain $A$ in a finite poset $P$ is the set of elements minimal with the property of having with each member of $A$ a common predecessor. The following is done: 1. The posets $P$ for which $A^{**}=A$ for all antichains are characterized. 2. The blocker $A^*$ of a symmetric antichain in the partition lattice is characterized. 3. Connections with the question of finding minimal size blocking sets for certain set families are discussed

Geometrically constructed bases for homology of partition lattices of types A, B and D

We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles \rho_{R_i} in the homology of the proper part \bar{L_A} of the intersection lattice such that {\rho_{R_i}}_{i=1,...,k} is a basis for \tilde H_{d-2}(\bar{L_A}). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.Comment: 29 pages, 4 figure

The Möbius function of factor order

AbstractIntervals in the factor ordering of a free monoid are investigated. It was shown by Farmer (1982) that such intervals (β, α) are contractible or homotopy spheres in case β is the empty word. We observe here that the same is true in general. This implies that the Möbius function of factor order takes values in {0, + 1, −1}. A recursive rule for this Möbius function is given, which allows efficient computation via the Knuth—Morris—Pratt algorithm.The Möbius function of subword order was studied in Björner (1990). We give here a simpler proof (a parity-changing involution) for its combinatorial interpretation

On complements in lattices of finite length

Let L be a lattice of finite length. It is shown that if L contains no 3-element interval then L is relatively complemented. Also, if for every xεL the set Cx=y ε L¦y covers x satisfies V Cx = 1, then L is complemented