Whitney Numbers Research Articles

Applications of the symmetric chain decomposition of the lattice of divisors

The symmetric chain decomposition of the lattice of divisors,D N, is applied to prove results about the strict unimodality of the Whitney numbers ofD N, about minimum interval covers for the union of consecutive rank-sets ofD N, and about the distribution of sums of vectors in which each vector can be included several times (an extension of the famous Littlewood-Offord problem)

Whitney numbers of some geometric lattices

It is a well known fact that the Whitney numbers of a sequence of ranked posets P0, P1, P2,-.of rank 0, l, 2 .... such that every r-ranked filter of P, is isomorphic to Pr are the connection constants between the sequence of powers and the sequence of the characteristic polynomials of the posets [-Dow]. We note here that, if the posets are indeed supersolvable geometric lattices, then the sequence of characteristic polynomials is persistent in the sense of [Dam2]. Thus, in Sections 3 and 4, we will be able to transfer the two term recursion and the explicit formula for connection constants [Daml] as well as several log-concavity properties of symmetric functions [Sag2] to the Whitney numbers of those lattices. Moreover, since the roots of the characteristic polynomial of a supersolvable lattice have been given a nice combinatorial meaning [Sta2], we can also give a simple semantics for those formulas. As a consequence of our results, we obtain (Section 5) unifying proofs of several properties enjoyed by Whitney numbers of Boolean algebras, subspace lattices, partition lattices, and Dowling lattices. It turns out that these lattices form the only infinite sequences of modularly complemented geometric lattices satisfying the conditions mentioned at the beginning.

Partition Lattice q -Analogs Related to q -Stirling Numbers

We construct a family of partially ordered sets (posets) that are q-analogs of the set partition lattice. They are different from the q-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the q-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for S[n, k] as sequences of lines in a vector space over the Galois field F q. Another member is constructed so as to mirror the partial order in the subspace lattice.

Flags and Whitney Numbers of Matroids

Let [Xi] be a saturated chain of flats in a rank-r simple matroid G and let ai be the number of points in Xi, but not in Xi − 1. We prove that the mth Whitney number wm(G) of the first kind (defined to be the absolute value of the sum ∑ μ(0̂, X) over all rank-m flats X) is greater than or equal to the coefficient of λr − m in the polynomial (λ − a1)(λ − a2) ··· (λ − ar). Equality occurs for any m in the range 2 ≤ m ≤ r if and only if all the flats Xi, are modular.

Sign-Coherent Identities for Characteristic Polynomials of Matroids

We derive an explicit formula for the difference χ(G;λ) − χ(G|X;λ)χ(Tx(G); λ)/(λ − 1), where χ(G;λ) is the characteristic polynomial of a simple matroid G, G|X is the restriction of G to a flat X in G, and Tx(G) is the complete principal truncation of G at the flat X. Two counting proofs of this formula are given. The first uses the critical problem and the second uses the broken-circuit complex. We also derive several inequalities involving Whitney numbers of the first kind and other numerical invariants.

Whitney numbers of the second kind of finite modular lattices

Let L be any finite modular lattice. We prove that each interval Jz L has a symmetric and unimodal sequence of Whitney numbers of the second kind iff L may be represented as a direct product of primary q-lattices. We use the term “primary” in the classical sense of Jbnsson and Monk [7], while the term “q-lattice” is from Stanley [lo]. Actually, in the “if” part of the proof, we prove that, upon denoting by n and 1 the dimension and the rank of J, the sequence of the Whitney numbers of J is symmetric and unimodal of the form

Whitney Numbers of the Second Kind for the Star Poset

The integers W 0 , ..., W t are called Whitney numbers of the second kind for a ranked poset if W k is the number of elements of rank k . The set of transpositions T = {(1, n ), (2, n ), ..., ( n - 1, n )} generates S n , the symmetric group. We define the star poset, a ranked poset the elements of which are those of S n and the partial order of which is obtained from the Cayley graph using T . We characterize minimal factorizations of elements of S n as products of generators in T and provide recurrences, generating functions and explicit formulae for the Whitney numbers of the second kind for the star poset.

Alternating Whitney sums and matchings in trees, part II

The number of k node subtrees of a tree is its kth Whitney number. This paper establishes quadratic bounds in the number of nodes on the alternating sum of the Whitney numbers weighted by k 2. The lower bound is achieved precisely for paths on an even number of nodes. The upper bound is achieved for Edmonds' alternating trees. A rooted alternating sum is shown to be related to the Gallai-Edmonds matching decomposition and the structure of maximum independent sets in the tree.

Alternating Whitney sums and matchings in trees, part 1

The number of k-node subtrees of a tree is its kth Whitney number. This paper investigates the behavior of certain alternating sums of these Whitney numbers and shows how they are related to the structure of maximum matchings in the tree. It is shown that the alternating sum of the Whitney numbers gives the maximum cardinality of an independent set of nodes. Moreover, a weighted alternating sum yields the number of nodes left uncovered by at least one maximum matching.

Two results about points, lines and planes

Given n points in three dimensional euclidean space, not all lying on aplane, let l be the number of lines determined by the points, and let p be the number of planes determined. We show that l 2⩾ cnp, where c>0. This is the weak version of the so-called Points-Lines-Planes conjecture (a conjecture of considerable interest to combinatorialists) being an instance of the conjectured log-concavity of the Whitney numbers. We also show that there is always a point incident with at least cl planes, where c>0, provided that the n points do not all lie on two skew lines. This result lends support to our conjecture, published in 1981, that n − 1 + p + 2 ⩾ 0.

Strong properties in partially ordered sets II

An [α, β)-normal poset with (α, β)-logarithmic concave Whitney numbers is a normal poset with logarithmic concave Whitney numbers, with the additional condition that, without mentioning trivial cases, in the definitional inequalities for normality and logarithmic concavity equality can only hold in the exceptional intervals [α, β) or (α, β), respectively. A theorem is proved, where some conditions for the posets P and Q are given such that P × Q is an [α, β)-normal poset with (α, β)-logarithmic concave Whitney numbers. Some corollaries are deduced from which the strong h-family property of many posets can be obtained.

Parenthesizations of finite distributive lattices

One of the nicer classes of finite partially ordered sets (posets) consists of the so-called symmetr ic chain orders. Besides possessing a variety of interesting structural propert ies, some of which we ment ion below, these posets include among their kind several important types of posets: the lattice B , of the subsets of an n e lement set [1, 3, 11, 12], the lattice Dn of divisors of an integer n [3, 6], the lattice Ln(q) of subspaces of an n-dimensional vector space over the finite field with q elements [8]. It is also conjectured that the lattice of partitions of integers, with part size and number of parts bounded, is a symmetric chain order [16, 17]. Here , we are interested in a general technique, originally used by Greene and Klei tman [6], for obtaining a chain decomposi t ion of a finite distributive lattice. This technique involves labeling the elements of the lattice with sequences of O's and l ' s and using a parenthesizat ion of the labels to induce a chain decomposi t ion of the lattice. With certain conditions on the labeling, this process yields a symmetr ic chain decomposit ion. W e obtain necessary and sufficient conditions for a lattice to allow such a labeling as well as present a collection of other structural propert ies of such lattices. (Actually, we work with the underlying poset of join-irreducibles.) As a related result, we characterize when another technique [1] for obtaining a chain decomposi t ion, using lexicographic partial matchings between elements of adjacent ranks, is the same as the method of parenthesization. To establish our basic definitions, let L be a poset. We say that x covers y in L if y < x and there does not exist a z in L such that y < z < x. L is a ranked poset if it possesses a rank function r f rom the elements of L into the nonnegative integers such that r(x) = 0 if x is minimal and r(x) = r (y )+ 1 if x covers y. If L has a rank function, the kth Whitney number WL(k) is the number of elements in L of rank k. The rank of L is defined to be the max imum of r(x) over all x in L, that is, the length of the longest chain in L.

On the Interpretation of Whitney Numbers Through Arrangements of Hyperplanes, Zonotopes, Non-Radon Partitions, and Orientations of Graphs

On the Interpretation of Whitney Numbers Through Arrangements of Hyperplanes, Zonotopes, Non-Radon Partitions, and Orientations of Graphs

The slimmest arrangements of hyperplanes

We show how the results of Dowling and Wilson on Whitney numbers in ‘The slimmest geometric lattices’ imply minimum values for the numbers of k-dimensional flats and d-dimensional cells of a projective d-arrangement of hyperplanes and for the number of d-cells missed by an extra hyperplane. Their theorems also characterize the extremal arrangements. We extend their lattice results to doubly indexed Whitney numbers; thence we obtain minima for the number of k-dimensional cells and the number of pairs of flats with x\(\subseteq\)y and dim x=k, dim y=l. The lower bounds are in terms of the rank and number of points of the geometric lattice, or the dimension d and the number of hyperplanes of the arrangement. The minima for k-cells were conjectured by Grunbaum; R. W. Shannon proved the minima for k-dimensional flats and cells and characterized attainment for the latter by a more strictly geometric, non-latticial technique.

On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs

The doubly indexed Whitney numbers of a finite, ranked partially ordered set L L are (the first kind) w i j = ∑ { μ ( x i , x j ) : x i , x j ∈ L {w_{ij}} = \sum {\{ \mu ({x^i},{x^j}):{x^i},{x^j} \in L} with ranks i , j } i,j\} and (the second kind) W i j = {W_{ij}} = the number of ( x i , x j ) ({x^i},{x^j}) with x i ⩽ x j {x^i} \leqslant {x^j} . When L L has a 0 0 element, the ordinary (simply indexed) Whitney numbers are w j = w 0 j {w_j} = {w_{0j}} and W j = W 0 j = W j j {W_j} = {W_{0j}} = {W_{jj}} . Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of k k -dimensional faces for any k k , of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope P P inside the visible boundary as seen from a distant point on a generating line of P P . The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley’s theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly q q sources (generalizing Rényi’s enumeration of permutations with q q "outstanding" elements). The number of totally cyclic orientations of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph’s having a unique source.

On the lattice of order ideals of an up-down poset

The combinatorial structure of the distributive lattice of order ideals of an up-down poset is studied. Two recursions are given for the Whitney numbers, and generating functions for the Whitney numbers are derived. In addition, an explicit nested chain decomposition is given for the lattice, the existence of which implies that the lattice satisfies the Sperner property and its generalizations, and has unimodal Whitney numbers.

The slimmest arrangements of hyperplanes: II. Basepointed geometric lattices and Euclidean arrangements

We calculate the minimum numbers of k -dimensional flats and cells of any Euclidean d -arrangement of n hyperplanes. The bounds are obtained by calculating lower bounds for the values of the doubly indexed Whitney numbers of a basepointed geometric lattice of rank r with n points. Additional geometric results concern the minimum number of cells of a Euclidean or projective arrangement met by a subspace in general position and the minimum number of non-Radon partitions of a Euclidean point set. We include remarks on the relationship between Euclidean arrangements and basepointed geometric lattices and on the minimum numbers of cells of arrangements with a bounded region.

From sets to functions: Three elementary examples

A sequence of binomial type is a basis for R [x] satisfying a binomial-like identity, e.g. powers, rising and falling factorials. Given two sequences of binomial type, the authors describe a totally combinatorial way of finding the change of basis matrix: to each pair of sequences is associated a poset whose Whitney numbers of the 1st and 2nd kind give the entries of the matrix and its inverse.

On chains and Sperner k-families in ranked posets, II

A ranked poset P satisfies condition S if for all k the set of elements of the k largest ranks in P is a Sperner k-family. It satisfies condition T if for all k there exist disjoint chains in P which each meet the k largest ranks and which cover the kth largest rank. Griggs employed the theory of saturated partitions to prove that if P satisfies S it also satisfies T, and that the converse is true for posets with unimodal Whitney numbers. Here we present new proofs of these facts which do not require the existence of saturated partitions. The first result is proven with a simple network flow argument and the second is proven directly.

On chains and Sperner k-families in ranked posets

A ranked poset P has the Sperner property if the sizes of the largest rank and of the largest antichain in P are equal. A natural strengthening of the Sperner property is condition S: For all k, the set of elements of the k largest ranks in P is a Sperner k-family. P satisfies condition T if for all k there exist disjoint chains in P each of which meets the k largest ranks and which covers the kth largest rank. It is proven here that if P satisfies S, it also satisfies T, and that the converse, although in general false, is true for posets with unimodal Whitney numbers. Conditions S and T and the Sperner property are compared here with two other conditions on posets concerning the existence of certain partitions of P into chains.