David E. Daykin

An Inequality for the Weights of Two Families of Sets, Their Unions and Intersections

then ~(A) fi(B) < 7(A v B) cS(A A B) for all A, B ~ S, (2) where e(A) = ~(a~A) e(a) and A v B = {awb; aeA, b~B} and A A B = {ac~b; a~A, b~B}. Since every distributive lattice can be embedded in the subsets of some set we get an immediate Corollary. If S is a distributive lattice and (2) holds whenever A, B each contain exactly one point of S then (2) always holds. Here S, A, B may be infinite. Our theorem contains as special cases results of Anderson, Daykin, Fortuin, Ginibre, Greene, Holley, Kasteleyn, Kleitman, Seymour, West and others 1. We discovered it whilst guests at the Mathematisches Forschungsinstit ut Oberwolfach and thank all concerned for their kindness to us.

Degrees giving independent edges in a hypergraph

For r -partite and for r -uniform hypergraphs bounds are given for the minimum degree which ensures d independent edges.

Graphs with Hamiltonian cycles having adjacent lines different colors

Let G be a complete graph Kp (or a complete bipartite graph Km,m) with its lines colored so that no point is on more than k lines of the same color. If p ≥ 17k (or m ≥ 25k) then G has a cycle of every possible size with adjacent lines different colors.

Lyndon-like and V-order factorizations of strings

We say a family W of strings is an UMFF if every string has a unique maximal factorization over W. Then W is an UMFF iff xy, yz ∈ W and y non-empty imply xyz ∈ W. Let L-order denote lexicographic order. Danh and Daykin discovered V-order, B-order and T-order. Let R be L, V, B or T. Then we call r an R-word if it is strictly first in R-order among the cyclic permutations of r. The set of R-words form an UMFF. We show a large class of B-like UMFF. The well-known Lyndon factorization of Chen, Fox and Lyndon is the L case, and it motivated our work.

PROPERTIES AND CONSTRUCTION OF UNIQUE MAXIMAL FACTORIZATION FAMILIES FOR STRINGS

We say a family of strings over an alphabet is an UMFF if every string has a unique maximal factorization over . Foundational work by Chen, Fox and Lyndon established properties of the Lyndon circ-UMFF, which is based on lexicographic ordering. Commencing with the circ-UMFF related to V-order, we then proved analogous factorization families for a further 32 Block-like binary orders. Here we distinguish between UMFFs and circ-UMFFs, and then study the structural properties of circ-UMFFs. These properties give rise to the complete construction of any circ-UMFF. We prove that any circ-UMFF is a totally ordered set and a factorization over it must be monotonic. We define atom words and initiate a study of u, v-atoms. Applications of circ-UMFFs arise in string algorithmics.

A generalization of Sperner's theorem

Some generalizations of Sperners theorem and of the LYM inequality are given to the case when A 1 ,… A t are t families of subsets of {1,…, m } such that a set in one family does not properly contain a set in another.

The number of meets between two subsets of a lattice

Let L be a lattice of divisors of an integer (isomorphically, a direct product of chains). We prove |A| |B| ⩽ |L| |A ∧ B| for any A, B ⊃ L, where |·| denotes cardinality and A ∧ B = {a ∧ b: a ϵ A, b ϵ B}. |A ∧ B| attains its minimum for fixed |A|, |B| when A and B are ideals. |·| can be replaced by certain other weight functions. When the n chains are of equal size k, the elements may be viewed as n-digit k-ary numbers. Then for fixed |A|, |B|, |A ∧ B| is minimized when A and B are the |A| and |B| smallest n-digit k-ary numbers written backwards and forwards, respectively. |A ∧ B| for these sets is determined and bounded. Related results are given, and conjectures are made.

A linear partitioning algorithm for Hybrid Lyndons using V-order

In this paper we extend previous work on unique maximal factorization families (UMFFs) and a total (but non-lexicographic) ordering of strings called V-order. We present new combinatorial results for V-order, in particular concatenation under V-order. We propose linear-time RAM algorithms for string comparison in V-order and for Lyndon-like factorization of a string into V-words. This asymptotic efficiency thus matches that of the corresponding algorithms for lexicographical order. Finally, we introduce Hybrid Lyndon words as a generalization of standard Lyndon words, and hence propose extensions of factorization algorithms to other forms of order.

Combinatorics of Unique Maximal Factorization Families (UMFFs)

Suppose a set W of strings contains exactly one rotation (cyclic shift) of every primitive string on some alphabet Σ. Then W is a circ-UMFF if and only if every word in Σ

Ordered Ranked Posets, Representations of Integers and Inequalities from Extremal Poset Problems

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