The number of meets between two subsets of a lattice

AbstractLet 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

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$^+$ has a unique maximal factorization over W. The classic circ-UMFF is the set of Lyndon words based on lexicographic ordering (1958). Duval (1983) designed a linear sequential Lyndon factorization algorithm; a corresponding PRAMparallel algorithmwas described by J. Daykin, Iliopoulos and Smyth (1994). Daykin and Daykin defined new circ-UMFFs based on various methods for totally ordering sets of strings (2003), and further described the structure of all circ-UMFFs (2008). Here we prove new combinatorial results for circ-UMFFs, and in particular for the case of Lyndon words. We introduce Acrobat and Flight Deck circ-UMFFs, and describe some of our results in terms of dictionaries. Applications of circ-UMFFs pertain to structured methods for concatenating and factoring strings over ordered alphabets, and those of Lyndon words are wide ranging and multidisciplinary

Erdös-Ko-Rado from Kruskal-Katona

AbstractThe Erdös-Ko-Rado theorem follows immediately from the Kruskal-Katona theorem

A linear partitioning algorithm for Hybrid Lyndons using -order

In this paper we extend previous work on unique maximal factorization families (UMFFs) and a total (but non-lexicographic) ordering of strings called VV-order. We present new combinatorial results for VV-order, in particular concatenation under VV-order. We propose linear-time RAM algorithms for string comparison in VV-order and for Lyndon-like factorization of a string into VV-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

String comparison and Lyndon-like factorization using V-order in linear time

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 describe linear-time algorithms for string comparison and Lyndon factorization based on V-order. We propose extensions of these algorithms to other forms of order

Generic algorithms for factoring strings

In this paper we describe algorithms for factoring words over sets of strings known as circ-UMFFs, generalizations of the well-known Lyndon words based on lexorder, whose properties were first studied in 1958 by Chen, Fox and Lyndon. In 1983 Duval designed an elegant linear-time sequential (RAM) Lyndon factorization algorithm; a corresponding parallel (PRAM) algorithm was described in 1994 by Daykin, Iliopoulos and Smyth. In 2003 Daykin and Daykin introduced various circ-UMFFs, including one based on V-words and V-ordering; in 2011 linear string comparison and sequential factorization algorithms based on V-order were given by Daykin, Daykin and Smyth. Here we first describe generic RAM and PRAM algorithms for factoring a word over any circ-UMFF; then we show how to customize these generic algorithms to yield optimal parallel Lyndon-like V-word factorization

An inequality for the weights of two families of sets, their unions and intersections

Ahlswede R, Daykin DE. An inequality for the weights of two families of sets, their unions and intersections. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 1978;43(3):183-185