Douglas G. Rogers

Rhyming schemes: Crossings and coverings

Some recent results on the enumeration of relations on finite totally ordered sets are unified by establishing correspondences between these relations and some types of rhyming scheme which are characterized by the planarity of certain graphical representations. These representations themselves and a systematic method of enumeration involving them are sketched. Interpretations and proofs of a number of combinatorial identities are obtained.

Critical Perfect Systems of Difference Sets

Publisher Summary This chapter describes critical perfect systems of difference sets. A key equality for difference triangles is that the sum of entries in the kth row from the apex is equal to the sum of entries in the k th row from the bottom. The theory of perfect systems involves, at several points, the allied notion of a complete permutation; a permutation π of the set N , of integers in modulus less than c , is complete when the set of differences, π ( i) – i , for | i | N c . Perfect systems of difference sets have applications in convolutional coding, missile guidance, and the layout of radio telescopes; they also give rise to certain cyclic block designs or, equivalently, in the guise of graceful labelings, to the decompositions of certain complete graphs into the edge-disjoint copies of the complete graph.

Irregular, extremal perfect systems of difference sets II

An (m, n; u, v; c)-system is a collection of components,m of valencyu − 1 andn of valencyv − 1, whose difference sets form a perfect system with thresholdc. A necessary condition for the existence of an (m, n; u, v; c)-system foru = 3 or 4 is thatm ≥ 2c − 1; and there are (2c − 1,n; 3, 6;c)-systems for all sufficiently largec at least whenn = 1 or 2. It is shown here that if there is a (2c − 1,n; u, 6;c)-system thenn = 0 whenu = 4 andn ≤ 2c − 1 whenu = 3. Moreover, if there is a (2c − 1,n; 3, 6;c)-system with a certain splitting property thenn ≤ c − 1, this last result being of possible interest in connection with the multiplication theorem for perfect systems.

The problem of irregular perfect systems of sets of iterated differences

Fors≥2, the set of iterated differences associated with the prescribed integersa(s, j), 1≤j≤s, is the set {a(i, j): 1≤j≤i≤s} wherea(i−1,j)=|a(i, j)−a(i, j+1)|, general problem raised by work of Kreweras and Loeb concerns the existence of partitions of runs of consecutive integers into full sets of iterated differences. In the regular case, where all the sets of iterated differences have the same valencys, it is known that such partitions do not exist at least fors>8. We find here that the problem is more challenging in the case where the sets have different valencies.

Uniform perfect systems of sets of iterated differences of size 4

A general problem raised by the work of Kreweras and Loeb concerns the existence of partitions of runs of consecutive integers into sets, known as sets of iterated differences, the s(s + 1)/2 elements of which can be expressed as certain linear combinations of s integer valued parameters. In this paper we study the case of four parameters. By examining properties already present in examples with only a single set, we build up the rudiments of an arithmetic of these partitions comparable to that for perfect systems of difference sets and complete permutations.

Further results on irregular, critical perfect systems of difference sets I: split systems

Abstract An (m, n;u, v;c)-system is a collection of components, m of valency u – 1 and n of valency v − 1, whose difference sets form a perfect system with threshold c. If there is an (m, n;3, 6;c)-system, then m ⩾ 2c − 1; and if there is a (2c − 1, n; 3, 6; c)-system, then 2c − 1 ⩾ n. For all sufficiently large c, there are (2c − 1, n; 3, 6; c)-systems at least when n = 1 or 2 and, in particular, (2c minus; 1, 1; 3, 6;c)-systems which have a certain splitting property enabling them to be pulled apart nicely. We show here that if, for some c and n, there is a (2c − 1, n; 3, 6; c)-system which splits at 3c + 6n − 1, then, in the first place, c − 1 ⩾ n, and, secondly, there is a (2c∗ − 1, n; 3, 6, c∗)- system with a split at 3c∗ + 6n − 1 for all sufficiently large c∗ depending on c and n. We then confirm the existence of such split systems at least when n = 1, 5, 6 and 7, finding also that they do not exist for n = 2, 3 or 4. We discuss the bearing of these results on the study of critical perfect systems and on the multiplication theorem for these systems. Another approach to (2c − 1, n; 3, 6, c)-systems, including the cases n = 2, 3 and 4, is considered in the sequel.

Regular Perfect Systems of Sets of Iterated Differences

Fors?2, a set {a(i,j):1 ?j?s+1 ?i?s} wherea(1,j), 1? j?s, are some prescribed integers anda(i+1,j) =|a(i,j) ?a(i,j+1)|, for 1?i