Class-uniformly resolvable designs with block sizes 2 and 3

In this article we consider restricted resolvable designs (RRP) with block sizes 2 and 3 that are class uniform. A characterization scheme is developed, based on the ratio a:b of pairs to triples, and necessary conditions are provided for the existence of these designs based on this characterization. We show asymptotic existence results when (a,b) = (1, 2n), n ≥ 1 and when (a,b) = (9, 2). We also study the specific cases when (a,b) = (1, 2n), 1 ≤ n ≤ 5, (a,b) = (3, 6u−2), u ≥ 1 and when (a,b) E {(1, 1), (3, 1), (7, 2), (3, 4), (9, 2)}. © 1996 John Wiley & Sons, Inc.     

Realizations for direct constructions of resolvable Steiner 2‐designs with block size 5

We consider direct constructions due to R. J. R. Abel and M. Greig, and to M. Buratti, for ({ν},5,1) balanced incomplete block designs. These designs are defined using the prime fields F_p for certain primes p, are 1‐rotational over G ⊕ F_p where G is a group of order 4, and are also resolvable under certain conditions. We introduce specifications to the constructions and, by means of character sum arguments, show that the constructions yield resolvable designs whenever p is sufficiently large. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:207–217, 2000     

Existence results for near resolvable designs

A near resolvable design, NRB(v, k), is a balanced incomplete block design whose block set can be partitioned into v classes such that each class contains every point of the design but one, and each point is missing from exactly one class. The necessary conditions for the existence of near resolvable designs are v ≡ 1 mod k and λ = k ? 1. These necessary conditions have been shown to be sufficient for k ? {2,3,4} and almost always sufficient for k ? {5,6}. We are able to show that there exists an integer n₀(k) so that NRB(v,k) exist for all v > n₀(k) and v ≡ 1 mod k. Using some new direct constructions we show that there are many k for which it is easy to compute an explicit bound on n₀(k). These direct constructions also allow us to build previously unknown NRB(v,5) and NRB(v,6). © 1995 John Wiley & Sons, Inc.     

General frame constructions for Z-cyclic triplewhist tournaments

Frames are useful in dealing with resolvable designs such as resolvable balanced incomplete block designs and triplewhist tournaments. Z-cyclic triplewhist tournament frames are also useful in the constructions of Z-cyclic triplewhist tournaments. In this paper, the concept of an (h₁,h₂,…,hn;u)-regular Z-cyclic triplewhist tournament frame is defined, and used to establish several quite general recursive constructions for Z-cyclic triplewhist tournaments. As corollaries, we are able to unify many known constructions for Z-cyclic triplewhist tournaments. As an application, some new Z-cyclic triplewhist tournament frames and Z-cyclic triplewhist tournaments are obtained. The known existence results of such designs are then extended.     

Pitch tournament designs and other BIBDs—existence results for the case ν = 8n

A pitch tournament is a resolvable or near resolvable(ν,8,7) BIBD that satisfies certain criteria in addition to theusual condition that ν ≡ 0 or 1 (mod 8). Here we establish that for the case ν = 8n the necessary condition forpitch tournaments is sufficient for all n > 1615, with at most 187 smaller exceptions. This complements our earlier study of the ν = 8n + 1 case, where we established sufficiency for all n > 224, with at most 28 smaller exceptions. The four missing cases for (ν,8,7) BIBDs are provided, namely ν∈{48,56,96,448}, thereby establishing that the necessary existence conditions are sufficient without exception. Some constructions for resolvable designs are also provided, reducing the existence question for (ν,8,7) RBIBDs to 21 possible exceptions. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 334–356, 2001     

On vector space partitions and uniformly resolvable designs

Let V _n (q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V _n (q) is a partition of V _n (q) if every nonzero vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

On the Convergence of Limit Periodic Continued Fractions K(an/1) where an→ - ¼. Part IV

Continued fractions K(a _n /b _n ) , where a _n , b _n ∈\smallbf C and a _n /b _n b _n-1 →-\frac 14 , may converge or diverge depending on how a _n /b _n b _n-1 approaches its limit. Due to equivalence transformations it suffices to study the special case where all b _n =1 . We shall prove that K(a _n /1) converges if a _n →-\frac 14 and there exists a set V\subseteq\smallbf C \cup{∈fty} with certain properties such that a _n /(1+V)\subseteq V for all n . We shall also summarize some other useful consequences of such value sets V . January 31, 2000. Date revised: July 28, 2000. Date accepted: August 16, 2000.     

The existence of doubly near resolvable (v,3,2)-BIBDs

The existence of doubly near resolvable (v,2,1)-BIBDs was established by Mullin and Wallis in 1975. In this article, we determine the spectrum of a second class of doubly near resolvable balanced incomplete block designs. We prove the existence of DNR(v,3,2)-BIBDs for v ≡ 1 (mod 3), v ≥ 10 and v ? {34,70,85,88,115,124,133,142}. The main construction is a frame construction, and similar constructions can be used to prove the existence of doubly resolvable (v,3,2)-BIBDs and a class of Kirkman squares with block size 3, KS₃(v,2,4). © 1994 John Wiley & Sons, Inc.     

On the Hamilton-Waterloo Problem

 The Hamilton-Waterloo problem asks for a 2-factorisation of K _v in which r of the 2-factors consist of cycles of lengths a ₁,a ₂,…,a _t and the remaining s 2-factors consist of cycles of lengths b ₁,b ₂,…,b _u (where necessarily ∑ _i=1 ^t a _i =∑ _j=1 ^u b _j =v). In this paper we consider the Hamilton-Waterloo problem in the case a _i =m, 1≤i≤t and b _j =n, 1≤j≤u. We obtain some general constructions, and apply these to obtain results for (m,n)∈{(4,6),(4,8),(4,16),(8,16),(3,5),(3,15),(5,15)}. Received: July 5, 2000     

Maximal resolvable packings and minimal resolvable coverings of triples by quadruples

Determination of maximal resolvable packing number and minimal resolvable covering number is a fundamental problem in designs theory. In this article, we investigate the existence of maximal resolvable packings of triples by quadruples of order v (MRPQS(v)) and minimal resolvable coverings of triples by quadruples of order v (MRCQS(v)). We show that an MRPQS(v) (MRCQS(v)) with the number of blocks meeting the upper (lower) bound exists if and only if v≡0 (mod 4). As a byproduct, we also show that a uniformly resolvable Steiner system URS(3, {4, 6}, {r₄, r₆}, v) with r₆≤1 exists if and only if v≡0 (mod 4). All of these results are obtained by the approach of establishing a new existence result on RH(6²ⁿ) for all n≥2. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 209–223, 2010     

Constructions for resolvable and related designs

Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with ₁=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.     

Extremal values of continuants and transcendence of certain continued fractions

We prove a criterion for the transcendence of continued fractions whose partial quotients are contained in a finite set {b₁,…,b_r} of positive integers such that the density of occurrences of b_i in the sequence of partial quotients exists for 1ir. As an application we study continued fractions [0,a₁,a₂,a₃,…] with a_n=1+([nθ]modd) where θ is irrational and d2 is a positive integer.     

Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line

We study ratio asymptotics, that is, existence of the limit of P_n+1(z)/P_n(z) (P_n= monic orthogonal polynomial) and the existence of weak limits of p_n² dμ (p_n=P_n/||P_n||) as n→∞ for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z₀ with Im(z₀)≠0 implies dμ is in a Nevai class (i.e., a_n→a and b_n→b where a_n,b_n are the off-diagonal and diagonal Jacobi parameters). For μ's with bounded support, we prove p_n² dμ has a weak limit if and only if lim b_n, lim a_2n, and lim a_2n+1 all exist. In both cases, we write down the limits explicitly.     

Rado Numbers fora(x+y)=bz

In the case of existence the smallest numberN=Ra_kis called a Rado number if it is guaranteed that anyk-coloring of the numbers 1, 2, …, Ncontains a monochromatic solution of a given system of linear equations. We will determine Ra_k(a, b) for the equationa(x+y)=bzifb=2 andb=a+1. Also, the case of monochromatic sequences {x_n} generated bya(x_n+x_n+1)=bx_n+2 is discussed.     

The existence of NRB[v,k] where k + 1 is an even prime power

The existence of NRB[v,k] where k ≥ 7 and k + 1 is an even prime power is considered. We will show that there exists an NRB[kn + 1, k] for all n > (3k)^b-1(kb)^3kb(b-1)+1, where k + 1 is an even prime power, k ≥ 7 and . The tools used to construct this bound include the frames extracted from a construction of J. X. Lu's for resolvable balanced incomplete block designs © 1998 John Wiley & Sons, Inc. J Combin Designs 6:43–49, 1998     

An elementary derivation of the asymptotics of partition functions

Let S_a,b = {an+b:n ≥ 0 } where n is an integer. Let P_a,b(n) denote the number of partitions of n into elements of S_a,b. In particular, we have the generating function,

Reconstituting the distribution of the components of a random Vector from distributions of certain statistics

Let (X₁, ..., X_n) be a random vector with independent components. It is proven in this paper that, under certain restrictions, the distributions of the pairS ₁=sup (a ₁X₁, ..., a_nX_n) andS ₂=sup (b₁X₁,...,b_nX_n) univocally define the distribution function of the components X_j.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 889–892, June, 1973.     

Constructions for near resolvable designs and BIBDs

The purpose of this article is twofold. First, it is shown that classical inversive planes of even order can be used to construct a class of 2—(2²ⁿ + 1, 2ⁿ, 2ⁿ—1) near resolvable designs, in which any two blocks have at most 2 points in common. Secondly, it is shown that a recursive construction method for BIBDs using resolvable BIBDs due to Shrikhande and Raghavarao can be extended by using near resolvable designs. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 227–231, 1999     

On the enumeration of parking functions by leading terms

Let x=(x₁,…,x_n) be a sequence of positive integers. An x-parking function is a sequence (a₁,…,a_n) of positive integers whose non-decreasing rearrangement b₁b_n satisfies b_ix₁++x_i. In this paper we give a combinatorial approach to the enumeration of (a,b,…,b)-parking functions by their leading terms, which covers the special cases x=(1,…,1), (a,1,…,1), and (b,…,b). The approach relies on bijections between the x-parking functions and labeled rooted forests. To serve this purpose, we present a simple method for establishing the required bijections. Some bijective results between certain sets of x-parking functions of distinct leading terms are also given.     

Attainability of the minimal exponential growth rate for free products of finite cyclic groups

We consider free products of two finite cyclic groups of orders 2 and n, where n is a prime power. For any such group ℤ₂ * ℤ _n = 〈a, b | a ² = b ⁿ = 1〉, we prove that the minimal growth rate α _n is attained on the set of generators {a, b} and explicitly write out an integer polynomial whose maximal root is α _n . In the cases of n = 3, 4, this result was obtained earlier by A. Mann. We also show that under sufficiently general conditions, the minimal growth rates of a group G and of its central extension [(G)\tilde]\tilde G coincide and that the attainability of one implies the attainability of the other. As a corollary, the attainability is proved for some cyclic extensions of the above-mentioned free products, in particular, for groups 〈a, b | a ² = b ⁿ 〉, which are groups of torus knots for odd n.