Václav Linek

Extended Langford Sequences with Small Defects

Ak-extended Langford sequence of defectdand lengthmis a sequences1,s2,?,s2m+1in whichsk=?, where?is the null symbol, and each other member of the sequence comes from the set {d,d+1,?,d+m?1}. Eachj?{d,d+1,?,d+m?1} occurs exactly twice in the sequence, and the two occurrences are separated by exactlyj?1 symbols. In this paper we prove that whend=2,3, the necessary conditions for the existence of such a sequence are sufficient.

Hooked k -extended Skolem sequences

Abstract A hooked k-extended Skolem sequence of order n is a sequence s1s2…s2n+2 in which sk = s2n+1 = e (e is the null symbol) and each j ϵ {1, 2, …, n} occurs exactly twice, the two occurrences separated by exactly j − 1 symbols. It is proved that, with the exception of (k, n) = (2, 1), such a sequence exists if and only if n ≡ 0, 1 (mod 4) for k even, and n ≡ 2, 3 (mod 4) for k odd. This result is then used to give an alternative proof of the existence of bicyclic Steiner triple systems.

The existence of (p,q)-extended Rosa sequences

A (p,q)-extended Rosa sequence is a sequence of length 2n+2 containing each of the symbols 0,1,...,n exactly twice, and such that two occurrences of the integer j>0 are separated by exactly j-1 symbols. We prove that, with two exceptions, the conditions necessary for the existence of a (p,q)-extended Rosa sequence with prescribed positions of the symbols 0 are sufficient. We also extend the result to @l-fold (p,q)-extended Rosa sequences; i.e., the sequences where every pair of numbers is repeated exactly @l times.

Note Concerning an Odd Langford Sequence

It is shown that the set {1, 2,? , 2 n+ 3} ? {p } can be partitioned into differences 1, 3,? , 2 n+ 1 precisely whenn? 1, p is odd and (n, p) ?= (1, 3). All sets whose elements are in arithmetic progression and which can be partitioned into differences that are again in arithmetic progression are classified.

HOOKED EXTENDED LANGFORD SEQUENCES OF SMALL AND LARGE DEFECTS

It is shown that for m = 2d +5, 2d+6, 2d+7 and d ≥ 1, the set {1, …, 2m + 1} − {k} can be partitioned into differences d, d + 1, …, d + m − 1 whenever (m, k) ≡ (0, 1), (1, d), (2, 0), (3, d+1) (mod (4, 2)) and 1 ≤ k ≤ 2m+1.It is also shown that for m = 2d + 5, 2d + 6, 2d + 7, and d ≥ 1, the set {1, …, 2m + 2} − {k, 2m + 1} can be partitioned into differences d, d + 1, … …, d + m − 1 whenever (m, k) ≡ (0, 0), (1, d + 1), (2, 1), (3, d) (mod (4, 2)) and k ≥ m + 2.These partitions are used to show that if m ≥ 8d + 3, then the set {1, … …, 2m+2}−{k, 2m+1} can be partitioned into differences d, d+1, …, d+m−1 whenever (m, k) ≡ (0, 0), (1, d+1), (2, 1), (3, d) (mod (4, 2)).A list of values m, d that are open for the existence of these partitions (which are equivalent to the existence of Langford and hooked Langford sequences) is given in the conclusion.