16 pages

There are n travellers who have k bicycles and they wish to complete a journey in the shortest possible time. We investigate optimal solutions of this problem where each traveller cycles for k n of the journey. Each solution is represented by an n × n binary matrix M with k non-zero entries in each row and column. We determine when such a matrix gives an optimal solution. This yields an algorithm deciding the question of optimality of complexity O(n 2 log n). We introduce three symmetries of matrices that preserve optimality, allowing identification of minimal non-optimal members of this class. An adjustment to optimal solutions that eliminates unnecessary handovers of cycles is established, which maintains all other features of the solution. We identify two mutually transpose solution types, the first uniquely minimises the number of handovers, while the second keeps the number of separate cohorts to three while bounding their overall separation, in the case 2k ≤ n, to under 2 n of the journey. There are n friends who have k bicycles between them and the group needs to reach its destination as soon as possible. How should they go about doing this? An early allusion to this problem is in the novel The Great House by Cynthia Harnett [2] . Here a pair of 17th century travelling companions with only one horse between them adopt the 'ride and tie' method for their journey from Henley-on-Thames to London. Assumptions Every person walks and cycles at the same speed as all the others, and cycling is faster than walking. We assume that the time required to swap from one form of locomotion to the other is negligible. For brevity, individual travellers will sometimes be referred to as 'he' while a set of travellers will be referred to as 'they'. Solution Suppose that we devise a scheme, we shall call it an optimal scheme, in which each traveller cycles for k n of the length of the journey and never stops moving forward at any stage. Each will then have cycled and walked the same distance as each of their companions and so all n friends will arrive at their destination simultaneously. We claim that, if it exists, such a scheme is truly optimal in that it delivers the entire group to its destination in the least possible time, and that any non-optimal scheme is inferior in this respect.

We call a pair (m, f ) of integers, m ≥ 1, 0 ≤ f ≤ m 2 , absolutely avoidable if there is n 0 such that for any pair of integers (n, e) with n > n 0 and 0 ≤ e ≤ n 2 there is a graph on n vertices and e edges that contains no induced subgraph on m vertices and f edges. Some pairs are clearly not absolutely avoidable, for example (m, 0) is not absolutely avoidable since any sufficiently sparse graph on at least m vertices contains independent sets on m vertices. Here we show that there are infinitely many absolutely avoidable pairs. We give a specific infinite set M such that for any m ∈ M , the pair (m, m 2 /2) is absolutely avoidable. In addition, among other results, we show that for any monotone integer function q(m), |q(m)| = O(m), there are infinitely many values of m such that the pair (m, m 2 /2 + q(m)) is absolutely avoidable.

On the size of an r

In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each class and is given by the entries of a new Schroder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim. We employ a variety of techniques to prove our results, including generating trees, direct bijections and the kernel method. For the latter, we make use of in a creative way what we are trying to show in three cases to aid in solving a system of functional equations satisfied by the associated generating functions.