Please cite this article in press as: Z. Ahmed et al., New congruences modulo 5 for the number of 2-color partitions, J. Number Theory (2015), http://dx.Abstract. Let p k (n) be the number of 2-color partitions of n where one of the colors appears only in parts that are multiples of k. In this paper, we find some interesting congruences modulo 5 for p k (n) for k ∈ {2, 3, 4} by employing Ramanujan's theta function identities and some identities for the Rogers-Ramanujan continued fraction. The congruence for p 2 (n) was earlier proved by Chen and Lin with the aid of modular forms.
The bottleneck traveling salesman problem is to find a Hamiltonian circuit that minimizes the largest cost of any of its arcs in a graph. A simple genetic algorithm (GA) using sequential constructive crossover has been developed to obtain heuristic solution to the problem. The hybrid GA incorporates 2-opt search, another proposed local search and immigration to the simple GA for obtaining better solution. The efficiency of our hybrid GA to the problem against two existing heuristic algorithms has been examined for some symmetric TSPLIB instances. The comparative study shows the effectiveness of our hybrid algorithm. Finally, we present solutions to the problem for asymmetric TSPLIB instances.
ACM Reference Format:Ahmed, Z. H. 2013. A hybrid genetic algorithm for the bottleneck traveling salesman problem. ACM Trans.
Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function C k,i (n) which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. He also proved that C 3,1 (9n + 3) ≡ C 3,1 (9n + 6) ≡ 0 (mod 3). Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for C 3,1 (n), C 6,1 (n) and C 6,2 (n). In this paper, we find new congruences for C 3,1 (n) modulo 4, 18 and 36, infinite families of congruences modulo 2 and 4 for C 8,2 (n), congruences modulo 2 and 3 for C 12,2 (n), C 12,4 (n), and congruences modulo 2 for C 24,8 (n) and C 48,16 (n). We use simple p-dissections of Ramanujan's theta functions.
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