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Abstract. In this paper, we consider the Fibonacci conditional sequence ff n g and the Lucas conditional sequence fl n g. We derive some properties of Fibonacci and Lucas conditional sequences by using the matrix method. Our results are elegant as the results for ordinary Fibonacci and Lucas sequences.

In a recent paper, we give 13-dissection and some congruences for modulo 13 for the partition generating function (1 − q r ) −1 by using a method of Kolberg. In this paper, by following similar course, we develop an algoritmic approach and give 11-dissection for the partition generating function (1 − q r ) −1 . Then we re-obtain the congruences given by Atkin and Swinnerton-Dyer. Mathematics Subject Classification: 11P83

Abstract. Let N (r, m, n) (resp. M (r, m, n)) denote the number of partitions of n whose ranks (resp. cranks) are congruent to r modulo m. Atkin and Swinnerton-Dyer gave the relations between the numbers N (r, m, mn+k) when m = 5, 7 and 0 ≤ r, k < m. Garvan gave the relations between the numbers M (r, m, mn + k) when m = 5, 7, and 11, 0 ≤ r, k < m. Here, we show that the methods of Atkin and Swinnerton-Dyer can be extended to prove the relations for the crank.

Garvan first defined certain``vector partitions'' and assigned to each such partition a``rank.'' Denoting by N V (r, m, n) the (weighted) count of the vector partitions of n with rank r modulo m, he gave a number of relations between the numbers N V (r, m, mn+k) when m=5, 7 and 11, 0 r, k