Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Permutation Patterns and Continued Fractions. | Permutation Patterns and Continued Fractions Aaron Robertson Colgate University Hamilton NY 13346 aaron@ Herbert S. Wilf University of Pennsylvania Philadelphia PA 19104-6395 wilf@ Doron Zeilberger Temple University Philadelphia PA 19122 zeilberg@ Abstract We find in the form of a continued fraction the generating function for the number of 132 -avoiding permutations that have a given number of 123 patterns and show how to extend this to permutations that have exactly one 132 pattern. We also find some properties of the continued fraction which is similar to though more general than those that were studied by Ramanujan. THE ELECTRONIC .JOURNAL OF COmBINATORICS 6 1999 R38 2 A 132 pattern resp. a 123 pattern in a permutation K of k letters is a triple 1 i j k n of indices for which K i K k K j resp. K i K j K k . Let fr n denote the number of permutations K of n letters that have no 132 patterns and exactly r 123 patterns. Our main result is the following. Theorem 1 The generating function for the fr n is E fr n z . -------- -------- r n 0 1z_ z 1-------------- 1 q 1 1 - zf 1 in which the nth numerator is zq 2 . We think it is remarkable that such a continued fraction encodes information about 132 -avoiding permutations. We will hrst prove the theorem and then study some consequences and generalizations. 1 The patterns Let the weight of a permutation K of k letters be A ql123 dtl12O l in which 123 k is the number of 123 patterns rising triples in K and 12 k is the number of rising pairs in K. Let P q z Í X weight K 2 where the sum extends over all 132 -avoiding permutations K. If K is a 132 -avoiding permutation on 1 2 . n n 0 and the largest element n is at the kth position . K k n then by letting K1 K i g_1 and K2 K i n 1 we have that every element in K1 must be larger than every element of K2 or else a 132 would be formed with the n serving as the 3 of the 132 . Hence K1 is a permutation of the
