On The Extended Patterns in a Simion-Schmidt's Bijection
Asep Juarna · Gunadarma University Repository (Gunadarma University) · 2006
In 1985 Simion and Schmidt gave a constructive bijection çô I F,_i —‘ S(123, 132,213), where F_1 is the set of all length (n — 1) binary strings having no two consecutive is, also known as the set of Fibonacci strings (of 2—tid order), and S(123, 132,213) is the set of all permutations of { 1, 2,. . . , n} that avoid all patterns in the set { 123, 132, 213}. In this paper we extend the set of patterns {123, 132, 213} while to generalize the domain (n—i) such that we get three following new bijections: : F_1 —, S(12.. .p, 132,213), : S(123,1p(p— 1)...2,213), and : —, S(123,132,(p—1)(p—2)...p), where ç actually is exactly same as the original mapping p due to Simion-Schmidt. Furthermore, we show that the three bijections are actually coml)inatorial isomorphisms. i.e., closeness )reserviI1g bijections. Since each domain of the l)ijections has known Gray code, therefore, through the corresponding combinatorial isomorphism. we construct similar Gray code for each of corresponding codomain.