Grid Classes and the Fibonacci Dichotomy for Restricted Permutations
نویسندگان
چکیده
We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length n in a permutation class is either at least as large as the nth Fibonacci number or is eventually polynomial.
منابع مشابه
On the effective and automatic enumeration of polynomial permutation classes
The “Fibonacci Dichotomy” of Kaiser and Klazar [15] was one of the first general results on the enumeration of permutation classes. It states that if there are fewer permutations of length n in a permutation class than the nth Fibonacci number, for any n, then the enumeration of the class is given by a polynomial for sufficiently large n. Since the Fibonacci Dichotomy was established for permut...
متن کاملPermutation Statistics and q-Fibonacci Numbers
In a recent paper, Goyt and Sagan studied distributions of certain set partition statistics over pattern restricted sets of set partitions that were counted by the Fibonacci numbers. Their study produced a class of q-Fibonacci numbers, which they related to q-Fibonacci numbers studied by Carlitz and Cigler. In this paper we will study the distributions of some Mahonian statistics over pattern r...
متن کاملRestricted Permutations, Fibonacci Numbers, and k-generalized Fibonacci Numbers
In 1985 Simion and Schmidt showed that the number of permutations in Sn which avoid 132, 213, and 123 is equal to the Fibonacci number Fn+1. We use generating function and bijective techniques to give other sets of pattern-avoiding permutations which can be enumerated in terms of Fibonacci or k-generalized Fibonacci numbers.
متن کاملRestricted Permutations Related to Fibonacci Numbers and k-Generalized Fibonacci Numbers
A permutation π ∈ Sn is said to avoid a permutation σ ∈ Sk whenever π contains no subsequence with all of the same pairwise comparisons as σ. In 1985 Simion and Schmidt showed that the number of permutations in Sn which avoid 123, 132, and 213 is the Fibonacci number Fn+1. In this paper we generalize this result in two ways. We first show that the number of permutations which avoid 132, 213, an...
متن کاملSymmetric Permutations Avoiding Two Patterns ∗
Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D4, the symmetry group of a square. We examine pattern-avoiding permutations with 180◦ rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our resu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Electr. J. Comb.
دوره 13 شماره
صفحات -
تاریخ انتشار 2006