Enumeration of 3-letter Patterns in Compositions
نویسندگان
چکیده
Let A be any set of positive integers and n ∈ N. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with m parts in A that have r occurrences of 3-letter patterns formed by two (adjacent) instances of levels, rises and drops. We also derive asymptotics for the number of compositions of n that avoid a given pattern. Finally, we obtain the generating function for the number of k-ary words of length m which contain a prescribed number of occurrences of a given pattern as a special case of our results.
منابع مشابه
Counting l-letter subwords in compositions
Let N be the set of all positive integers and let A be any ordered subset of N. Recently, Heubach and Mansour enumerated the number of compositions of n with m parts in A that contain the subword τ exactly r times, where τ ∈ {111, 112, 221, 123}. Out aims are (1) to generalize the above results, i.e., to enumerate the number of compositions of n with m parts in A that contain an l-letter subwor...
متن کاملCounting `-letter subwords in compositions
Let N be the set of all positive integers and let A be any ordered subset of N. Recently, Heubach and Mansour enumerated the number of compositions of nwithm parts inA that contain the subword τ exactly r times, where τ ∈ {111, 112, 221, 123}. Our aims are (1) to generalize the above results, i.e., to enumerate the number of compositions of n with m parts in A that contain an `-letter subword, ...
متن کاملConsecutive Patterns: From Permutations to Column-Convex Polyominoes and Back
We expose the ties between the consecutive pattern enumeration problems as sociated with permutations, compositions, column-convex polyominoes, and words. Our perspective allows powerful methods from the contexts of compositions, column convex polyominoes, and of words to be applied directly to the enumeration of per mutations by consecutive patterns. We deduce a host of new consecutive patt...
متن کاملThe Frequency of Summands of a Particular Size in Palindromic Compositions
A composition of a positive integer n consists of an ordered sequence of positive integers whose sum is n. A palindromic composition is one for which the sequence is the same from left to right as from right to left. This paper shows various ways of generating all palindromic compositions, counts the number of times each integer appears as a summand among all the palindromic compositions of n, ...
متن کاملRestricted Dumont permutations, Dyck paths, and noncrossing partitions
We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that use...
متن کامل