Compositions of n with no occurrence of k

A composition of n is an ordered collection of one or more positive integers whose sum is n. The number of summands is called the number of parts of the composition. A palindromic composition or palindrome is a composition in which the summands are the same in the given or in reverse order. Compositions may be viewed as tilings of 1-by-n rectangles with 1-by-i rectangles, 1 i n ≤ ≤. We count the number of compositions and the number of palindromes of n that do not contain any occurrence of a particular positive integer k. We also count the total number of occurrences of each positive integer among all the compositions of n without occurrences of k. This counting problem corresponds to the number of rectangles of each allowable size among the tilings of length n without 1-by-k tiles. Finally we count the number of compositions without k having a fixed number of parts, and explore some patterns involving the number of parts in compositions without k. 1. Introduction A composition of n is an ordered collection of one or more positive integers whose sum is n. The number of summands is called the number of parts of the composition. A palindromic composition or palindrome is a composition in which the summands are the same in the given or in reverse order. Compositions may be viewed as tilings of 1-by-n rectangles with 1-by-i rectangles, 1 i n ≤ ≤. In this view, a palindromic composition is one corresponding to a symmetric tiling. Because of the relation of compositions to tilings, we sometimes refer to a composition of n as a composition of length n. Grimaldi [5] explores the question of how many compositions of n exist when no 1's are allowed in the composition. In [4], the authors explore the question of how many compositions of n exist when no 2's are allowed in the composition. In this paper we explore the general question of how many compositions of n exist when no k's are allowed in the composition. Related to this question we will also explore how many of these compositions are palindromes. We count the number of compositions and the number of palindromes without k, as well as the total number of occurrences of each positive integer among all the compositions of n with no k's. The preceding two counting problems correspond respectively to the number of 1-by-n …