Compositions of N with No Occurrence of K
نویسنده
چکیده
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 1by-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.
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