Integer sequences constructed by partitions of positive integers

In this thesis, a connection has been established between the partition theory and the Fibonacci Numbers which have attracted the attention of scientists from the past to the present. The partition of positive integers, which is the basis of the partition theory and their composition have been studied in detail. Especially in this study, sets have been defined for partitions and compositions and new operations have been defined on these sets. With the help of these operations, relationships have been established between the compositions of consecutive positive integers. Algebraic and combinatorial operations have been performed on the defined sets, and new relations have been obtained for the compositions. Some partitions in the literature, which have special cases such as partitions with limited number of summands, partitions with odd integer summands, partitions whose summands are odd and distinct integers, have also been examined for the concept of composition. Compositions, restricted compositions and odd compositions have been examined separately with their special cases and generating functions have been found for the number of related compositions. The relationship between odd compositions and Fibonacci numbers have been examined. Many number sequences such as Fibonacci numbers have been produced with the help of partition theory and generating functions, which are the focus of attention and the subject of study for many mathematicians today. The summand size of the compositions have been constrained, and the generating function, for the number of compositions constrained by this restriction, have been found. With this generating function, n-nacci numbers have been generated. In this thesis, a color catalogue has been created for the compositions, and patterns have been obtained for the compositions by adhering to this catalogue. While obtaining the patterns, different patterns have been examined by introducing different restrictions for the summands of the compositions. In addition, generating functions have been found for the number of these patterns. With these generating functions, new number sequences have been brought to the literature. It has been observed that the composition pattern of each positive integer is within the composition pattern of the integer that is one more than itself. It has been shown that there is a golden ratio between the n-color composition patterns of consecutive integers. As a result, the studies obtained in this thesis have a high potential to be used in the fields of mathematics, physics, computational science and engineering, and even architecture.