Relations among special numbers, special functions, and de montmort numbers (Derangement numbers), by permutations of the elements of a set

In this thesis, the concept of Derangement, which holds a significant place in mathematics, particularly in the Theory of Combinatorial Analysis, is examined. A derangement is defined as a permutation of a set in which no element appears in its original position. In other words, derangements are permutations without fixed points. These numbers were first studied in 1713 by the French mathematician Pierre Rémond de Montmort \cite{montmort1713}, and after his death, they were sometimes referred to as "de Montmort numbers." Today, however, they are more commonly known as derangement numbers. This thesis investigates derangement numbers and polynomials, exploring their relationships with other special numbers and polynomials through new identities and associated graphs. The focus is particularly on their connections with generating functions, permutations, and graphical representations. Initially, permutation groups are discussed, followed by an introduction to the family of derangement numbers and polynomials. Several new equations involving generating functions are presented. Using these equations, the relationships between derangement numbers/polynomials and other special number families are examined. Furthermore, the study includes derivatives, integrals, various properties, graphical representations, and numerical tables related to the relevant polynomial families.