  This paper shows how the study of colored compositions of integers reveals
some unexpected and original connection with the Invert operator. The Invert
operator becomes an important tool to solve the problem of directly counting
the number of colored compositions for any coloration. The interesting
consequences arising from this relationship also give an immediate and simple
criterion to determine whether a sequence of integers counts the number of some
colored compositions. Applications to Catalan and Fibonacci numbers naturally
emerge, allowing to clearly answer to some open questions. Moreover, the
definition of colored compositions with the "black tie" provides
straightforward combinatorial proofs to a new identity involving multinomial
coefficients and to a new closed formula for the Invert operator. Finally,
colored compositions with the "black tie" give rise to a new combinatorial
interpretation for the convolution operator, and to a new and easy method to
count the number of parts of colored compositions.
