Colored compositions, Invert operator and elegant compositions with the 'black tie'

摘要

This paper shows how the study of colored compositions of integers revealssome unexpected and original connection with the Invert operator. The Invertoperator becomes an important tool to solve the problem of directly countingthe number of colored compositions for any coloration. The interestingconsequences arising from this relationship also give an immediate and simplecriterion to determine whether a sequence of integers counts the number of somecolored compositions. Applications to Catalan and Fibonacci numbers naturallyemerge, allowing to clearly answer to some open questions. Moreover, thedefinition of colored compositions with the "black tie" providesstraightforward combinatorial proofs to a new identity involving multinomialcoefficients and to a new closed formula for the Invert operator. Finally,colored compositions with the "black tie" give rise to a new combinatorialinterpretation for the convolution operator, and to a new and easy method tocount the number of parts of colored compositions.