Colored compositions, Invert operator and elegant compositions with the “black tie”

Abstract

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.

Fibonacci colored compositions and applications

“…This means that (W n ) n∈N is the invert transform of (w n ) n∈N . Colored compositions appear naturally in a wide variety of combinatorial problems and are therefore of special interest (see e.g [1,2,3,5,6,10]). For a comprehensive study of compositions, we refer to the book by S. Heubach and T. Mansour [4].…”

The combinatorics of weighted vector compositions

“…We call a vector composition f -weighted, for a function f : N N → N, when each part of 'size' m may occur in one of f (m) different colors in the composition. For example, for N = 2 and f : and f (x) = 0 for all other x ∈ N 2 , there are seven distinct f -weighted vector compositions of ℓ = (1, 2), namely: where ♦ distinguishes between the two values of (1,1). For fixed number k ≥ 0 of parts, we denote the number of distinct f -weighted vector compositions of ℓ ∈ N N by k ℓ f .…”

“…where ♦ distinguishes between the two values of (1,1). For fixed number k ≥ 0 of parts, we denote the number of distinct f -weighted vector compositions of ℓ ∈ N N by k ℓ f .…”

“…and analogously for P (S(ℓ)) (ℓ). 1 For a scalar a and a vector b, we write a|b, when a|b i for all components b i of b.…”

“…Recently, they have attracted attention in the form of so-called scolor compositions, for which f is specified as identity function, that is, f (s) = s [2,21,32,39,41]. More general f have been considered in [1,7,12,17,16,25,40], to name just a few. Results on standard integer compositions, i.e., where f is the indicator function on N − {0} or a subset thereof, are found in [23].…”