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

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 determinewhether 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. © 2014 Elsevier B.V. All rights reserved. 1. Colored compositions and the Invert operator The compositions of integer numbers correspond to ordered partitions [14] in the following sense: any composition of an integer n is a sequence of integers (called parts) whose sum is n, univocally determined by the order of its parts. In [1] and [2], n-color partitions and n-color compositions have been introduced, respectively. In an n-color composition a part of sizem occurs withm different colors, i.e., we havem different parts of sizem. Recently, colored compositions have been studied in different works, see, e.g., [3,11,4,12]. In the following, we extend the study to general colored compositions where each part can occur with any number of colors (i.e., a part of size m can occur with j different colors, where j can be any integer number). We give combinatorial interpretations of colored compositions from a different point of view by using the Invert operator, the ordinary complete Bell polynomials and linear recurrence sequences. Let us fix some definitions. Definition 1. Wedefine the coloration X to be the sequence X = (xi)∞i=1 of non-negative integers, where any xi is the number of colors of the part i. If xi = 0, it means that we do not use the integer i in the compositions. Moreover, we define • L(X) as the set of all colored compositions with coloration X; we call an element of this set a composition (for shortness) and indicate it with bold letters, e.g., b ∈ L(X) ∗ Corresponding author. E-mail addresses:[email protected] (M. Abrate), [email protected] (S. Barbero), [email protected] (U. Cerruti), [email protected], [email protected] (N. Murru). http://dx.doi.org/10.1016/j.disc.2014.06.026 0012-365X/© 2014 Elsevier B.V. All rights reserved. 2 M. Abrate et al. / Discrete Mathematics 335 (2014) 1–7 • Ln(X) as the set of colored compositions of nwith coloration X • An(X) = |Ln(X)|; A = A(X) = (An(X))∞n=1 is the sequence of the number of colored compositions of nwith coloration X , for n = 1, 2, . . . • p(b) as the number of the parts of the composition b • Pn(X) =  b∈Ln(X) p(b) • r(b) = p(b) + 1 as the number of break-points • Rn(X) =  b∈Ln(X) r(b). Remark 1. Let us observe that An(X) can be viewed as a polynomial in x1, . . . , xn (since parts of size greater than n cannot be used in the composition of n). Thus, sometimes we will write An(x1, . . . , xn) instead of An(X). Moreover, we set A0(X) = 1, for any coloration X , meaning that we can compose the number 0 only if we do not use any composition. Definition 2. The Invert operator I transforms a sequence a = (an)∞n=0 into a sequence b = (bn) ∞ n=0 as follows: