Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.5 |

Department of Mathematics and Informatics

University of Banja Luka

Banja Luka, 78000

Republic of Srpska, Bosnia and Herzegovina

**Abstract:**

For an arithmetic function *f*_{0},
we consider the number *c*_{m}(*n*,*k*) of weighted compositions of *n* into *k* parts, where the weights are the values of the (*m*-1)^{th} invert transform of *f*_{0}. We connect *c*_{m}(*n*,*k*) with *c*_{1}(*n*,*k*) via Pascal matrices. We then relate *c*_{m}(*n*,*k*) to the number of certain restricted words over a finite alphabet. In addition, we develop a method which transfers some properties of restricted words over a finite alphabet to words over a larger alphabet.

Several examples illustrate our findings. Some examples concern binomial coefficients and Fibonacci numbers. Some examples also extend the classical results about weighted compositions of Hoggatt and Lind. In each example, we derive an explicit formula for *c*_{m}(*n*,*k*).

(Concerned with sequences A000027 A006130 A030528 A037027 A054456 A125662 A154929 A207823 A207824 A249139.)

Received October 3 2016; revised versions received March 26 2017; June 1 2017.
Published in *Journal of Integer Sequences*, June 25 2017.

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