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

Some Formulas for Numbers of Restricted Words

Milan Janjić
Department of Mathematics and Informatics
University of Banja Luka
Banja Luka, 78000
Republic of Srpska, Bosnia and Herzegovina


For an arithmetic function f0, we consider the number cm(n,k) of weighted compositions of n into k parts, where the weights are the values of the (m-1)th invert transform of f0. We connect cm(n,k) with c1(n,k) via Pascal matrices. We then relate cm(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 cm(n,k).

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(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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