Journal of Integer Sequences, Vol. 19 (2016), Article 16.7.3

Binomial Coefficients and Enumeration of Restricted Words

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


We derive partial solutions for a recently-posed problem of the enumeration of restricted words. We obtain several explicit formulas in which the number of restricted words is expressed in terms of the binomial coefficients. These results establish relations between the partial Bell polynomials and the binomial coefficients. In particular, we link the r-step Fibonacci numbers, the binomial coefficients, and the partitions of a positive integer into at most r parts. Also, we prove that several well-known classes of integers can be interpreted in terms of compositions. We finish the paper with an extension of a recent result about Euler-type identities for integer compositions.

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(Concerned with sequences A000027 A000125 A000217 A000270 A000290 A000326 A000384 A000566 A000578 A000930 A000931 A001590 A002411 A002414 A003269 A005251 A005252 A005253 A008778 A008779 A008998 A008999 A017817 A017898 A049856 A051937 A052541 A052917 A052920 A052927 A071675 A079960 A079976 A096000 A099524 A108750 A108758 A117760 A124304 A126030 A159284 A180177 A191238 A198295 A205553 and A253189.)

Received January 7 2016; revised versions received June 13 2016; June 28 2016; July 10 2016. Published in Journal of Integer Sequences, August 29 2016.

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