Geodesic
Binomial coefficients and enumeration of restricted words
Journal of integer sequences, Tome 19 (2016) no. 7
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.
Classification : 05A10, 11B39
Keywords: binary word, integer composition, restricted word, enumeration 
@article{JIS_2016__19_7_a2,
     author = {Janji\'c,  Milan},
     title = {Binomial coefficients and enumeration of restricted words},
     journal = {Journal of integer sequences},
     year = {2016},
     volume = {19},
     number = {7},
     zbl = {1348.05016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_7_a2/}
}
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Janjić,  Milan. Binomial coefficients and enumeration of restricted words. Journal of integer sequences, Tome 19 (2016) no. 7. http://geodesic.mathdoc.fr/item/JIS_2016__19_7_a2/