A composition of a positive integer n is a finite sequence π1π2...πm of positive integers such that π1+...+πm = n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if πi+1 ≥ πi+d (respectively, πi < πi+1 < πi + d). Recently, Brennan and Knopfmacher determined the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. In this paper, we find an explicit formula for the multi-variable generating function for the number of compositions of n according to the number of parts, ascents of size d or more, ascents of size less than d, descents and levels. Also, we extend the results of Brennan and Knopfmacher to the case of ascents of size less than d. More precisely, we determine the mean, variance and limiting distribution of the number of ascents of size less than d in the set of compositions of n.
@article{bwmeta1.element.doi-10_2478_s11533-010-0078-4,
author = {Maisoon Falah and Toufik Mansour},
title = {Ascents of size less than d in compositions},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {196-203},
zbl = {1229.05018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0078-4}
}
Maisoon Falah; Toufik Mansour. Ascents of size less than d in compositions. Open Mathematics, Tome 9 (2011) pp. 196-203. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0078-4/
[1] Brennan C., Knopfmacher A., The distribution of ascents of size d or more in compositions, Discrete Math. Theor. Comput. Sci., 2009, 11(1), 1–10 | Zbl 1193.68179
[2] Carlitz L., Restricted compositions, Fibonacci Quart., 1976, 14(3), 254–264 | Zbl 0338.05005
[3] Flajolet P., Prodinger H., Level number sequences for trees, Discrete Math., 1987, 65(2), 149–156 http://dx.doi.org/10.1016/0012-365X(87)90137-3
[4] Flajolet P., Sedgewick R., Analytic Combinatorics, Cambridge University Press, Cambridge, 2009
[5] Goulden I.P., Jackson D.M., Combinatorial Enumeration, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1983
[6] Heubach S., Mansour T., Counting rises, levels, and drops in compositions, Integers, 2005, 5(1), A11 | Zbl 1077.05003
[7] Heubach S., Mansour T., Combinatorics of Compositions and Words, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, 2009 | Zbl 1184.68373
[8] Knopfmacher A., Prodinger H., On Carlitz compositions, European J. Combin., 1998, 19(5), 579–589 http://dx.doi.org/10.1006/eujc.1998.0216 | Zbl 0902.05004