# Ascents of size less than d in compositions

Open Mathematics (2011)

• Volume: 9, Issue: 1, page 196-203
• ISSN: 2391-5455

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## Abstract

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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.

## How to cite

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Maisoon Falah, and Toufik Mansour. "Ascents of size less than d in compositions." Open Mathematics 9.1 (2011): 196-203. <http://eudml.org/doc/269765>.

@article{MaisoonFalah2011,
abstract = {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.},
author = {Maisoon Falah, Toufik Mansour},
journal = {Open Mathematics},
keywords = {Compositions; Distributions; Generating functions; Ascents; Descents; Levels; compositions; ascents; descents; generating functions},
language = {eng},
number = {1},
pages = {196-203},
title = {Ascents of size less than d in compositions},
url = {http://eudml.org/doc/269765},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Maisoon Falah
AU - Toufik Mansour
TI - Ascents of size less than d in compositions
JO - Open Mathematics
PY - 2011
VL - 9
IS - 1
SP - 196
EP - 203
AB - 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.
LA - eng
KW - Compositions; Distributions; Generating functions; Ascents; Descents; Levels; compositions; ascents; descents; generating functions
UR - http://eudml.org/doc/269765
ER -

## References

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1. [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 Zbl1193.68179
2. [2] Carlitz L., Restricted compositions, Fibonacci Quart., 1976, 14(3), 254–264 Zbl0338.05005
3. [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. [4] Flajolet P., Sedgewick R., Analytic Combinatorics, Cambridge University Press, Cambridge, 2009
5. [5] Goulden I.P., Jackson D.M., Combinatorial Enumeration, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1983
6. [6] Heubach S., Mansour T., Counting rises, levels, and drops in compositions, Integers, 2005, 5(1), A11 Zbl1077.05003
7. [7] Heubach S., Mansour T., Combinatorics of Compositions and Words, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, 2009 Zbl1184.68373
8. [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 Zbl0902.05004

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