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• # Artykuł - szczegóły

## Open Mathematics

2011 | 9 | 1 | 196-203

## Ascents of size less than d in compositions

EN

### Abstrakty

EN
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.

EN

196-203

wydano
2011-02-01
online
2010-12-30

### Twórcy

autor
• University of Haifa
autor
• University of Haifa

### Bibliografia

• [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
• [2] Carlitz L., Restricted compositions, Fibonacci Quart., 1976, 14(3), 254–264
• [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
• [7] Heubach S., Mansour T., Combinatorics of Compositions and Words, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, 2009
• [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