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Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

100%

Blecher A., Brennan C., Mansour T.

Open Mathematics

2012

tom 10

nr 2

788-796

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”. From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.

Ascents of size less than d in compositions

75%

Falah M., Mansour T.

Open Mathematics

2011

tom 9

nr 1

196-203

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.

Ograniczanie wyników

2 Open Mathematics

2 Mansour T.

1 Blecher A.

1 Brennan C.

1 Falah M.

1 2012

1 2011

Wyniki wyszukiwania

Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Ascents of size less than d in compositions