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

Open Mathematics (2012)

• Volume: 10, Issue: 2, page 788-796
• ISSN: 2391-5455

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## Abstract

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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.

## How to cite

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Aubrey Blecher, Charlotte Brennan, and Toufik Mansour. "Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions." Open Mathematics 10.2 (2012): 788-796. <http://eudml.org/doc/269231>.

@article{AubreyBlecher2012,
abstract = {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.},
author = {Aubrey Blecher, Charlotte Brennan, Toufik Mansour},
journal = {Open Mathematics},
keywords = {Compositions; Partitions; Generating functions; compositions; partitions; generating functions},
language = {eng},
number = {2},
pages = {788-796},
title = {Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions},
url = {http://eudml.org/doc/269231},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Aubrey Blecher
AU - Charlotte Brennan
AU - Toufik Mansour
TI - Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 788
EP - 796
AB - 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.
LA - eng
KW - Compositions; Partitions; Generating functions; compositions; partitions; generating functions
UR - http://eudml.org/doc/269231
ER -

## References

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1. [1] Andrews G., Eriksson K., Integer Partitions, Cambridge University Press, Cambridge, 2004 Zbl1073.11063
2. [2] Andrews G., Concave compositions, Electron. J. Combin., 2011, 18(2), #6 Zbl1229.05029
3. [3] Blecher A., Compositions of positive integers n viewed as alternating sequences of increasing/decreasing partitions, Ars Combin. (in press) Zbl1289.05015
4. [4] Heubach S., Mansour T., Combinatorics of Compositions and Words, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, 2010 Zbl1184.68373
5. [5] MacMahon P., Combinatory Analysis, Cambridge University Press, Cambridge, 1915–1916, reprinted by Chelsea, New York, 1960
6. [6] Mansour T., Shattuck M., Yan S.H.F., Counting subwords in a partition of a set, Electron. J. Combin., 2010, 17(1), #19 Zbl1193.05025
7. [7] Stanley R., Enumerative Combinatorics. I, Cambridge Stud. Adv. Math., 49, Cambridge University Press, Cambridge, 1997

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