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

• Autorzy: Aubrey Blecher , Charlotte Brennan , Toufik Mansour
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Compositions; Partitions; Generating functions

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 788-796

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Aubrey Blecher

• University of the Witwatersrand,

autor

Charlotte Brennan

• University of the Witwatersrand,

autor

Toufik Mansour

• University of Haifa,

Bibliografia

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

Identyfikatory

DOI

10.2478/s11533-011-0100-5