Five guidelines for partition analysis with applications to lecture hall-type theorems. (English) Zbl 1182.11049
Landman, Bruce (ed.) et al., Combinatorial number theory. Proceedings of the 2nd integers conference 2005 in celebration of the 70th birthday of Ronald Graham, Carrollton, GA, USA, October 27--30, 2005. Berlin: Walter de Gruyter (ISBN 978-3-11-019029-8/hbk). 131-155 (2007) and Integers 7, No. 2, Paper A9 (2007).
This investigation continues the authors’ earlier work [Ramanujan J. 8, No. 3, 357--381 (2004; Zbl 1071.05007
)] on nonnegative integer solutions to linear inequalities as they relate to enumeration of integer partitions and compositions. Five guidelines are developed to compute the generating function for the solutions, using an approach that emphasizes deriving recurrences. In their previous papers they used an invertible matrix $C$ that handled the cases of Hickerson partitions, partitions with $r$th differences nonnegative, and various others. For some cases this “$C$-matrix” approach fails, so here they develop the guidelines to derive a recurrence for the generating function of an infinite family of constraint systems. This is applied to the problem of enumerating anti-lecture hall compositions [Discrete Math. 263, No. 1--3, 275--280 (2003; Zbl 1019.05004
)] and truncated lecture-hall partitions [J. Comb. Theory, Ser. A 108, No. 2, 217--245 (2004; Zbl 1061.05009
)]. These methods are seen to be preferable to other work involving $q$-series. Finally, the authors point out how their guidelines relate to McMahon’s partition analysis. For the entire collection see [Zbl 1108.00016
|11P81||Elementary theory of partitions|
|05A17||Partitions of integers (combinatorics)|