Counting corners in compositions and set partitions presented as bargraphs. Issue 6 (3rd June 2018)

• Record Type: Journal Article
• Authors: Mansour, Toufik; Shabani, Armend Sh.; Shattuck, Mark
• Is Part Of: Journal of difference equations and applications. Volume 24:Issue 6(2018)
• Page Start: 992
• Page End: 1015
• Publication Date: 2018-06-03
• Subjects: Bargraphs -- combinatorial statistic -- composition -- set partitions; 05A15 -- 05A05 -- 05A18; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2018.1444760
• Languages: English
• ISSNs: 1023-6198

Abstract

Abstract: In this paper, we consider statistics on compositions of a fixed number and set partitions of a fixed size represented geometrically as bargraphs. By a corner of a bargraph, we mean a vertex along its polygonal boundary, where types of corners are identified by the sequence of steps directly before and after. Here, we find a generating function formula of the joint distribution for the statistics on compositions of n recording the number of corners of types uh or dh . We also find an explicit formula for the total number of corners of either type in all compositions of n, supplying both algebraic and combinatorial proofs. We then determine the joint distribution on compositions for corners of typeu h i for all i less than some fixed number as well as the comparable distributions for corners of typeu h i andh i d on set partitions of size n having k blocks. In deriving our results, we solve various types of difference equations that are satisfied by the generating functions, using inductive arguments to determine closed form expressions.