Limit shapes of stable configurations of a generalized Bulgarian solitaire
(English)In: Order, ISSN 0167-8094, E-ISSN 1572-9273, ISSN 0167-8094Article in journal (Other academic) Submitted
Bulgarian solitaire is played on n cards divided into several piles; a move consists of picking one card from each pile to form a new pile. In a recent generalization, -Bulgarian solitaire, the number of cards you pick from a pile is some function of the pile size, such that you pick cards from a pile of size h. Here we consider a special class of such functions. Let us call well-behaved if and if both and are non-decreasing functions of h. Well-behaved -Bulgarian solitaire has a geometric interpretation in terms of layers at certain levels being picked in each move. It also satisfies that if a stable configuration of n cards exists it is unique. Moreover, if piles are sorted in order of decreasing size () then a configuration is convex if and only if it is a stable configuration of some well-behaved -Bulgarian solitaire. If sorted configurations are represented by Young diagrams and scaled down to have unit height and unit area, the stable configurations corresponding to an infinite sequence of well-behaved functions () may tend to a limit shape . We show that every convex with certain properties can arise as the limit shape of some sequence of well-behaved . For the special case when for , these limit shapes are triangular (in case ), or exponential (in case ), or interpolating between these shapes (in case ).
Place, publisher, year, edition, pages
Bulgarian solitaire, Limit shape
Research subject Mathematics/Applied Mathematics
IdentifiersURN: urn:nbn:se:mdh:diva-35013OAI: oai:DiVA.org:mdh-35013DiVA: diva2:1081125
FunderSwedish Research Council, 2010-5565, 621-2009-6090