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Limit shapes of stable configurations of a generalized Bulgarian solitairePrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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(English)In: Order, ISSN 0167-8094, E-ISSN 1572-9273, ISSN 0167-8094Article in journal (Other academic) Submitted
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer Netherlands.
##### Keywords [en]

Bulgarian solitaire, Limit shape
##### National Category

Discrete Mathematics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

URN: urn:nbn:se:mdh:diva-35013OAI: oai:DiVA.org:mdh-35013DiVA, id: diva2:1081125
#####

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##### Funder

Swedish Research Council, 2010-5565, 621-2009-6090Available from: 2017-03-13 Created: 2017-03-13 Last updated: 2017-09-28Bibliographically approved
##### In thesis

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 ).

1. Processes on Integer Partitions and Their Limit Shapes$(function(){PrimeFaces.cw("OverlayPanel","overlay1082060",{id:"formSmash:j_idt777:0:j_idt781",widgetVar:"overlay1082060",target:"formSmash:j_idt777:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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