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Processes on Integer Partitions and Their Limit ShapesPrimeFaces.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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2017 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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

Västerås: Mälardalen University , 2017.
##### Series

Mälardalen University Press Dissertations, ISSN 1651-4238 ; 223
##### National Category

Mathematics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

URN: urn:nbn:se:mdh:diva-35023ISBN: 978-91-7485-316-2 (print)OAI: oai:DiVA.org:mdh-35023DiVA, id: diva2:1082060
##### Public defence

2017-05-05, Delta, Mälardalens högskola, Västerås, 13:15 (English)
##### Opponent

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt777",{id:"formSmash:j_idt777",widgetVar:"widget_formSmash_j_idt777",multiple:true}); Available from: 2017-03-15 Created: 2017-03-15 Last updated: 2017-09-28Bibliographically approved
##### List of papers

This thesis deals with processes on integer partitions and their limit shapes, with focus on deterministic and stochastic variants on one such process called *Bulgarian solitaire*. The main scientific contributions are the following.

**Paper I:** Bulgarian solitaire is a dynamical system on integer partitions of *n* which converges to a unique fixed point if *n*=1+2+...+*k* is a triangular number. There are few results about the structure of the game tree, but when *k* tends to infinity the game tree itself converges to a structure that we are able to analyze. Its level sizes turns out to be a bisection of the Fibonacci numbers. The leaves in this tree structure are enumerated using Fibonacci numbers as well. We also demonstrate to which extent these results apply to the case when *k* is finite.

**Paper II:** 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, *σ*-Bulgarian solitaire, the number of cards you pick from a pile is some function *σ* of the pile size, such that you pick *σ*(*h*) < *h* cards from a pile of size *h*. Here we consider a special class of such functions. Let us call *σ* well-behaved if *σ*(1) = 1 and if both *σ*(*h*) and *h − σ*(

**Paper III:** We introduce *p _{n}-random q_{n}-proportion Bulgarian solitaire* (0 <

**Paper IV:** We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an *up chain* or *down chain*). The second type toggles between two adjacent rank levels (an *up-and-down chain*). We introduce two compatibility concepts between the up-directed transition probabilities (an *up rule*) and the down-directed (a *down rule*), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.

1. Level Sizes of the Bulgarian Solitaire Game Tree$(function(){PrimeFaces.cw("OverlayPanel","overlay1081111",{id:"formSmash:j_idt977:0:j_idt999",widgetVar:"overlay1081111",target:"formSmash:j_idt977:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Limit shapes of stable configurations of a generalized Bulgarian solitaire$(function(){PrimeFaces.cw("OverlayPanel","overlay1081125",{id:"formSmash:j_idt977:1:j_idt999",widgetVar:"overlay1081125",target:"formSmash:j_idt977:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. An exponential limit shape of random *q*-proportion Bulgarian solitaire$(function(){PrimeFaces.cw("OverlayPanel","overlay1081141",{id:"formSmash:j_idt977:2:j_idt999",widgetVar:"overlay1081141",target:"formSmash:j_idt977:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Markov chains on graded posets: Compatibility of up-directed and down-directed transition probabilities$(function(){PrimeFaces.cw("OverlayPanel","overlay1081145",{id:"formSmash:j_idt977:3:j_idt999",widgetVar:"overlay1081145",target:"formSmash:j_idt977:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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