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Jonsson, Markusorcid.org/0000-0003-1242-3599

Open this publication in new window or tab >>Markov chains on graded posets: Compatibility of up-directed and down-directed transition probabilities### Jonsson, Markus

### Kimmo, Eriksson

### Sjöstrand, Jonas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Order, ISSN 0167-8094, E-ISSN 1572-9273, ISSN 0167-8094, no 1, p. 93-109Article in journal (Refereed) Published
##### Abstract [en]

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

Springer Netherlands, 2018
##### Keywords

Graded poset, Markov chain, Young diagram, Young's lattice, Limit shape
##### National Category

Discrete Mathematics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

urn:nbn:se:mdh:diva-35015 (URN)10.1007/s11083-016-9420-1 (DOI)000427496600006 ()2-s2.0-85017160313 (Scopus ID)
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##### Funder

Swedish Research Council, 2010-5565, 621-2009-6090
Available from: 2017-03-13 Created: 2017-03-13 Last updated: 2018-04-05Bibliographically approved

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.

Kungliga Tekniska Högskolan, Sweden.

We consider two types of discrete-time Markov chains where thestate space is a graded poset and the transitionsare 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 betweenup-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.

Open this publication in new window or tab >>Level Sizes of the Bulgarian Solitaire Game Tree### Eriksson, Henrik

### Jonsson, Markus

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: The Fibonacci quarterly, ISSN 0015-0517, ISSN 0015-0517, Vol. 55, no 3, p. 243-251Article in journal (Refereed) Published
##### Abstract [en]

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

The Fibonacci Association, 2017
##### Keywords

Bulgarian solitaire
##### National Category

Discrete Mathematics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

urn:nbn:se:mdh:diva-35012 (URN)000412356200006 ()2-s2.0-85030458141 (Scopus ID)
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##### Funder

Swedish Research Council, 2010-5565
Available from: 2017-03-13 Created: 2017-03-13 Last updated: 2018-02-03Bibliographically approved

Kungliga Tekniska Högskolan, Sweden.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.

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 astructure 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.

Open this publication in new window or tab >>Processes on Integer Partitions and Their Limit Shapes### Jonsson, Markus

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 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:nbn:se:mdh:diva-35023 (URN)978-91-7485-316-2 (ISBN)
##### Public defence

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

### Jonasson, Johan

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##### Supervisors

### Eriksson, Kimmo

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Available from: 2017-03-15 Created: 2017-03-15 Last updated: 2017-09-28Bibliographically approved

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.

Chalmers tekniska högskola.

Mälardalen University, School of Education, Culture and Communication.

Open this publication in new window or tab >>Limit shapes of stable configurations of a generalized Bulgarian solitaire### Jonsson, Markus

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.### Kimmo, Eriksson

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.### Sjöstrand, Jonas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); (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

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

Discrete Mathematics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

urn:nbn:se:mdh:diva-35013 (URN)
#####

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##### Funder

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

Kungliga Tekniska Högskolan, Stockholm.

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 ).