| 1. |
- Alm, Sven Erick, et al.
(författare)
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Correlations for Paths in Random Orientations of G(n, p) and G(n, m)
- 2011
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Ingår i: Random structures & algorithms (Print). - : Wiley. - 1042-9832 .- 1098-2418. ; 39:4, s. 486-506
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Tidskriftsartikel (refereegranskat)abstract
- We study random graphs, both G(n, p) and G(n, m), with random orientations on the edges. For three fixed distinct vertices s, a, b we study the correlation, in the combined probability space, of the events {a -> s} and {s -> b}. For G(n, p), we prove that there is a p(c) = 1/2 such that for a fixed p < p(c) the correlation is negative for large enough n and for p > p(c) the correlation is positive for large enough n. We conjecture that for a fixed n >= 27 the correlation changes sign three times for three critical values of p. For G(n, m) it is similarly proved that, with p = m/((n)(2)), there is a critical p(c) that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any l directed edges in G(n, m), is thought to be of independent interest. We present exact recursions to compute P(a -> s) and P(a -> s, s -> b). We also briefly discuss the corresponding question in the quenched version of the problem.
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| 2. |
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| 3. |
- Janson, Svante, 1955-, et al.
(författare)
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Proportionella val inom kommunfullmäktige
- 2019
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Vi diskuterar två olika problem som kan uppstå vid proportionella val i kommunfullmäktige och regionfullmäktige når ett parti försöker en kupp genom att utan samtycke gå i kartell med ett annat parti vid val till nämnd eller styrelse, vilket aktualiserades i åtminstone ett par fall hösten 2018. Det första problemet är vad sådana oönskade valkarteller kan få för effekter, och vilka möjligheter det finns för ett parti att skydda sig från att bli del i en oönskad valkartell. Det andra problemet är att i en sådan valkartell kan ett parti genom att splittra upp sina kandidater strategiskt på flera olika valsedlar få fler platser i en nämnd är vad som är proportionellt. Detta andra problem bottnar i att lagen om proportionella val stipulerar att Thieles metod skall användas för fördelning inom kartellen. På detta problem finns en enkel matematisk lösning och vi argumenterar för att man skall byta till Phragméns metod som används för motsvarande val till utskott i riksdagen.
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| 4. |
- Janson, Svante, et al.
(författare)
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The Probability Of The Alabama Paradox
- 2012
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 49:3, s. 773-794
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Tidskriftsartikel (refereegranskat)abstract
- Hamilton's method is a natural and common method to distribute seats proportionally between states (or parties) in a parliament. In the USA it has been abandoned due to some drawbacks, in particular the possibility of the Alabama paradox, but it is still in use in many other countries. In this paper we give, under certain assumptions, a closed formula for the asymptotic probability, as the number of seats tends to infinity, that the Alabama paradox occurs given the vector p(l), ..., p(m) of relative sizes of the states. From the formula we deduce a number of consequences. For example, the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1/e and on average approximately 0.123.
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| 5. |
- Alm, Sven Erick, et al.
(författare)
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First critical probability for a problem on random orientations in G(n,p)
- 2014
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Ingår i: Electronic Journal of Probability. - 1083-6489. ; 19, s. 69-
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Tidskriftsartikel (refereegranskat)abstract
- We study the random graph G (n,p) with a random orientation. For three fixed vertices s, a, b in G(n,p) we study the correlation of the events {a -> s} (there exists a directed path from a to s) and {s -> b}. We prove that asymptotically the correlation is negative for small p, p < C-1/n, where C-1 approximate to 0.3617, positive for C-1/n < p < 2/n and up to p = p(2)(n). Computer aided computations suggest that p(2)(n) = C-2/n, with C-2 approximate to 7.5. We conjecture that the correlation then stays negative for p up to the previously known zero at 1/2; for larger p it is positive.
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| 6. |
- Aas, Erik, et al.
(författare)
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Limiting directions for random walks in classical affine Weyl groups
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Let be a finite Weyl group and the corresponding affine Weyl group. A random element of can be obtained as a reduced random walk on the alcoves of . By a theorem of Lam (Ann. Probab. 2015), such a walk almost surely approaches one of many directions. We compute these directions when is , and and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types and in the affirmative and for type in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process called the -TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.
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| 7. |
- Aas, E., et al.
(författare)
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The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps
- 2019
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Ingår i: Journal of Physics A. - : Institute of Physics Publishing (IOPP). - 1751-8113 .- 1751-8121. ; 52:35
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Tidskriftsartikel (refereegranskat)abstract
- We consider a finite one-dimensional totally asymmetric simple exclusion process with four types of particles, {1, 0, 1, }, in contact with reservoirs. Particles of species 0 can neither enter nor exit the lattice, and those of species are constrained to lie at the first and last site. Particles of species 1 enter from the left reservoir into either the first or second site, move rightwards, and leave from either the last or penultimate site. Conversely, particles of species 1 enter from the right reservoir into either the last or penultimate site, move leftwards, and leave from either the first or last site. This dynamics is motivated by a natural random walk on the Weyl group of type D. We compute the exact nonequilibrium steady state distribution using a matrix ansatz building on earlier work of Arita. We then give explicit formulas for the nonequilibrium partition function as well as densities and currents of all species in the steady state, and derive the phase diagram.
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| 8. |
- Alexandersson, Per, 1987-, et al.
(författare)
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Promotion and cyclic sieving on families of SSYT
- 2021
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Ingår i: Arkiv för matematik. - : International Press of Boston. - 0004-2080 .- 1871-2487. ; 59:2, s. 247-274
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Tidskriftsartikel (refereegranskat)abstract
- We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family contains skew shapes, consisting of disjoint rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier results by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in the form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occurs with the same frequency.
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| 9. |
- Alexandersson, Per, et al.
(författare)
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Refined Catalan and Narayana cyclic sieving
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of “ears”, non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.
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| 10. |
- Hultman, Axel, et al.
(författare)
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From Bruhat intervals to intersection lattices and a conjecture of Postnikov
- 2009
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 116:3, s. 564-580
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Tidskriftsartikel (refereegranskat)abstract
- We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w is an element of (sic)(n). is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.
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