On the largest sizes of certain simultaneous core partitions with distinct parts

Motivated by Amdeberhan's conjecture on $(t,t+1)$-core partitions with distinct parts, various results on the numbers, the largest sizes and the average sizes of simultaneous core partitions with distinct parts were obtained by many mathematicians recently. In this paper, we derive the largest sizes of $(t,mt\pm 1)$-core partitions with distinct parts, which verifies a generalization of Amdeberhan's conjecture. We also prove that the numbers of such partitions with the largest sizes are at most $2$.Comment: 9 page

Core partitions with distinct parts

Simultaneous core partitions have attracted much attention since Anderson's work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on simultaneous core partitions with distinct parts. The generating function of $t$-core partitions with distinct parts is obtained. We also prove the results on the number, the largest size and the average size of $(t, t + 1)$-core partitions. This gives a complete answer to a conjecture of Amdeberhan, which is partly and independently proved by Straub, Nath and Sellers, and Zaleski recently.Comment: 8 page

When Does the Set of (a,b,c)(a, b, c)-Core Partitions Have a Unique Maximal Element?

In 2007, Olsson and Stanton gave an explicit form for the largest $(a, b)$-core partition, for any relatively prime positive integers $a$ and $b$, and asked whether there exists an $(a, b)$-core that contains all other $(a, b)$-cores as subpartitions; this question was answered in the affirmative first by Vandehey and later by Fayers independently. In this paper we investigate a generalization of this question, which was originally posed by Fayers: for what triples of positive integers $(a, b, c)$ does there exist an $(a, b, c)$-core that contains all other $(a, b, c)$-cores as subpartitions? We completely answer this question when $a$, $b$, and $c$ are pairwise relatively prime; we then use this to generalize the result of Olsson and Stanton.Comment: 8 pages, 2 figure

Generalized Fleming-Viot processes with immigration via stochastic flows of partitions

The generalized Fleming-Viot processes were defined in 1999 by Donnelly and Kurtz using a particle model and by Bertoin and Le Gall in 2003 using stochastic flows of bridges. In both methods, the key argument used to characterize these processes is the duality between these processes and exchangeable coalescents. A larger class of coalescent processes, called distinguished coalescents, was set up recently to incorporate an immigration phenomenon in the underlying population. The purpose of this article is to define and characterize a class of probability-measure valued processes called the generalized Fleming-Viot processes with immigration. We consider some stochastic flows of partitions of Z_{+}, in the same spirit as Bertoin and Le Gall's flows, replacing roughly speaking, composition of bridges by coagulation of partitions. Identifying at any time a population with the integers $\mathbb{N}:=\{1,2,...\}$, the formalism of partitions is effective in the past as well as in the future especially when there are several simultaneous births. We show how a stochastic population may be directly embedded in the dual flow. An extra individual 0 will be viewed as an external generic immigrant ancestor, with a distinguished type, whose progeny represents the immigrants. The "modified" lookdown construction of Donnelly-Kurtz is recovered when no simultaneous multiple births nor immigration are taken into account. In the last part of the paper we give a sufficient criterion for the initial types extinction.Comment: typos and corrections in reference