Core Partitions Research Articles

Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

Arithmetic density and new congruences for $$\pmb 3$$-core partitions

Arithmetic density and new congruences for $$\pmb 3$$-core partitions

The Maximum Hook Length of $d$-Distinct Simultaneous Core Partitions

We exactly determine the maximum possible hook length of $(s,t)$-core partitions with $d$-distinct parts when there are finitely many such partitions. Moreover, we provide an algorithm to construct a $d$-distinct $(s,t)$-core partition with this maximum possible hook length.

Distinct Features Based on Partitioning of the Endophytic Fungi of Cereals and Other Grasses.

Endophytic fungi form a significant part of the plant mycobiome. Defining core members is crucial to understanding the assembly mechanism of fungal endophytic communities (FECs) and identifying functionally important community members. We conducted a meta-analysis of FECs in stems of wheat and five wild cereal species and generated a landscape of the fungal endophytic assemblages in this group of plants. The analysis revealed that several Ascomycota members and basidiomycetous yeasts formed an important compartment of the FECs in these plants. We observed a weak spatial autocorrelation at the regional scale and high intrahost variations in the FECs, suggesting a space-related heterogeneity. Accordingly, we propose that the heterogeneity among subcommunities should be a criterion to define the core endophytic members. Analysis of the subcommunities and meta-communities showed that the core and noncore members had distinct roles in various assembly processes, such as stochasticity, universal dynamics, and network characteristics, within each host. The distinct features identified between the community partitions of endophytes aid in understanding the principles that govern the assembly and function of natural communities. These findings can assist in designing synthetic microbiomes. IMPORTANCE This study proposes a novel method for diagnosing core microbiotas based on prevalence of community members in a meta-community, which could be determined and supported statistically. Using this approach, the study found stratification in community assembly processes within fungal endophyte communities (FECs) in the stems of wheat and cereal-related wild species. The core and noncore partitions of the FECs exhibited certain degrees of determinism from different aspects. Further analysis revealed abundant and consistent interactions between rare taxa, which might contribute to the determinism process in noncore partitions. Despite minor differences in FEC compositions, wheat FECs showed distinct patterns in community assembly processes compared to wild species, suggesting the effects of domestication on FECs. Overall, our study provided a new approach for identifying core microbiota and provides insights into the community assembly processes within FECs in wheat and related wild species.

Combinatorics on Bounded Free Motzkin Paths and its Applications

In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections between a set of bounded cornerless Motzkin paths and a set of $t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths and a set of self-conjugate $t$-core partitions. As an application, we get explicit formulas for the number of ordinary and self-conjugate $t$ core partitions with a fixed number of corners.

Results on bar-core partitions, core shifted Young diagrams, and doubled distinct cores

Self-conjugate core partitions, bar-core partitions, core shifted Young diagrams (or CSYDs), and doubled distinct core partitions have been studied independently. In this paper, we show that doubled distinct cores and CSYDs are special cases of bar-core partitions. From this fact together with the abacus construction, we find path interpretations of bar-cores, CSYDs, and doubled distinct cores on (s,t)-cores and (s,s+d,s+2d)-cores, respectively. As a result, we get the number of these core partitions.

On the largest sizes of $$(s,qs\pm 1)$$-core partitions with parts of the same parity

On the largest sizes of $$(s,qs\pm 1)$$-core partitions with parts of the same parity

Strong Players and Stable Coalition Structures in PMAS Profit Game

In a non-negative profit game that possesses a Population Monotonic Allocation Scheme (PMAS), being a member of a larger coalition implies that your profit cannot decrease. In this paper, we refer to such games as PMAS profit games. As population monotonicity is a nice and desirable property that encourages formation of larger coalitions and implies stability of the grand coalition, we explore if this special feature of PMAS games can help in identifying additional stable coalition structures under different stability concepts in cooperative games—namely, core partitions, the von Neumann–Morgenstern (vNM) stable set, the largest consistent set, and the equilibrium process of coalition formation (EPCF)—and in developing relationships between coalition structures that are stable under these different stability concepts. We first define two special classes of players for PMAS profit games—extreme and strong players—and use them to develop an algorithm for construction of stable (core) partitions. We also use extreme players to identify absorbing states for equilibrium processes of coalition formation with high level of farsightedness. We then explore the impact of population monotonicity on the relationship between stable coalition structures under abovementioned stability concepts. While we are able to obtain some results related to stability of the grand coalition and to establish relationships between stable coalition structures under different stability notions that are consistent with the existing body of knowledge, population monotonicity in general does not add enough for strengthening of the existing results. However, we are able to show a couple of more general result that hold for arbitrary cooperative TU profit games. That is, we show that the members of vNM farsighted stable sets are core partitions, and that core partitions are members of a vNM stable sets. Moreover, we show that the members of vNM farsighted stable sets are EPCF-stable partitions.

Humans and the core partition: An agent-based modeling experiment.

Although strategic coalition formation is traditionally modeled using cooperative game theory, behavioral game theorists have repeatedly shown that outcomes predicted by game theory are different from those generated by actual human behavior. To further explore these differences, in a cooperative game theory context, we experiment to compare the outcomes resulting from human participants' behavior to those generated by a cooperative game theory solution mechanism called the core partition. Our experiment uses an interactive simulation of a glove game, a particular type of cooperative game, to collect the participant's decision choices and their resultant outcomes. Two different glove games are considered, and the outputs from 62 trial games are analyzed. The experiment's outcomes show that core coalitions, which are coalitions in a core partition, are found in about 42% of games. Though this number may seem low, a trial's outcome is more complex than whether the human player finds a core coalition or not. Finding the core coalition depends on factors such as the other possible feasible solutions and the payoffs available from these solutions. These factors, and the complexity they generate, are discussed in the paper.

On the unique core partition of coalition formation games: correction to İnal (2015)

On the unique core partition of coalition formation games: correction to İnal (2015)

Multiplication theorems for self-conjugate partitions

In 2011, Han and Ji proved addition-multiplication theorems for integer partitions, from which they derived modular analogues of many classical identities involving hook-length. In the present paper, we prove addition-multiplication theorems for the subset of self-conjugate partitions. Although difficulties arise due to parity questions, we are almost always able to include the BG-rank introduced by Berkovich and Garvan. This gives us as consequences many self-conjugate modular versions of classical hook-lengths identities for partitions. Our tools are mainly based on fine properties of the Littlewood decomposition restricted to self-conjugate partitions.Mathematics Subject Classifications: 05A15, 05A17, 05A19, 05E05, 05E10, 11P81Keywords: Hook-length formulas, BGP-ranks, Integers partitions, Littlewood decomposition, core partitions

Corners of self-conjugate (s,s + 1)-cores and [formula omitted]-cores

In this paper, we describe a result on self-conjugate (s,s+1)-core partitions with the fixed number of corners. We also define shifted corners of a distinct partition and find formulas for the number of (s‾,s+1‾)-core partitions and the number of (s,s+1)-core shifted Young diagrams with the fixed number of shifted corners.

The average size of [formula omitted]-core partitions

In [1], a poset Pt,t+1=⋃1≤i≤t−1{x∈N|(i−1)(t+1)+1≤x≤it−1} is introduced by Anderson, whose partial order is specified by requiring that x covers y if x−y=t or t+1. Then we follow the poset method used by Stanley and Zanello [13] to obtain the sum of elements of all ideals I∈J(Pt,t+1). Furthermore, we obtain the sum of the sizes of all (tˆ,t+1ˆ)-core partitions, and deduce the average size of (tˆ,t+1ˆ)-core partitions.

A survey on t-core partitions

$t$-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on $t$-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on $t$-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of $t$-cores in number theoretic aspects of partition theory. This includes the modularity of $t$-core partition generating functions, the existence of $t$-core partitions, asymptotic formulas and arithmetic properties of $t$-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of $t$-core partitions, which include relations between core partitions and self-conjugate core partitions, a $t$-core crank explaining Ramanujan's partition congruences, and relations with class numbers.

An Empirical Distribution of the Number of Subsets in the Core Partitions of Hedonic Games

An Empirical Distribution of the Number of Subsets in the Core Partitions of Hedonic Games

Generating Empirical Core Size Distributions of Hedonic Games Using a Monte Carlo Method

Hedonic games have gained popularity over the last two decades, leading to several research articles that have used analytical methods to understand their properties better. In this paper, a Monte Carlo method, a numerical approach, is used instead. Our method includes a technique for representing, and generating, random hedonic games. We were able to create and solve, using core stability, millions of hedonic games with up to 16 players. Empirical distributions of the hedonic games’ core sizes were generated, using our results, and analyzed for games of up to 13 players. Results from games of 14–16 players were used to validate our research findings. Our results indicate that core partition size might follow the gamma distribution for games with a large number of players.

Sizes of simultaneous core partitions

There is a well-studied correspondence by Jaclyn Anderson between partitions that avoid hooks of length s or t and certain binary strings of length s+t. Using this map, we prove that the total size of a random partition of this kind converges in law to Watson's U2 distribution, as conjectured by Doron Zeilberger.

Self-conjugate $$(s,s+d,\dots ,s+pd)$$-core partitions and free Motzkin paths

Self-conjugate $$(s,s+d,\dots ,s+pd)$$-core partitions and free Motzkin paths

On t-core and self-conjugate (2t − 1)-core partitions in arithmetic progressions

We extend recent results of Ono and Raji, relating the number of self-conjugate 7-core partitions to Hurwitz class numbers. Furthermore, we give a combinatorial explanation for the curious equality 2sc7(8n+1)=c4(7n+2). We also conjecture that an equality of this shape holds if and only if t=4, proving the cases t∈{2,3,5} and giving partial results for t>5.

Refined Counting of Core Partitions into $d$-Distinct Parts

Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers have enumerated $(s,ms\pm 1)$-core partitions into distinct parts. We generalize their result in several directions by including the number of parts of these partitions, by considering $d$-distinct partitions, and by allowing more general $(s,ms\pm r)$-core partitions. As an application of our approach, we obtain the average and maximum number of parts of these core partitions.