Central limit theorems for some set partition statistics

Chern, Bobbie; Diaconis, Persi; Kane, Daniel M.; Rhoades, Robert C.

doi:10.1016/j.aam.2015.06.008

Cited by 14 publications

(10 citation statements)

References 13 publications

(23 reference statements)

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“…The component structure of random combinatorial objects such as permutations and mappings are studied in great detail in [3]. For work on the related subject of random set partitions see [4,8,14]. Conditional on |A 2 |, the distribution of the process of cycle counts (S 1 , S 2 , .…”

Section: The Components Of Injectionsmentioning

confidence: 99%

The Component Counts of Random Injections

Stark

2021

Electron. J. Combin.

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A model of random injections is defined which has domain $A\cup B$ and codomain $A\cup C$, where $A$, $B$ and $C$ are mutually disjoint finite sets such that $|B|\leqslant |C|$. The model encompasses both random permutations, which is the case $B=C=\varnothing$, and random maximum matchings of a complete bipartite graph, which is the case $A=\varnothing$. The possible components of random injections are cycles and paths. Results on the counts of cycles and paths of different sizes are obtained for this model.

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Section: The Components Of Injectionsmentioning

confidence: 99%

The Component Counts of Random Injections

Stark

2021

Electron. J. Combin.

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“…, x (m) ) of these elements such that CLTs for local and global patterns in other structures than the Ising model have attracted attention in the literature. We mention Markov chains (see [29,9,8] and references therein), patterns in random permutations (see [2,31] for global patterns and [12,2,30] for local patterns) and arc configurations in random set-partitions (CLTs for the number of arcs of size 1, which is a local pattern, and the number of crossings, which is a global pattern, were given in [3]). Note that Markov chains are (discrete) one-dimensional Markov random fields, while the random permutation model is a non-Markovian twodimensional model (when considering patterns, we think of permutations as permutation matrices).…”

Section: 2mentioning

confidence: 99%

Weighted dependency graphs and the Ising model

Dousse

Féray

2019

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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Weighted dependency graphs have been recently introduced by the second author, as a toolbox to prove central limit theorems. In this paper, we prove that spins in the d-dimensional Ising model display such a weighted dependency structure. We use this to obtain various central limit theorems for the number of occurrences of local and global patterns in a growing box.2010 Mathematics Subject Classification. 82B20,60F05.

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“…For both objects, some notion of patterns have been studied in the literature, see [AAA + 01] and [CDKR14], respectively. In both settings, a CLT is only known for the simplest kind of patterns: inversions in multiset permutations, where the central limit theorem was established by Canfield, Janson and Zeilberger [CJZ11] (see also [Thi16]) and crossings in set partitions, from the work of Chern, Diaconis, Kane and Rhoades [CDKR15]. The methods used in these papers do not seem to be generalizable to longer patterns.…”

Section: Introductionmentioning

confidence: 99%