Mixtures and limits of symmetric random integer partitions

METRON ◽  
2012 ◽  
Vol 70 (2-3) ◽  
pp. 207-217
Author(s):  
Mauro Gasparini
Keyword(s):  
Integer Partitions ◽  
Random Integer

A General Asymptotic Scheme for the Analysis of Partition Statistics

2014 ◽  
Vol 23 (6) ◽  
pp. 1057-1086 ◽  
Author(s):  
PETER J. GRABNER ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Generating Function ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Higher Moments ◽  
Integer Partitions ◽  
Classical Singularity ◽  
Random Integer ◽  
Partition Statistics ◽  
New Statistics

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

scholarly journals Sampling parts of random integer partitions: a probabilistic and asymptotic analysis

2015 ◽  
Vol 25 (1) ◽  
pp. 79-95
Author(s):  
Ljuben Mutafchiev
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Asymptotic Analysis ◽  
Integer Partitions ◽  
Random Partition ◽  
Limiting Distributions ◽  
Sampling Plans ◽  
Random Integer ◽  
Sampling Procedures ◽  
The Relationship

Abstract Let λ be a partition of the positive integer n, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of λ at random. They obtained limiting distributions of the multiplicity μn = μn(λ) of the randomly-chosen part as n → ∞. The asymptotic behavior of the part size σn = σn(λ), under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of (μn; σn), as n → ∞, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for (μn; σn). Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of μn and σn.

scholarly journals Sampling part sizes of random integer partitions

2014 ◽  
Vol 37 (2) ◽  
pp. 329-343
Author(s):  
Ljuben Mutafchiev
Keyword(s):  
Integer Partitions ◽  
Random Integer

scholarly journals Counting monster potentials

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Riccardo Conti ◽  
Davide Masoero
Keyword(s):  
Excited States ◽  
Ground State ◽  
Hermite Polynomials ◽  
Large Momentum ◽  
Integer Partitions ◽  
Complex Equilibria ◽  
State Potential

Abstract We study the large momentum limit of the monster potentials of Bazhanov-Lukyanov-Zamolodchikov, which — according to the ODE/IM correspondence — should correspond to excited states of the Quantum KdV model.We prove that the poles of these potentials asymptotically condensate about the complex equilibria of the ground state potential, and we express the leading correction to such asymptotics in terms of the roots of Wronskians of Hermite polynomials.This allows us to associate to each partition of N a unique monster potential with N roots, of which we compute the spectrum. As a consequence, we prove — up to a few mathematical technicalities — that, fixed an integer N , the number of monster potentials with N roots coincides with the number of integer partitions of N , which is the dimension of the level N subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

scholarly journals Probabilistic Divide-and-Conquer: A New Exact Simulation Method, With Integer Partitions as an Example

2016 ◽  
Vol 25 (3) ◽  
pp. 324-351 ◽  
Author(s):  
RICHARD ARRATIA ◽  
STEPHEN DeSALVO
Keyword(s):  
Success Probability ◽  
Simulation Method ◽  
New Method ◽  
Divide And Conquer ◽  
Integer Partitions ◽  
Rejection Sampling ◽  
Substantial Fraction ◽  
Recursive Scheme ◽  
Rejection Cost ◽  
Sampling Cost

We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order n3/4 to a constant. We show other examples for which a non-recursive, one-time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost.We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collector's problem, in order to obtain a cost that is sublinear in the number of samples.

scholarly journals Facets of the Generalized Cluster Complex and Regions in the Extended Catalan Arrangement of Type $A$

10.37236/3169 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Susanna Fishel ◽  
Myrto Kallipoliti ◽  
Eleni Tzanaki
Keyword(s):  
Simple Root ◽  
Type A ◽  
Cluster Complex ◽  
Integer Partitions ◽  
Positive Part

In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and that of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, the map which we define bijects facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid\left\langle v,\alpha \right\rangle=m\}$ as separating wall. As a result, it restricts to a bijection between the set of facets of the positive part of $\Delta^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}^m(A_n)$. Our map is a composition of two bijections in which integer partitions in an $m$-dilated $n$-staircase shape come into play.

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