Integer partitions

2021 ◽  
pp. 388-392
Keyword(s):  
Integer Partitions

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.

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.

Integer partitions

2010 ◽  
pp. 339-353
Author(s):  
Jörg Arndt
Keyword(s):  
Integer Partitions

scholarly journals A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

2019 ◽  
Vol 7 ◽  
Author(s):  
DANIEL M. KANE ◽  
ROBERT C. RHOADES
Keyword(s):  
Asymptotic Behavior ◽  
Modular Form ◽  
Generating Function ◽  
Generating Functions ◽  
Asymptotic Properties ◽  
Bootstrap Percolation ◽  
Asymptotic Result ◽  
Integer Partitions ◽  
Mock Modular Form ◽  
Precise Asymptotic

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of$n$without$k$consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without$k$consecutive parts. Andrews showed that when$k=2$, the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For$k>2$, the asymptotic properties of the generating functions have proved more difficult to obtain. Using$q$-series identities and the$k=2$case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case$k=3$was given by Zagier.This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.

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