Part-products of random integer compositions

2021 ◽  
Author(s):  
Caroline J. Shapcott
Keyword(s):  
Random Integer

scholarly journals Output Sum of Transducers: Limiting Distribution and Periodic Fluctuation

10.37236/5026 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Clemens Heuberger ◽  
Sara Kropf ◽  
Helmut Prodinger
Keyword(s):  
Periodic Function ◽  
Error Term ◽  
Fourier Coefficients ◽  
Higher Dimensions ◽  
Expected Value ◽  
Hölder Continuous ◽  
Periodic Fluctuation ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Random Integer

As a generalization of the sum of digits function and other digital sequences, sequences defined as the sum of the output of a transducer are asymptotically analyzed. The input of the transducer is a random integer in $[0, N)$. Analogues in higher dimensions are also considered. Sequences defined by a certain class of recursions can be written in this framework.Depending on properties of the transducer, the main term, the periodic fluctuation and an error term of the expected value and the variance of this sequence are established. The periodic fluctuation of the expected value is Hölder continuous and, in many cases, nowhere differentiable. A general formula for the Fourier coefficients of this periodic function is derived. Furthermore, it turns out that the sequence is asymptotically normally distributed for many transducers. As an example, the abelian complexity function of the paperfolding sequence is analyzed. This sequence has recently been studied by Madill and Rampersad.

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.

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

Probabilistic Number Theory, the GEM/Poisson-Dirichlet Distribution and the Arc-sine Law

1997 ◽  
Vol 6 (1) ◽  
pp. 57-77 ◽  
Author(s):  
ULRICH MARTIN HIRTH
Keyword(s):  
Number Theory ◽  
Dirichlet Distribution ◽  
Prime Factorization ◽  
Number Of Components ◽  
Probabilistic Number Theory ◽  
Probabilistic Number ◽  
Prime Factors ◽  
Random Integer ◽  
The Mean ◽  
Sine Law

The prime factorization of a random integer has a GEM/Poisson-Dirichlet distribution as transparently proved by Donnelly and Grimmett [8]. By similarity to the arc-sine law for the mean distribution of the divisors of a random integer, due to Deshouillers, Dress and Tenenbaum [6] (see also Tenenbaum [24, II.6.2, p. 233]), – the ‘DDT theorem’ – we obtain an arc-sine law in the GEM/Poisson-Dirichlet context. In this context we also investigate the distribution of the number of components larger than ε which correspond to the number of prime factors larger than nε.

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