Part-products of random integer compositions

2021 ◽  
Author(s):  
Caroline J. Shapcott
Keyword(s):  
10.37236/5026 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Clemens Heuberger ◽  
Sara Kropf ◽  
Helmut Prodinger

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.


2014 ◽  
Vol 23 (6) ◽  
pp. 1057-1086 ◽  
Author(s):  
PETER J. GRABNER ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER

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.


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

1997 ◽  
Vol 6 (1) ◽  
pp. 57-77 ◽  
Author(s):  
ULRICH MARTIN HIRTH

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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