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1.
2.
Ljuben R. Mutafchiev 《Monatshefte für Mathematik》2002,9(3):313-325
For a partition , of a positive integer n chosen uniformly at random from the set of all such partitions, the kth excess is defined by if . We prove a bivariate local limit theorem for as . The whole range of possible values of k is studied. It turns out that ρ and η
k
are asymptotically independent and both follow the doubly exponential (extreme value) probability law in a suitable neighbourhood
of . 相似文献
3.
Ljuben Mutafchiev 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2018,88(1):255-272
Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where \(Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}\) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around \(x=1\). The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series. 相似文献
4.
We study the asymptotic behavior of the maximal multiplicity μn = μn(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμn/(6n)1/2 converges weakly to max jXj/j as n→∞, where X1, X2, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05 相似文献
5.
We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ
n
= τ
n
(ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ
n
− c
0
n
2/3)/c
1
n
1/3 log1/2
n converges weakly, as n → ∞, to the standard normal distribution, where c
0 = ζ(2)/ [2ζ(3)]2/3, c
1 = √(1/3/) [2ζ(3)]1/3 and ζ(s) = Σ
j=1∞
j
−s
.
Partial support given by the National Science Fund of the Bulgarian Ministry of Education and Science, grant No. VU-MI-105/2005. 相似文献
6.
Mapping patterns may be represented by unlabelled directed graphs in which each point has out-degree one. Assuming uniform probability distribution on the set of all mapping patterns onn points, we obtain limit distributions of some characteristics associated with the graphs of mapping patterns (connected and disconnected), asn. In particular, we study the number of points belonging to cycles, the number of cycles and components having prescribed (fixed) number of points and the total number of components. 相似文献
7.
L.R. Mutafchiev 《Acta Mathematica Hungarica》2000,87(1-2):47-69
Let Y
s,n
denote the number of part sizes ≧ s in a random and uniform partition of the positive integer n that are counted without multiplicity. For s = λ(6n)1/2/π + o(n
1/4), 0 ≦ λ < ∞, as n → ∞, we establish the weak convergence of Y
s,n
to a Gaussian distribution in the form of a central limit theorem. The mean and the standard deviation are also asymptotically
determined.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
8.
We study the asymptotic behavior of the maximal multiplicity Mn = Mn(σ) of the block sizes in a set partition σ of [n] = {1,2,…,n}, assuming that σ is chosen uniformly at random from the set of all such partitions. It is known that, for large n, the blocks of a random set partition are typically of size W = W(n), with WeW = n. We show that, over subsequences {nk}k ≥ 1 of the sequence of the natural numbers, , appropriately normalized, converges weakly, as k→∞, to , where Z1 and Z2 are independent copies of a standard normal random variable. The subsequences {nk}k ≥ 1, where the weak convergence is observed, and the quantity u depend on the fractional part fn of the function W(n). In particular, we establish that . The behavior of the largest multiplicity Mn is in a striking contrast to the similar statistic of integer partitions of n. A heuristic explanation of this phenomenon is also given. 相似文献
9.
George P. Yanev Ljuben Mutafchiev 《Methodology and Computing in Applied Probability》2006,8(2):223-233
We associate with a Bienaymé-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer
, define a complete
-ary tree to be the family tree of a deterministic branching process with offspring generating function
. We study the random variables
and
counting the number of disjoint complete
-ary subtrees, rooted at the ancestor, and having height
and
, respectively. Dekking (1991) and Pakes and Dekking (1991) find recursive relations for
and
involving the offspring probability generation function (pgf) and its derivatives. We extend their results determining the
probability distributions of
and
. It turns out that they can be expressed in terms of the offspring pgf, its derivatives, and the above probabilities. We
show how the general results simplify in case of fractional linear, geometric, Poisson, and one-or-many offspring laws.
相似文献
10.
Ljuben R. Mutafchiev 《Monatshefte für Mathematik》2002,136(4):313-325
For a partition , of a positive integer n chosen uniformly at random from the set of all such partitions, the kth excess is defined by if . We prove a bivariate local limit theorem for as . The whole range of possible values of k is studied. It turns out that ρ and η
k
are asymptotically independent and both follow the doubly exponential (extreme value) probability law in a suitable neighbourhood
of .
Received February 6, 2001; in revised form February 25, 2002 Published online August 5, 2002 相似文献