In this note we describe a general principle for selecting at random from a collection of combinatorial objects, where “at random” means in such a way that each of the objects has equal probability, a priori, of being selected. We apply this principle by displaying two algorithms, the first of which will select a random partition of an integer n, and the second will choose a random partition of...

Abstract Let $$A\subset \mathbb {N}_{+}$$ A ⊂ N + and by $$P_{A}(n)$$ P ( n ) denotes the number of partitions an integer n into parts from set A...

where (θ1, θ2, . . .)≡ θ are real nonnegative numbers, rj (π) denotes the number of j -cycles in π [we always have ∑ j jrj (π) = n] and hn is the normalization. We are mainly interested in the distribution of cycle lengths in the limit n→∞ and in how these lengths depend on the set of parameters θ . The probability P is really a probability on sequences r = (r1, r2, . . .) that satisfy ∑ j jrj ...

We consider a Degree-Corrected Planted-Partition model: a random graph on n nodes with two equal-sized clusters. The model parameters are two constants a, b > 0 and an i.i.d. sequence of weights (φu) n u=1, with finite second moment Φ . Vertices u and v are joined by an edge with probability φuφv n a when they are in the same class and with probability φuφv n b otherwise. We prove that it is in...

We extend the circle theorem on the zeros of the partition function to a continuum system. We also calculate the exact zeros of the partition function for a finite system where the probability distribution for the order parameter is given by two asymmetric Gaussian peaks. For the temperature driven first order transition in the thermodynamic limit, the locus and the angular density of zeros are...

Abstract This paper develops optimal designs when it is not feasible for every cluster to be represented in a sample as stratified design, by assuming equal probability two-stage sampling where clusters are small areas. The allocation methods surveys small-area estimates priority. We seek efficient allocations the aim minimize linear combination of mean squared errors composite area estimators ...

Bisection Problem is to patition a given graph G into subgraphs G1 and G2 of equal size with minimum number of cut edges, i.e., edges between G1 and G2. This problem has been studied extensively by many researchers [1, 2, 4], and several algorithms have been proposed and analyzed both mathematically and experimentally. Here we propose yet another algorithm for this problem. In this paper, we co...

In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in commutative, free and Boolean cases. The way we define (generalized) cumulants ...