Related papers: Telescoping Sums, Permutations, and First Occurren…
Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic…
Consider a random permutation $\pi\in{\cal S}_n$. In this paper, perhaps best classified as a contribution to discrete probability distribution theory, we study the {\it first} occurrence $X=X_n$ of a I-II-III-pattern, where "first" is…
We consider the distribution of cycles in two models of random permutations, that are related to one another. In the first model, cycles receive a weight that depends on their length. The second model deals with permutations of points in…
The primes or prime polynomials (over finite fields) are supposed to be distributed `irregularly' , despite nice asymptotic or average behavior. We provide some conjectures/guesses/hypotheses with `evidence' of surprising symmetries in…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high…
Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into…
A predictive distribution over a sequence of $N+1$ events is said to be "frequency mimicking" whenever the probability for the final event conditioned on the outcome of the first $N$ events equals the relative frequency of successes among…
We studied two probabilistic models of the distribution of primes in the natural number [1].The paper considers the third probabilistic model of the distribution of primes in the natural number. The author proved that the results obtained…
Numerical study of the distribution of the Riemann zeros differences following the work [1] shows the significance of the function for which the prime sum expression is proposed. Computational results related to this definition explored…
As observers of the universe we are physical systems within it. If the universe is very large in space and/or time, the probability becomes significant that the data on which we base predictions is replicated at other locations in…
A general piecewise (including pointwise) probability distribution with space-saving notation and its hierarchical particular cases are considered. The explicit closed-form normalization, expectation, and variance formulas along with the…
Three-dimensional random tessellations that are stable under iteration (STIT tessellations) are considered. They arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the…
We study the distribution of the number of permutations with a given periodic up-down sequence w.r.t. the last entry, find exponential generating functions and prove asymptotic formulas for this distribution.
Maximum-entropy distributions are shown to appear in the probability calculus as approximations of a model by exchangeability or a model by sufficiency, the former model being preferable. The implications of this fact are discussed,…
This is the first installment in a series of papers devoted to examining certain aspects of the asymptotic value distribution and distribution of zeros manifested by members of a broad class of linear combinations of L-functions in the…
We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time…
We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines…
We derive functional equations for distributions of six classical statistics (ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, and right-to-left minima) on separable and irreducible separable…
Let $T\$ be a stopping time associated with a sequence of independent random variables $Z_{1},Z_{2},...$ . By applying a suitable change in the probability measure we present relations between the moment or probability generating functions…