Related papers: Partitions of an Integer into Powers
We characterize which graph invariants are partition functions of a spin model over the complex numbers, in terms of the rank growth of associated `connection matrices'.
This paper addresses the general problem of modelling and learning rank data with ties. We propose a probabilistic generative model, that models the process as permutations over partitions. This results in super-exponential combinatorial…
Dynamic regression trees are an attractive option for automatic regression and classification with complicated response surfaces in on-line application settings. We create a sequential tree model whose state changes in time with the…
The cyclotomic matrix is commonly used to arrange cyclotomic numbers in a convenient format. A natural question is whether the structure of the matrix can reflect properties of these numbers. In this article, we examine cyclotomic numbers…
Dynamic probabilistic networks are a compact representation of complex stochastic processes. In this paper we examine how to learn the structure of a DPN from data. We extend structure scoring rules for standard probabilistic networks to…
Given an ideal $I$ in a commutative ring $A$, a divided power structure on $I$ is a collection of maps $\{\gamma_n \colon I \to A\}_{n \in \mathbb{N}}$, subject to axioms that imply that it behaves like the family $\{x \mapsto…
We introduce a new partial order on the set of all antichains of a fixed size in any poset. When applied to minuscule posets, these partial orders give rise to distributive lattices that appear in the branching rules for minuscule…
Distributed representations (such as those based on embeddings) and discrete representations (such as those based on logic) have complementary strengths. We explore one possible approach to combining these two kinds of representations. We…
The number partitioning problem consists of partitioning a sequence of positive numbers ${a_1,a_2,..., a_N}$ into two disjoint sets, ${\cal A}$ and ${\cal B}$, such that the absolute value of the difference of the sums of $a_j$ over the two…
We demonstrate that statistics of certain classes of set partitions is described by generating functions related to the Burgers, Ibragimov--Shabat and Korteweg--de Vries integrable hierarchies.
Number theory is considered, by proposing quantum mechanical models and string-like models at zero and finite temperatures, where the factorization of number into prime numbers is viewed as the decay of particle into elementary particles…
We study the enumeration problem of higher dimensional partitions, a natural generalisation of classical integer partitions. We show that their counting problem is equivalent to the enumeration of simpler classes of higher dimensional…
A novel integer value-sorting technique is proposed replacing bucket sort, distribution counting sort and address calculation sort family of algorithms. It requires only constant amount of additional memory. The technique is inspired from…
The symbolic representation of a number should be considered as a data structure, and the choice of data structure depends on the arithmetic operations that are to be performed. Numbers are almost universally represented using position…
In the first part we associate a periodic sequence to a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved
Ferrers diagrams are used to visually represent integer partitions. We describe a way to use Ferrers diagrams to uniquely represent integers in terms of their prime factors. This leads to a lower bound on the number of primes less than a…
A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields, independently of Ford a.a., who investigated Pratt trees for primality tests. It leads to a…
We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in…
We develop the partitioning technique for quantum discrete systems. The graph consists of several subgraphs: a central graph and several branch graphs, with each branch graph being rooted by an individual node on the central one. We show…
We review briefly the concepts underlying complex systems and probability distributions. The later are often taken as the first quantitative characteristics of complex systems, allowing one to detect the possible occurrence of regularities…