Related papers: Partitions of an Integer into Powers
A simple and computationally efficient scheme for tree-structured vector quantization is presented. Unlike previous methods, its quantization error depends only on the intrinsic dimension of the data distribution, rather than the apparent…
In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a…
We apply the geometric construction of solutions of some variational problems of combinatorics to estimate the number of partitions and of plane partitions of an integer.
In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…
A statistical model for the fragmentation of a conserved quantity is analyzed, using the principle of maximum entropy and the theory of partitions. Upper and lower bounds for the restricted partitioning problem are derived and applied to…
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
Recently, many authors have investigated how various partition statistics distribute as the size of the partition grows. In this work, we look at a particular statistic arising from the recent rejuvenation of MacMahon's partition analysis.…
We consider probability distributions with constant rate on partially ordered sets, generalizing distributions in the usual reliability setting that have constant failure rate. In spite of the minimal algebraic structure, there is a…
In the paper we present a description of complex systems in terms of self-organization processes of prime integer relations. A prime integer relation is an indivisible element made up of integers as the basic constituents following a single…
Let $v\geq6$ be an integer with $v\equiv2 \pmod 4$. In this paper, we introduce a new partitioning of the set of all $3$-subsets of a $v$-set into some simple trades.
We introduce a class of stochastic integer sequences. In these sequences, every element is a sum of two previous elements, at least one of which is chosen randomly. The interplay between randomness and memory underlying these sequences…
Mass partition problems describe the partitions we can induce on a family of measures or finite sets of points in Euclidean spaces by dividing the ambient space into pieces. In this survey we describe recent progress in the area in addition…
In this paper, we use the transference principle to investigate the representation of sufficiently large positive integers as the sum of prime powers and integer powers, where the primes are drawn from a positive density subset of the set…
For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We…
The lattice of the set partitions of $[n]$ ordered by refinement is studied. Given a map $\phi: [n] \rightarrow [n]$, by taking preimages of elements we construct a partition of $[n]$. Suppose $t$ partitions $p_1,p_2,\dots,p_t$ are chosen…
A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can…
The exactly solvable four-vertex model with the fixed boundary conditions in the presence of inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal…
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
In this note, we represent integers in a type of factoradic notation. Rather than use the corresponding Lehmer code, we will view integers as permutations. Given a pair of integers n and k, we give a formula for n mod k in terms of the…