Related papers: Counting birthday collisions using partitions

We present an algorithm to compute the number of solutions of the (constrained) number partitioning problem. A concrete implementation of the algorithm on an Ising-type quantum computer is given.

As an attempt to bridge the gap between the probabilistic world of classical information theory and the combinatorial world of zero-error information theory, this paper studies the performance of randomly generated codebooks over discrete…

We develop a new numerical algorithm to model collisional cascades in debris disks. Because of the large dynamical range in particle masses, we solve the integro-differential equations describing erosive and catastrophic collisions in a…

Using the index theory of seaweed algebras, we explore various new integer partition statistics. We find relations to some well-known varieties of integer partitions as well as a surprising periodicity result.

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$…

We show the value of mass-momentum diagrams for analyzing collision problems in classical mechanics in one dimension. Collisions are characterized by the coefficient of restitution and the momentum of the interacting particles both before…

Rota's Twelvefold Way gave formulas for the numbers of partitions which could be formed in twelve scenarios. This proposed AMM article expands Rota's 4 x 3 table. The resulting 6 x 5 table considers a broader collection of…

In the paper, the authors present several new relations and applications for the combinatorial sequence that counts the possible partitions of a finite set with the restriction that the size of each block is contained in a given set. One of…

We recall the notion of fractional enumeration and immediately focus on the fractional counting of integer partitions, where each partition gets credit equal to the reciprocal of the product of its parts. We raise two intriguing questions…

Robustly handling collisions between individual particles in a large particle-based simulation has been a challenging problem. We introduce particle merging-and-splitting, a simple scheme for robustly handling collisions between particles…

We consider the problem of learning an unknown partition of an $n$ element universe using rank queries. Such queries take as input a subset of the universe and return the number of parts of the partition it intersects. We give a simple…

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…

In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For…

We use the Circle Method to derive asymptotic formulas for functions related to the number of parts of partitions in particular residue classes.

We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of numbers, illustrate how different mathematical fields play…

We introduce Partiti, the puzzle that will run in Mathematics Magazine in 2018, and use the opportunity to recall some basic properties of integer partitions.

We study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such…

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a…

Given are the facets of an abstract (finite) simplicial complex SC. We show how to partition SC into few pieces, each one compactly encoded by the use of wildcards. Such a representation is useful for the optimization of a target function…

Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase…