相关论文: Global structure of integer partitions sequences
In this work, we propose a simple but effective method to interpret black-box machine learning models globally. That is, we use a compact binary tree, the interpretation tree, to explicitly represent the most important decision rules that…
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…
In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to…
The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the…
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of…
A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb N$ is finitely colored, there must be some $\vec{x}$ with entries from $\mathbb N$ such that all entries of $A\vec{x}$ are in some color class. In…
The basic result of this note is a statement about the existence of families of partitions of the set of natural numbers with some favourable properties, the n-optimal matrices of partitions. We use this to improve a decomposition result…
For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…
Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let the representation function $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. In…
A probabilistic characterization of the dominance partial order on the set of partitions is presented. This extends work in "Symmetric polynomials and symmetric mean inequalities". Electron. J. Combin., 20(3): Paper 34, 2013. Let $n$ be a…
The role of integrable systems in string theory is discussed. We remind old examples of the correspondence between stringy partition functions or effective actions and integrable equations, based on effective application of the matrix model…
We consider several types of internal queries, that is, questions about fragments of a given text $T$ specified in constant space by their locations in $T$. Our main result is an optimal data structure for Internal Pattern Matching (IPM)…
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…
Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this article, we will consider the types of partitions with restrictions on consecutive parts. We will show that such partitions are…
Motility-induced phase separation (MIPS) is a purely non-equilibrium phenomenon in which self-propelled particles phase separate without any attractive interactions. One surprising feature of MIPS is the emergence of polar, nematic, and…
Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be…
A classical method for partition generating functions is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with n copies of n are presented.
To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add…
A hierarchy on a set $S$, also called a total partition of $S$, is a collection $\mathcal{H}$ of subsets of $S$ such that $S \in \mathcal{H}$, each singleton subset of $S$ belongs to $\mathcal{H}$, and if $A, B \in \mathcal{H}$ then $A \cap…
We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…