A Fine Rediscovery

A partition identity

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

The study of integer partitions is a significant area in number theory, intertwining arithmetic, algebra, and combinatorics. This theoretical study delves into the arithmetic and algebraic properties of integer partitions, examining their structure, relationships, and applications. An integer partition refers to a way of expressing a positive integer as the sum of other positive integers, where the order of summands does not matter. The paper explores the generating functions that encapsulate the partition function, focusing on their deep connections to modular forms and q-series. We examine the partition identities, such as Euler's pentagonal number theorem and their implications for partition theory. Furthermore, the study investigates algebraic approaches, such as the use of symmetric functions and Young tableaux, to provide insights into the combinatorial nature of partitions. The paper also emphasizes the arithmetic properties of partitions, such as congruences and partition identities modulo certain integers. Through rigorous analysis, we explore the role of integer partitions in solving Diophantine equations, their applications in statistical mechanics, and their contribution to the understanding of the structure of the partition lattice. Finally, the study draws connections to other areas of mathematics, such as q-algebras and representation theory, highlighting the interplay between algebraic structures and partition theory.

Combinatorics of counting finite fuzzy subsets

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit shapes, based in part on a method of Petrov. We also mention a few notable corollaries---for example, we obtain a limit shape for so-called "overpartitions'' by a simple DeSalvo-Pak-type transfer. To aid in the proof of these limit shapes, we prove an asymptotic formula for the number of partitions of the integer n into distinct parts where the largest part is at most t times the square root of n for fixed t. Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres' asymptotic formula for distinct parts partitions when instead the number of parts is bounded by t times the square root of n. The probabilistic approach is equivalent to a circle method/saddle-point method calculation, but it makes some of the steps more intuitive and even predicts the shape of the asymptotic formula, to some degree. Finally, motivated by certain problems concerning Rogers-Ramanujan-type identities, we give combinatorial proofs of three families of inequalities among certain types of integer partitions.

This paper considers the problem of synthesizing application-specific network-on-chip (NoC) architectures. We propose two heuristic algorithms called CLUSTER and DECOMPOSE that can systematically examine different set partitions of communication flows, and we propose Rectilinear-Steiner-tree (RST) based algorithms for generating an efficient network topology for each group in the partition. Different evaluation functions in fitting with the implementation backend and the corresponding implementation technology can be incorporated into our solution framework to evaluate the implementation cost of the set partitions and RST topologies generated. In particular, we experimented with an implementation cost model based on the power consumption parameters of a 70 nm process technology where leakage power is a major source of energy consumption. Experimental results on a variety of NoC benchmarks showed that our synthesis results can on average achieve a 6.92 x reduction in power consumption over the best standard mesh implementation. To further gauge the effectiveness of our heuristic algorithms, we also implemented an exact algorithm that enumerates all distinct set partitions. For the benchmarks where exact results could be obtained, our CLUSTER and DECOMPOSE algorithms on average can achieve results within 1% and 2% of exact results, with execution times all under 1 second whereas the exact algorithms took as much as 4.5 hours.

We introduce a new combinatorial optimization problem in this paper, called the Minimum Common Integer Partition (MCIP) problem, which was inspired by computational biology applications including ortholog assignment and DNA fingerprint assembly. A partition of a positive integer n is a multiset of positive integers that add up to exactly n, and an integer partition of a multiset S of integers is defined as the multiset union of partitions of integers in S. Given a sequence of multisets S 1, ⋯, S k of integers, where k ≥ 2, we say that a multiset is a common integer partition if it is an integer partition of every multiset S i , 1≤ i≤ k. The MCIP problem is thus defined as to find a common integer partition of S 1, ⋯, S k with the minimum cardinality. It is easy to see that the MCIP problem is NP-hard since it generalizes the well-known Set Partition problem. We can in fact show that it is APX-hard. We will also present a $\frac{5}{4}$ -approximation algorithm for the MCIP problem when k = 2, and a $\frac{3k(k-1)}{3k-2}$ -approximation algorithm for k ≥ 3.

Wide partitions, Latin tableaux, and Rota's basis conjecture

Using the slow triangle map (a type of multi-dimensional continued fraction algorithm), we exhibit a method for generating any number of new identities for subsets of integer partitions.

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where $$Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}$$ is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around $$x=1$$ . The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

We prove a theorem which adds a new companion to the Rogers–Ramanujan identities. This new companion counts partitions with different type of constraints on even and odd parts. Generalizing this theorem, we obtain two families of partition identities.

Let $n$ be a nonnegative integer and $\mathbb{A}=\{a_1,\ldots,a_k\}$ be a multiset with $k$ positive integers such that $a_1\leqslant\cdots\leqslant a_k$. In this paper, we give a recursive formula for partitions and distinct partitions of positive integer $n$ with respect to a multiset $\mathbb{A}$. We also consider the extension of the twelvefold way. By using this notion, we solve the nonintersecting circles problem, which asks to evaluate the number of ways to draw $n$ nonintersecting circles in the plane regardless of their sizes. The latter also enumerates the number of unlabeled rooted trees with $n+1$ vertices.

Avoidance of partitions of a three-element set

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$, while the second set contains partitions with smallest part at least $s+1$. Let $G_{L,s}(q)$ be the generating series whose coefficient of $q^n$ is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu (2019). Previous results concentrated on the nonnegativity of $G_{L,s}(q)$ in the cases $s=1$ and $s=2$. In the present paper, we show the eventual positivity of $G_{L,s}(q)$ for general $s$ and also find a precise nonnegativity result for the case $s=3$.