A partition identity

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.

This article explores the history of the two results in integer partitions known as Stanley’s theorem and Elder’s theorem. While history has credited Richard Stanley with the discovery of the results, we note that Nathan Fine had established these results among a host of other partition identities over a decade earlier. In tribute to Fine, analogues in the sets of odd partitions and distinct partitions are presented.

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

Combinatorics of counting finite fuzzy subsets

The integral sum number of complete bipartite graphs Kr, s

The number of ways to distribute r identical objects into n identical boxes can usually be obtained by a method of generating function or by a recursive formula. In this paper, for another approach, it is shown that we can obtain this number by using generalization of Polya’s theorem. From this, we can also find the number of partitions of integer n as a sum of k positive integers. Computing for the results is discussed.

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].

In this paper we study, pointed integer partition defined as a pair where is an integer partition of n - m, and m is a non-negative integer ≤ n. Shellability of pointed integer partition with Mobius values -1 and +1 denoted by . We determine the cardinality of and for 1 ≤ n ≤ 10 and n ≥ 1 respectively and compute the Mobius number of for 1 ≤ n ≤ 6. We have shown hat admit an EL-labeling which is EL-shellable.

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.

In the maximal ring O of octonion integers (“octavians”), let O m be the set of elements of norm m. Motivated by a question of H. P. Rehm, we ask whether, for a given positive rational integer m, there exists a set {α i } of norm-m octavians such that O m is partitioned by cosets O 1α i . For the case when m = 2, we characterize all possible intersections O 1α ∩ O 1α′, where α and α′ are any norm-2 octavians, and we show that all norm-2 partitioning sets are equivalent under the automorphisms of the underlying E 8 lattice mod 2. We then provide conditions sufficient to ensure that certain collections of one-sided cosets of O 1 partition O m , leading to partitioning sets in the new cases m = 4, 8, and 2 k, k odd.

The classification of tensor categories of two-colored noncrossing 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.

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.

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.

Using sequences of finite length with positive integer entries and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their q-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between finite differences of the coefficients of generalized Galois numbers and integer partitions with kinds is established.