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Let A , B be subsets of a finite abelian group G. Suppose that A + B does not contain a unique sum, i.e., there is no g ∈ G with a unique representation g = a + b , a ∈ A , b ∈ B. From such sets A , B , sparse linear systems over the rational numbers arise. We obtain a new determinant bound on invertible submatrices of the coefficient matrices of these linear systems. Under the condition that | A | + | B | is small compared to the order of G , these bounds provide essential information on the Smith Normal Form of these coefficient matrices. We use this information to prove that A and B admit coset partitions whose parts have properties resembling those of A and B. As a consequence, we improve previously known sufficient conditions for the existence of unique sums in A + B and show how our structural results can be used to classify sets A and B for which A + B does not contain a unique sum when | A | + | B | is relatively small. Our method also can be applied to subsets of abelian groups which have no unique differences. [ABSTRACT FROM AUTHOR]

Let G(v, E) be a graph with vertex set v(G) and edge set E(G). The maximum order of the partitions of the vertex set represented by {v1, v2, ..., vk }, is called upper domatic number where each class of partition dominates one another (i.e.) vi dominates vj or vj dominates vi or both (i ≠ j). Upper domatic number is denoted by D(G). In this paper we determine the upper domatic number of certain graphs such as double fan graph DFn, friendship graph Fn and helm graph Hn. [ABSTRACT FROM AUTHOR]

A proper total coloring of a graph G is said to be equitable, if the numberof elements (vertices and edges) in any two color classes differ by at most one, for which therequired minimum number of colors is called the equitable total chromatic number. In thispaper, we determine equitable total chromatic numbers of double star graph, bistar graph and splitting graph of star graph. [ABSTRACT FROM AUTHOR]

Information retrieval is intended to help people who are constantly looking for information. Partitioning the video into frames is the first stage in video information retrieval. Most video frames are brief and do not provide much information about the image content. On the other hand, scene border recognition or video fragmentation into scenes provides a better understanding of the video scene by clustering images based on similar image content. This paper is about video scene identification, specifically video formation mining for template matching with deep characteristics. The study proposed and created a workflow that included phases for frame extraction, finding similarities between consecutive frames, grouping frames, identifying key frames, and seeing detection by merging the relevant frames. Python's OpenCV generates the frames. The process is evaluated using scene identification metrics. The results show that scene detection and quality are significant, as measured by several criteria. In addition, we examined and studied current recognition and analysis criteria. Furthermore, our proposed methodologies have been thoroughly tested on various public scene video datasets, and they outperform some state-of-the-art approaches. This work's findings can be used to create real-time conceptual video interpretations. [ABSTRACT FROM AUTHOR]

We consider a class of tripartite systems for which two d-dimensional QFTs are cross-coupled via a third d + 1-dimensional "messenger" QFT. We analyse in detail the example of a pair of one-dimensional matrix quantum mechanics) coupled via a two-dimensional theory of the BF-type and compute its partition function and simple correlators. This construction is extendible in higher dimensions) using a Chern-Simons "messenger" theory. In all such examples, the exact partition function acquires a form, speculated to correspond to systems dual to Euclidean wormholes and the cross correlators are sufficiently soft and consistent with analogous gravitational calculations. Another variant of the tripartite system is studied, where the messenger theory is described by a non-self-interacting (matrix)-field, reaching similar conclusions. While the Euclidean theories we consider are perfectly consistent, the two possible analytic continuations into Lorentzian signature (messenger vs. boundary QFT directions) of the tripartite models, reveal physical features and "pathologies" resembling those of the expected Lorentzian gravitational backgrounds. [ABSTRACT FROM AUTHOR]

Let X be a nonempty set and T X the full transformation semigroup on X. For a partition P of X, we study semigroups T (X , P) = { f ∈ T X ∣ (∀ X i ∈ P) (∃ X j ∈ P) X i f ⊆ X j } , Σ (X , P) = { f ∈ T (X , P) ∣ (∀ X i ∈ P) X f ∩ X i ≠ ∅ } , and Γ (X , P) = { f ∈ T X ∣ (∀ X i ∈ P) (∃ X j ∈ P) X i f = X j } under composition. We give necessary and sufficient conditions for Γ (X , P) to be the semigroup of all closed selfmaps on X endowed with the topology having P as a basis. For finite X, we characterize unit-regular elements of T (X , P) and Σ (X , P) . We discuss set inclusions between Γ (X , P) and certain semigroups of selfmaps that preserve P . We characterize idempotents and regular elements of Γ (X , P) . For finite X, we also prove that all regular elements of Γ (X , P) are unit-regular. We finally count the number of elements, idempotents, and regular elements of Γ (X , P) for finite X. [ABSTRACT FROM AUTHOR]

We obtain estimates for the number p d (n) of (d − 1) -dimensional integer partitions of a number n. It is known that the two-sided inequality C 1 (d) n 1 − 1 / d < log p d (n) < C 2 (d) n 1 − 1 / d is always true and that C 1 (d) > 1 whenever log n > 3 d. However, establishing the " right " dependence of C 2 on d remained an open problem. We show that if d is sufficiently small with respect to n , then C 2 does not depend on d , which means that log p d (n) is up to an absolute constant equal to n 1 − 1 / d. Besides, we provide estimates of p d (n) for different ranges of d in terms of n , which give the asymptotics of log p d (n) in each case. [ABSTRACT FROM AUTHOR]

In 2003, Tuan showed a finiteness theorem for block-transitive point-imprimitive 3- (k (k − 1) 2 + 1 , k , λ) designs. As a generalization of this result, we consider the block-transitive point-imprimitive 3- (v , k , λ) designs with v < k (k − 1) 2 + 1. Let D = (P , B) be a 3- (v , k , λ) design admitting G as a block-transitive point-imprimitive automorphism group, and G preserve a partition C of the points into d imprimitivity classes of size c , here v = w k (k − 1) s + 1 , s , w are two positive integers with g c d (s , w) = 1 and 2 w < s. We prove that, for a given positive integer s , there are only finitely many numbers v such that there exist nontrivial block-transitive point-imprimitive 3- (v , k , λ) designs with c , d ≥ 3. Moreover, we obtain the classification for this type of designs when s ≤ 10. [ABSTRACT FROM AUTHOR]

Given a real number α ∈ (0 , 1) , we define the Webster sequence of density α to be W α = (⌈ (n − 1 / 2) / α ⌉) n ∈ N , where ⌈ x ⌉ is the ceiling function. It is known that if α and β are irrational with α + β = 1 , then W α and W β partition N. On the other hand, an analogous result for three-part partitions does not hold: There does not exist a partition of N into sequences W α , W β , W γ with α , β , γ irrational. In this paper, we consider the question of how close one can come to a three-part partition of N into Webster sequences with prescribed densities α , β , γ. We show that if α , β , γ are irrational with α + β + γ = 1 , there exists a partition of N into sequences of densities α , β , γ , in which one of the sequences is a Webster sequence and the other two are "almost" Webster sequences that are obtained from Webster sequences by perturbing some elements by at most 1. We also provide two efficient algorithms to construct such partitions. The first algorithm outputs the n th element of each sequence in O (1) time and the second algorithm gives the assignment of the m th positive integer to a sequence in O (1) time. We show that the results are best-possible in several respects. Moreover, we describe applications of these results to apportionment and optimal scheduling problems. [ABSTRACT FROM AUTHOR]

In this paper, the author gives two linear homogeneous recurrence relations to compute the values of Euler's partition function p(n) when n is odd. These recurrences do not involve the generalized pentagonal numbers and provide new connections between the partition function p(n) and the positive divisor function σ x (n) for x = 0 and x = 1. Some relations involving the divisor functions. σ x (n) and the parity of the partition function p(n) are presented in this context. [ABSTRACT FROM AUTHOR]

Let q be a fixed prime power and let V(n,q) denote a vector space of dimension n over the Galois field with q elements. A subspace partition (also called "vector space partition") of V(n,q) is a collection of subspaces of V(n,q) with the property that every nonzero element of V(n,q) appears in exactly one of these subspaces. Given positive integers a,b,n such that 1≤a

We consider the statement of a terminal control problem that partitions system state coordinates into two types: slowly varying coordinates, which occur in the boundary conditions, and control loop coordinates. We present a generalization of the theorem on the derivative of the mismatches in the boundary conditions. The system is discretized by using this generalization. A criterion for the resulting discrete system to be reducible to a linear system by a change of the control is proved. This result is used to synthesize a model predictive controller for the system. The algorithm has been numerically simulated for the problem of putting the center of mass of a rocket stage into a desired orbit in vacuum. [ABSTRACT FROM AUTHOR]

For each integer partition q with d parts, we denote by Δ (1 , q) the lattice simplex obtained as the convex hull in R d of the standard basis vectors along with the vector - q . For q with two distinct parts such that Δ (1 , q) is reflexive and has the integer decomposition property, we establish a characterization of the lattice points contained in Δ (1 , q) . We then construct a Gröbner basis with a squarefree initial ideal of the toric ideal defined by these simplices. This establishes the existence of a regular unimodular triangulation for reflexive 2-supported Δ (1 , q) having the integer decomposition property. As a corollary, we obtain a new proof that these simplices have unimodal Ehrhart h ∗ -vectors. [ABSTRACT FROM AUTHOR]

We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set X ⊆ Z 2 , there are two naturally associated monoids: F X , the monoid of all X-walks/paths; and A X , the monoid of all endpoints of X-walks starting from the origin O. For each A ∈ A X , write π X (A) for the number of X-walks from O to A. Calculating the numbers π X (A) is a classical problem, leading to Fibonacci, Catalan, Motzkin, Delannoy and Schröder numbers, among many other well-studied sequences and arrays. Our main results give relationships between finiteness properties of the numbers π X (A) , geometrical properties of the step set X, algebraic properties of the monoid A X , and combinatorial properties of a certain bi-labelled digraph naturally associated to X. There is an intriguing divergence between the cases of finite and infinite step sets, and some constructions rely on highly non-trivial properties of real numbers. We also consider the case of walks constrained to stay within a given region of the plane. Several examples are considered throughout to highlight the sometimes-subtle nature of the theoretical results. [ABSTRACT FROM AUTHOR]

Partitions appear across physics like Bose-Einstein statistics, dimer coverings, Potts model, crystal growth, Vasiliev theory, quasicrystals, Penrose-like random tilings, and so on. Since the general analytical study of partitions is rather tough, so statistical study is immensely preferred. This paper derives an asymptotic formula to evaluate the m -th raw moment of the multiplicity of a part for a broad category of partitions given by a specific generating function. In addition to being a multifold refinement/generalization of results in the literature, this work allows to calculate useful statistical quantities like mean and variance of the number of appearance of a part which is of deeper significance to analysts. The obtained expression is numerically validated to be in close agreement with the actual values. This study instigates the development of a general paradigm where partition theory is exposed to more sophisticated statistical analysis, thus, opening new avenues of research in this field. [ABSTRACT FROM AUTHOR]

A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and 0-large generic extensions of the constructible universe L. More specifically, if a ∈ 2 ω is either Sacks generic or 0-large generic real over L, then it is true in L[a] that there is a lightface Π 2 1 equivalence relation Q on the Π 2 1 set U = 2 ω \ L with exactly two equivalence classes, and both those classes are non-OD sets. [ABSTRACT FROM AUTHOR]

A famous conjecture of Parkin-Shanks predicts that p (n) is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the "striking" fact that, if for any A ≡ ± 1 (mod 6) the multipartition function p A (n) has positive odd density, then so does p (n). Similarly, the positive odd density of any p A (n) with A ≡ 3 (mod 6) would imply that of p 3 (n). Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two methods to express the natural powers of 2 as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients as a sum over partitions. The second method invokes the central binomial coefficients and the logarithmic differentiation of their generating function. Some experimental results suggest the existence of other methods of decomposing the power of 2 as sums over partitions. [ABSTRACT FROM AUTHOR]

One of the highest causes of death in many countries in this modern era is cardiovascular diseases. There are a few symptoms that linked significantly to the cardiovascular diseases. The symptoms and diseases relationship can be represented by neutrosophic set values. In general, neutrosophic set gives remarkable contribution in denoising, clustering, segmentation, and classification in handling data of many real applications including in medical field. This study aims to analyse the cardiovascular disease risks by a new distance-based similarity measure motivated from intuitionistic fuzzy set theory. The proof for all the properties is presented clearly. Then, a case study is conducted by using the data on the severity level of the six symptoms found in two different patients. The neutrosophic data are analysed to determine the patients' possibility of having any one or combination of the three types of cardiovascular diseases. A comparative study involving three common distance measures is conducted. The results show that the similarity indexes for all measures of both patients are less than 0.5. This situation can further conclude as both patients are possibly not suffering from cardiovascular diseases. [ABSTRACT FROM AUTHOR]

While presenting a combinatorial point of view to the class of equistable graphs, Miklavič and Milanič pointed out the inclusions among the classes of equistable, general partition and triangle graphs. Orlin conjectured that the first two classes were equivalent, but in 2014, Milanič and Trotignon showed that the three classes are all distinct, leading to the following hierarchy: general partition graphs ⊂ equistable graphs ⊂ triangle graphs. In this paper, we solve an open problem from Anbeek et al. (1997) by showing that all the above classes are equivalent when restricted to planar graphs. Additionally, we present short proofs on some structural properties of the general partition, equistable and triangle classes. In particular, we show that the general partition and triangle classes are both closed under the operations of substitution, induction and contraction of modules which allow us to recover several results in the literature. [ABSTRACT FROM AUTHOR]

We describe the qFunctions Mathematica package for q -series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q -difference equations and recurrences for given q -difference and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations on q -differential, q -difference equations and recurrences, such as switching between these forms, finding the greatest common divisor of recurrences, and formal substitutions. The package qFunctions has implementations that automate the recurrence system creation of the weighted words approach as well as a scheme on cylindric partitions. We also discuss a slight extension of the classical method of weighed words and present some new sum-product identities. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite and σ is some partition of the set of all primes P (that is, = {i | i 2 I}, where P = [i2Ii and i \ j =; for all i 6= j). A subgroup A of a group G is said to be -subnormal in G if there is a subgroup chain A = A0 A1 · · · An = G such that either Ai-1 E Ai or Ai/(Ai-1)Ai is a j -group for some j = j(i) for all i = 1, . . ., n. In this review, we discuss some known results of the theory of -subnormal sub-groups and also some open questions in this line research. [ABSTRACT FROM AUTHOR]

The isometric path cover (partition) problem of a graph consists of finding a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality of a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric k-path partition problem for k x 3 are NP-complete on general graphs. Fisher and Fitzpatrick in [The isometric number of a graph, J. Combin. Math. Combin. Comput. 38 (2001) 97{110] have shown that the isometric path cover number of the (r x r)-dimensional grid is d2r=3e. We show that the isometric path cover (partition) number of the (r x s)-dimensional grid is s when r x s(s - 1). We establish that the isometric path cover (partition) number of the (r x r)-dimensional torus is r when r is even and is either r or r+1 when r is odd. Then, we demonstrate that the isometric path cover (partition) number of an r-dimensional Benes network is 2r. In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and multi-dimensional grids. We apply two different techniques to achieve these results. [ABSTRACT FROM AUTHOR]

The character integral representation of one loop partition functions is useful to establish the relation between partition functions of conformal fields on Weyl equivalent spaces. The Euclidean space Sa × AdSb can be mapped to Sa+b provided Sa and AdSb are of the same radius. As an example, to begin with, we show that the partition function in the character integral representation of conformally coupled free scalars and fermions are identical on Sa × AdSb and Sa+b. We then demonstrate that the partition function of higher derivative conformal scalars and fermions are also the same on hyperbolic cylinders and branched spheres. The partition function of the four-derivative conformal vector gauge field on the branched sphere in d = 6 dimension can be expressed as an integral over 'naive' bulk and 'naive' edge characters. However, the partition function of the conformal vector gauge field on S q 1 × AdS5 contains only the 'naive' bulk part of the partition function. This follows the same pattern which was observed for the partition of conformal p-form fields on hyperbolic cylinders. We use the partition function of higher derivative conformal fields on hyperbolic cylinders to obtain a linear relationship between the Hofman-Maldacena variables which enables us to show that these theories are non-unitary. [ABSTRACT FROM AUTHOR]

For fixed m and R ⊆ { 0 , 1 , ... , m − 1 } , take A to be the set of positive integers congruent modulo m to one of the elements of R, and let p A (n) be the number of ways to write n as a sum of elements of A. Nathanson proved that log p A (n) ≤ (1 + o (1)) π 2 n | R | / 3 m using a variant of a remarkably simple method devised by Erdős in order to bound the partition function. In this short note, we describe a simpler and shorter proof of Nathanson's bound. [ABSTRACT FROM AUTHOR]

We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let $p_{j,k,m} (n)$ be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for $m \geq 2$. We prove that $p_{1,0,m} (n)$ is in general larger than $p_{0,1,m} (n)$. We also obtain asymptotic formulas for $p_{1,0,m}(n)$ and $p_{0,1,m}(n)$ for $m \geq 2$. [ABSTRACT FROM AUTHOR]

We investigate the sum of the parts in all the partitions of n into distinct parts and give two infinite families of linear inequalities involving this sum. The results can be seen as new connections between partitions and divisors. [ABSTRACT FROM AUTHOR]

For any positive integer k ≥ 1 , let p 1 / k (n) be the number of solutions of the equation n = [ a 1 k ] + ⋯ + [ a d k ] with integers a 1 ≥ ⋯ ≥ a d ≥ 1 , where [ t ] is the integral part of real number t. Recently, Luca and Ralaivaosaona gave an asymptotic formula for p 1 / 2 (n). In this paper, we give an asymptotic development of p 1 / k (n) for all k ≥ 1. Moreover, we prove that the number of such partitions is even (respectively, odd) infinitely often. [ABSTRACT FROM AUTHOR]

Let f : ℤ + → ℤ + be a polynomial with the property that corresponding to every prime p there exists an integer ℓ such that p ∤ f (ℓ). In this paper, we establish some equidistributed results between the number of partitions of an integer n whose parts are taken from the sequence { f (ℓ) } ℓ = 1 ∞ and the number of parts of those partitions which are in a certain arithmetic progression. [ABSTRACT FROM AUTHOR]

We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given 1-Lipschitz map u : R n → R m , m ≤ n , we define and prove the existence of a partition of R n , up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of u is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension m , the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag. [ABSTRACT FROM AUTHOR]

Semantic segmentation has been a major topic in research and industry in recent years. However, due to the computation complexity of pixel-wise prediction and backpropagation algorithm, semantic segmentation has been demanding in computation resources, resulting in slow training and inference speed and large storage space to store models. Existing schemes that speed up segmentation network change the network structure and come with noticeable accuracy degradation. However, neural network quantization can be used to reduce computation load while maintaining comparable accuracy and original network structure. Semantic segmentation networks are different from traditional deep convolutional neural networks (DCNNs) in many ways, and this topic has not been thoroughly explored in existing works. In this paper, we propose a new quantization framework for training and inference of segmentation networks, where parameters and operations are constrained to 8-bit integer-based values for the first time. Full quantization of the data flow and the removal of square and root operations in batch normalization give our framework the ability to perform inference on fixed-point devices. Our proposed framework is evaluated on mainstream semantic segmentation networks like FCN-VGG16 and DeepLabv3-ResNet50, achieving comparable accuracy against floating-point framework on ADE20K dataset and PASCAL VOC 2012 dataset. [ABSTRACT FROM AUTHOR]

Partitions, the partition function p(n), and the hook lengths of their Ferrers–Young diagrams are important objects in combinatorics, number theory, and representation theory. For positive integers n and t, we study p t e (n) (resp. p t o (n) ), the number of partitions of n with an even (resp. odd) number of t-hooks. We study the limiting behavior of the ratio p t e (n) / p (n) , which also gives p t o (n) / p (n) , since p t e (n) + p t o (n) = p (n) . For even t, we show that lim n → ∞ p t e (n) p (n) = 1 2 , and for odd t, we establish the non-uniform distribution lim n → ∞ p t e (n) p (n) = 1 2 + 1 2 (t + 1) / 2 if 2 ∣ n , 1 2 - 1 2 (t + 1) / 2 otherwise. Using the Rademacher circle method, we find an exact formula for p t e (n) and p t o (n) , and this exact formula yields these distribution properties for large n. We also show that for sufficiently large n, the sign of p t e (n) - p t o (n) is periodic. [ABSTRACT FROM AUTHOR]

For positive integers L ≥ 3 and s, Berkovich and Uncu (Ann Comb 23:263–284, 2019) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval { s , ... , L + s } . Further restrictions are placed on the sets by specifying impermissible parts as well as a minimum part. The authors proved their conjecture for the cases s = 1 and s = 2 . In the present article, we prove their conjecture for general s by proving a stronger theorem. We also prove other related conjectures found in the same paper. [ABSTRACT FROM AUTHOR]

We show that the determinant of the co-exact p-form on spheres and anti-de Sitter spaces can be written as an integral transform of bulk and edge Harish-Chandra characters. The edge character of a co-exact p-form contains characters of anti-symmetric tensors of rank lower to p all the way to the zero-form. Using this result we evaluate the partition function of p-forms and demonstrate that they obey known properties under Hodge duality. We show that the partition function of conformal forms in even d + 1 dimensions, on hyperbolic cylinders can be written as integral transforms involving only the bulk characters. This supports earlier observations that entanglement entropy evaluated using partition functions on hyperbolic cylinders do not contain contributions from the edge modes. For conformal coupled scalars we demonstrate that the character integral representation of the free energy on hyperbolic cylinders and branched spheres coincide. Finally we propose a character integral representation for the partition function of p-forms on branched spheres. [ABSTRACT FROM AUTHOR]

In this paper, we show that the difference between the number of parts in the odd partitions of n and the number of parts in the distinct partitions of n satisfies Euler's recurrence relation for the partition function p(n) when n is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of n is also derived. In this context, we conjecture that for k > 0, the series has non-negative coefficients. [ABSTRACT FROM AUTHOR]

The minimal excludant, or "mex" function, on a set S of positive integers is the least positive integer not in S. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely p t , t (n) , and provide complete parity characterizations of p 1 , 1 (n) and p 3 , 3 (n) . In this article, we study the parity of p t , t (n) when t = 2 α , 3 · 2 α for all α ≥ 1 . We prove that p 2 α , 2 α (n) and p 3 · 2 α , 3 · 2 α (n) are almost always even for all α ≥ 1 . Using a result of Ono and Taguchi on nilpotency of Hecke operators, we also find infinite families of congruences modulo 2 satisfied by p 2 α , 2 α (n) and p 3 · 2 α , 3 · 2 α (n) for all α ≥ 1 . [ABSTRACT FROM AUTHOR]

This article concerns Ehresmann structures in the partition monoid P X. Since P X contains the symmetric and dual symmetric inverse monoids on the same base set X , it naturally contains the semilattices of idempotents of both submonoids. We show that one of these semilattices leads to an Ehresmann structure on P X while the other does not. We explore some consequences of this (structural/combinatorial and representation theoretic), and in particular characterise the largest left-, right- and two-sided restriction submonoids. The new results are contrasted with known results concerning relation monoids, and a number of interesting dualities arise, stemming from the traditional philosophies of inverse semigroups as models of partial symmetries (Vagner and Preston) or block symmetries (FitzGerald and Leech): "surjections between subsets" for relations become "injections between quotients" for partitions. We also consider some related diagram monoids, including rook partition monoids, and state several open problems. [ABSTRACT FROM AUTHOR]

A folklore conjecture on the partition function asserts that the density of odd values of p (n) is 1 2. In general, for a positive integer t , let p t (n) be the t -multipartition function and δ t be the density of the odd values of p t (n). It is widely believed that δ t exists. Given an odd integer a and an integer b depending on a and t , Judge and Zanello framed an infinite family of conjectural congruence relations on p t (a n + b) (mod 2) which establishes a striking connection between δ a and δ 1. As a special case t = 1 , it implies that δ 1 > 0 if (3 , a) = 1 and δ a > 0. This conjecture was proved for several values of a by Judge, Keith and Zanello. In this paper we prove that the conjecture is true for a = ℓ α is a prime power with ℓ ≥ 5 and a = 3. [ABSTRACT FROM AUTHOR]

Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra ℂ R n and the subalgebra ℂ I k of the partition algebra ℂ A k (n) acting on (ℂ n) ⊗ k . In this paper, we consider a subalgebra ℂ I k + 1 2 of ℂ I k + 1 such that there is Schur–Weyl duality between the actions of ℂ R n − 1 and ℂ I k + 1 2 on (ℂ n) ⊗ k . This paper studies the representation theory of partition algebras ℂ I k and ℂ I k + 1 2 for rook monoids inductively by considering the multiplicity free tower ℂ I 1 ⊂ ℂ I 3 2 ⊂ ℂ I 2 ⊂ ⋯ ⊂ ℂ I k ⊂ ℂ I k + 1 2 ⊂ ⋯. Furthermore, this inductive approach is established as a spectral approach by describing the Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of ℂ I k and ℂ I k + 1 2 . Also, we describe the Jucys–Murphy elements of ℂ R n which play a central role in the demonstration of the actions of Jucys–Murphy elements of ℂ I k and ℂ I k + 1 2 . [ABSTRACT FROM AUTHOR]

Let G be a finite group. The set of all supercharacter theories of G forms a lattice, where the join operation coincides with the join operation on the lattice of partitions of G , with partial order given by refinement. The meet operation is more complicated however, and seems difficult to describe. In this paper, we outline algorithms for determining the coarsest supercharacter theory whose associated partition is finer than a given partition. One of the primary applications is to compute the supercharacters and superclasses for the meet of two supercharacter theories. [ABSTRACT FROM AUTHOR]

Two closely related discrete probability distributions are introduced. In each case the support is a set of vectors in R n obtained from the partitions of the fixed positive integer n. These distributions arise naturally when considering equally-likely random permutations on the set of n letters. For one of the distributions, the expectation vector and covariance matrix is derived. For the other distribution, conjectures for several elements of the expectation vector are provided. [ABSTRACT FROM AUTHOR]

Nested Sampling (NS) is a powerful athermal statistical mechanical sampling technique that directly calculates the partition function, and hence gives access to all thermodynamic quantities in absolute terms, including absolute free energies and absolute entropies. NS has been used predominately to compute the canonical (NVT) partition function. Although NS has recently been used to obtain the isothermal-isobaric (NPT) partition function of the hard sphere model, a general approach to the computation of the NPT partition function has yet to be developed. Here, we describe an isobaric NS (IBNS) method which allows for the computation of the NPT partition function of any atomic system. We demonstrate IBNS on two finite Lennard-Jones systems and confirm the results through comparison to parallel tempering Monte Carlo. Temperature-entropy plots are constructed as well as a simple pressure-temperature phase diagram for each system. We further demonstrate IBNS by computing part of the pressure-temperature phase diagram of a Lennard-Jones system under periodic boundary conditions. [ABSTRACT FROM AUTHOR]

The concept of Lucky k-polynomials and in particular Lucky X-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky X-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind. [ABSTRACT FROM AUTHOR]

For each prime p ≡ 1 (mod 4) consider the Legendre character χ = (⋅ p). Let p ± (n) be the number of partitions of n into parts λ > 0 such that χ (λ) = ± 1. Petersson proved a beautiful limit formula for the ratio of p + (n) to p − (n) as n → ∞ expressed in terms of important invariants of the real quadratic field K = ℚ (p). But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz–Cesàro theorem. In this paper, we suggest an approach to address Grosswald's conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman–Erdős. [ABSTRACT FROM AUTHOR]

For each integer k \geq 2, we determine a sharp bound on mad(G) such that V (G) can be partitioned into sets I and Fk, where I is an independent set and G[Fk] is a forest in which each component has at most k vertices. For each k we construct an infinite family of examples showing our result is the best possible. Our results imply that every planar graph G of girth at least 9 (resp., 8, 7) has a partition of V (G) into an independent set I and a set F such that G[F] is a forest with each component of order at most 3 (resp., 4, 6). Hendrey, Norin, and Wood asked for the largest function g(a, b) such that if mad(G) < g(a, b), then V (G) has a partition into sets A and B such that mad(G[A]) < a and mad(G[B]) < b. They specifically asked for the value of g(1, b), i.e., the case when A is an independent set. Previously, the only values known were g(1, 4/3) and g(1, 2). We find g(1, b) whenever 4/3 < b < 2. [ABSTRACT FROM AUTHOR]

We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid over the set of positive kth powers provided the equation has at least (1 + o(1))k log k variables. We thus completely describe which diagonal forms are partition regular and which are not, given sufficiently many variables. In addition, we prove a supersaturated version of Rado's theorem for a linear equation restricted either to squares minus one or to logarithmically-smooth numbers. [ABSTRACT FROM AUTHOR]

We present a minimization algorithm for non-deterministic finite state automata that finds and merges bisimulation-equivalent states. The bisimulation relation is computed through partition aggregation, in contrast to existing algorithms that use partition refinement. The algorithm simultaneously generalises and simplifies an earlier one by Watson and Daciuk for deterministic devices. We show the algorithm to be correct and run in time O n 2 r 2 Σ , where n is the number of states of the input automaton M , r is the maximal out-degree in the transition graph for any combination of state and input symbol, and Σ is the size of the input alphabet. The algorithm has a higher time complexity than derivatives of Hopcroft's partition-refinement algorithm, but represents a promising new solution approach that preserves language equivalence throughout the computation process. Furthermore, since the algorithm essentially computes the maximal model of a logical formula derived from M , optimisation techniques from the field of model checking become applicable. [ABSTRACT FROM AUTHOR]