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In this work, we establish two interesting partition identities involving the minimal odd excludant, which has attracted great attention in recent years. In particular, we find a strong refinement of Euler's celebrated theorem that the number of partitions of an integer into odd parts equals the number of partitions of that integer into distinct parts. [ABSTRACT FROM AUTHOR]

Let N k (m , n) denote the number of partitions of n with Garvan k -rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k -rank for k = 1 and k = 2 , respectively. In this paper, we prove that the sequences (N k (m , n)) | m | ≤ n − k − 71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently. [ABSTRACT FROM AUTHOR]

Ridesharing is an effective approach to alleviate traffic congestion. In most existing works, drivers and passengers are assigned prices without considering the constraints of desired benefits. This paper investigates ridesharing by formulating a matching and pricing problem to maximize the total payoff of drivers, with the constraints of desired benefit and quality of experience. An efficient algorithm is proposed to solve the formulated problem based on coalitional double auction. Secondary pricing based strategy and sacrificed minimum bid based strategy are proposed to support the algorithm. This paper also proves that the proposed algorithm can achieve a Nash-stable coalition partition in finite steps, and the proposed two strategies guarantee truthfulness, individually rational and budget balance. Extensive simulation results on the real-world dataset of taxi trajectory in Beijing city show that the proposed algorithm outperforms the existing ones, in terms of average total payoff of drivers while meeting the benefits of passengers. [ABSTRACT FROM AUTHOR]

Given a set A of non-negative integers and a set B of positive integers, we are interested in computing all sets C (of positive integers) that are minimal in the family of sets K (of positive integers) such that (i) K contains no elements generated by non-negative integer linear combinations of elements in A and (ii) for any partition of an element in B there is at least one summand that belongs to K. To solve this question, we translate it into a numerical semigroups problem. [ABSTRACT FROM AUTHOR]

An algorithm for obtaining the Chaundy-Bullard identity for a vector partition function with weight that uses computer algebra methods is proposed. To automate this process in Maple, an algorithm was developed and implemented that calculates the values of the vector partition function with weight by finding non-negative solutions of systems of linear Diophantine equations that are used to form the identities involved. The algorithm's input data is represented by the set of integer vectors that form a pointed lattice cone and by some point from this cone, and the Chaundy-Bullard identity for the vector partition function with weight is its output. The code involved is stored in the depository and is ready-to-use. An example demonstrating the algorithm's operation is given. [ABSTRACT FROM AUTHOR]

Given an integer partition of n , we consider the impartial combinatorial game LCTR in which moves consist of removing either the left column or top row of its Young diagram. We show that for both normal and misère play, the optimal strategy can consist mostly of mirroring the opponent's moves. We also establish that both LCTR and Downright are domestic as well as returnable, and on the other hand neither tame nor forced. For both games, those structural observations allow for computing the Sprague–Grundy value any position in O (log (n)) time, assuming that the time unit allows for reading an integer, or performing a basic arithmetic operation. This improves on the previously known bound of O (n) due to Ilić (2019). We also cover some other complexity measures of both games, such as state–space complexity, and number of leaves and nodes in the corresponding game tree. • We study the impartial combinatorial game LCTR and its misère variant Downright. • There are two allowable moves: remove the left column or top row of a Young diagram. • For both normal and misère play, the optimal move often mirrors the oppononent. • Both LCTR and Downright are domestic and returnable, and neither tame nor forced. • We describe an algorithm to compute the Sprague–Grundy value in O (log (n)) time. • We cover state–space complexity, and number of leaves and nodes in the game tree. [ABSTRACT FROM AUTHOR]

The new logic of partitions is dual to the usual Boolean logic of subsets (usually presented only in the special case of the logic of propositions) in the sense that partitions and subsets are category-theoretic duals. The new information measure of logical entropy is the normalized quantitative version of partitions. The new approach to interpreting quantum mechanics (QM) is showing that the mathematics (not the physics) of QM is the linearized Hilbert space version of the mathematics of partitions. Or, putting it the other way around, the math of partitions is a skeletal version of the math of QM. The key concepts throughout this progression from logic, to logical information, to quantum theory are distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. The distinctions of a partition are the ordered pairs of elements from the underlying set that are in different blocks of the partition and logical entropy is defined (initially) as the normalized number of distinctions. The cognate notions of definiteness and distinguishability run throughout the math of QM, e.g., in the key non-classical notion of superposition (=ontic indefiniteness) and in the Feynman rules for adding amplitudes (indistinguishable alternatives) versus adding probabilities (distinguishable alternatives). [ABSTRACT FROM AUTHOR]

We introduce two lattice growth models: aggregation of l -dimensional boxes and aggregation of partitions with l parts. We describe properties of the models: the parameter set of aggregations, the moments of the random variable of the number of growth directions, asymptotical behavior of proportions of the most frequent transitions of two- and three-dimensional self-aggregations. [ABSTRACT FROM AUTHOR]

A generic rectangular partition is a partition of a rectangle into a finite number of rectangles provided that no four of them meet at a point. A graph H is called dual of a plane graph G if there is one-to-one correspondence between the vertices of G and the regions of H, and two vertices of G are adjacent if and only if the corresponding regions of H are adjacent. A plane graph is a rectangularly dualizable graph if its dual can be embedded as a rectangular partition. A rectangular dual R of a plane graph G is a partition of a rectangle into n-rectangles such that (i) no four rectangles of R meet at a point, (ii) rectangles in R are mapped to vertices of G, and (iii) two rectangles in R share a common boundary segment if and only if the corresponding vertices are adjacent in G. In this paper, we derive a necessary and sufficient for a rectangularly dualizable graph G to admit a unique rectangular dual upto combinatorial equivalence. Further we show that G always admits a slicible as well as an area-universal rectangular dual. [ABSTRACT FROM AUTHOR]

For any finite group 퐺 and a positive integer 푚, we define and study a Schur ring over the direct power G m , which gives an algebraic interpretation of the partition of G m obtained by the 푚-dimensional Weisfeiler–Leman algorithm. It is proved that this ring determines the group 퐺 up to isomorphism if m ≥ 3 , and approaches the Schur ring associated with the group Aut ⁡ (G) acting on G m naturally if 푚 increases. It turns out that the problem of finding this limit ring is polynomial-time equivalent to the group isomorphism problem. [ABSTRACT FROM AUTHOR]

Robinson space is a dissimilarity space (X, d) (i.e., a set X of size n and a dissimilarity d on X) for which there exists a total order < on X such that x < y < z implies that d(x, z) \geq max\{ d(x, y), d(y, z)\}. Recognizing if a dissimilarity space is Robinson has numerous applications in seriation and classification. An mmodule of (X, d) (generalizing the notion of a module in graph theory) is a subset M of X which is not distinguishable from the outside of M; i.e., the distance from any point of X \setminus M to all points of M is the same. If p is any point of X, then \{ p\}, and the maximal-by-inclusion mmodules of (X, d) not containing p define a partition of X, called the copoint partition. In this paper, we investigate the structure of mmodules in Robinson spaces and use it and the copoint partition to design a simple and practical divide-and-conquer algorithm for recognition of Robinson spaces in optimal O(n²) time. [ABSTRACT FROM AUTHOR]

Andrews and Newman introduced the mex-function mex A , a (λ) for an integer partition λ of a positive integer n as the smallest positive integer congruent to a modulo A that is not a part of λ . They then defined p A , a (n) to be the number of partitions λ of n satisfying mex A , a (λ) ≡ a (mod 2 A) . They found the generating function for p t , t (n) and p 2 t , t (n) for any positive integer t, and studied their arithmetic properties for some small values of t. In this article, we study the partition function p m t , t (n) for all positive integers m and t. We show that for sufficiently large X, the number of all positive integers n ≤ X such that p m t , t (n) is an even number is at least O (X / 3) for all positive integers m and t. We also prove that for sufficiently large X, the number of all positive integers n ≤ X such that p m p , p (n) is an odd number is at least O (log log X) for all m ≢ 0 (mod 3) and all primes p ≡ 1 (mod 3) . Finally, we establish identities connecting the ordinary partition function to p m t , t (n) . [ABSTRACT FROM AUTHOR]

Let A = (a i) i = 1 ∞ be a non-decreasing sequence of positive integers and let k ∈ N + be fixed. The function p A (n , k) counts the number of partitions of n with parts in the multiset { a 1 , a 2 , ... , a k }. We find out a new Bessenrodt-Ono type inequality for the function p A (n , k). Further, we discover when and under what conditions on k , { a 1 , a 2 , ... , a k } and N ∈ N + , the sequence (p A (n , k)) n = N ∞ is log-concave. Our proofs are based on the asymptotic behavior of p A (n , k) — in particular, we apply the results of Netto and Pólya-Szegő as well as the Almkavist's estimation. [ABSTRACT FROM AUTHOR]

Partitioning, in and of itself, is an NP-hard problem. Prior to the Artificial Intelligence (AI)-based solutions, it was solved in the 1970s by optimization-based strategies. However, AI-based solutions appeared in the 1980s in a pioneering way, by using a Learning Automaton (LA)-motivated strategy known as the so-called Object Migrating Automaton (OMA). Although the OMA and its derivatives have been used in numerous applications since then, the basic kernel has remained the same. Because the number of possible partitions in a partitioning problem can be combinatorially exponential and the underlying tasks are NP-hard, the most advanced OMA algorithms could, until recently, only solve issues involving equally sized groups. Due to our recent innovations cited in the body of this paper, the enhanced OMA now also handles non-equally sized groups. Earlier, we had presented in Omslandseter (Pattern Anal Appl, 2023), a comprehensive survey of the state-of-the-art enhancements of the best-known OMA. We believe that these results will be the benchmark for a few decades and that it will be very hard to beat these results. This is a companion paper, intended to augment the contents of Omslandseter (Pattern Anal Appl, 2023). In this paper, we first discuss the OMA's prior applications, its historical and current innovations, and the OMA-based algorithms' relevance to societal needs. We also provide well-specified guidelines for future researchers so that they can use them for unresolved tasks, and also develop further advancements. [ABSTRACT FROM AUTHOR]

Let ℓ ≥ 5 be prime. For the partition function p (n) and 5 ≤ ℓ ≤ 31 , Atkin found a number of examples of primes Q ≥ 5 such that there exist congruences of the form p (ℓ Q 3 n + β) ≡ 0 (mod ℓ). Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every ℓ. In this paper, for a wide range of c ∈ F ℓ , we prove congruences of the form p (ℓ Q 3 n + β 0) ≡ c ⋅ p (ℓ Q n + β 1) (mod ℓ) for infinitely many primes Q. For a positive integer r , let p r (n) be the r -colored partition function. Our methods yield similar congruences for p r (n). In particular, if r is an odd positive integer for which ℓ > 5 r + 19 and 2 r + 2 ≢ 2 ± 1 (mod ℓ) , then we show that there are infinitely many congruences of the form p r (ℓ Q 3 n + β) ≡ 0 (mod ℓ). Our methods involve the theory of modular Galois representations. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, we study the number of partitions of a positive integer m into at most n parts in a given set A. We prove that such a number is bounded by the n-th Fibonacci number F(n) for any m and some family of sets A including sets of powers of an integer. Then, in the second part of the paper, we provide new results in bounding the cohomology of the simple algebraic group SL2 with coefficients in Weyl modules. [ABSTRACT FROM AUTHOR]

Data clustering is an unsupervised technique that can be used to partition the data into groups based on the similarities of the retrieved objects using different distance metrics like Euclidean, cosine, etc. In contrast to Euclidean, the cosine computes the objects similarity by considering both the magnitude and direction of the data vectors. As a result, it performed far better than a standard Euclidean distance metric in applications involving real-time data clustering. The initial k-value (clustering tendency) is required by top clustering techniques like k-means and hierarchical approaches to determine the clusters' quality. Users with knowledge can assign the k-value. However, sometimes the right k-value in such algorithms may need to be assigned. After a thorough review of the work, it was discovered that the visual technique known as visual assessment of (cluster) tendency (VAT) effectively addresses the clustering tendency issue. It uses the Euclidean metric to find the similarity features in its algorithm. Another enhanced visual technique, cosine-based VAT(cVAT), outperformed the VAT for text data and speech clustering applications. However, the similarity features are extracted about a single viewpoint in cVAT. This paper develops the multi-viewpoints-based cosine similarity measure (MVPCSM) for a more informative assessment. Instead of using a single reference point like a typical cosine measure, the MVPCSM generates precise similarity characteristics using several views. The performance of the existing and proposed technique (MVPCSM-VAT) is evaluated using clustering accuracy (CA) and normalized mutual information (NMI). It has been demonstrated that the proposed MVPCSM-VAT is 15-25% more efficient than VAT and cVAT in terms of the parameters of CA and NMI. The proposed method successfully obtains more quality data clusters than MVS-VAT. [ABSTRACT FROM AUTHOR]

Suppose s and t are coprime positive integers, and let σ be an s-core partition and τ a t-core partition. In this paper, we consider the set P σ , τ (n) of partitions of n with s-core σ and t-core τ . We find the smallest n for which this set is non-empty, and show that for this value of n the partitions in P σ , τ (n) (which we call (σ , τ) -minimal partitions) are in bijection with a certain class of (0, 1)-matrices with s rows and t columns. We then use these results in considering conjugate partitions: we determine exactly when the set P σ , τ (n) consists of a conjugate pair of partitions, and when P σ , τ (n) contains a unique self-conjugate partition. [ABSTRACT FROM AUTHOR]

In this paper, we give an iterated approach to concern with the positivity of \begin{equation*} \det \ (p(n-i+j))_{1\leq i,j\leq k}, \end{equation*} where p(n) is the partition function. We first apply a general method to prove that for given k_1,k_2,m_1,m_2, one can find a threshold N(k_1,k_2,m_1,m_2) such that for n>N(k_1,k_2,m_1,m_2), \begin{equation*} \begin {vmatrix} p(n-k_1+m_1) & p(n+m_1) & p(n+m_1+m_2)\\ p(n-k_1) & p(n) & p(n+m_2)\\ p(n-k_1-k_2) & p(n-k_2) & p(n-k_2+m_2) \end{vmatrix}>0. \end{equation*} Based on this result, we will prove that for n\geq 656, \det \ (p(n-i+j))_{1\leq i,j\leq 4}>0. Employing the same technique, we will show that determinants ({\bar p}(n-i+j))_{1\leq i,j\leq k} are positive for k=3 \text { and } 4 for overpartition {\bar p}(n). Furthermore, we will give an outline of how to prove the positivity of \det \ (p(n-i+j))_{1\leq i,j\leq k} for general k. [ABSTRACT FROM AUTHOR]

We discuss the double scaling limit of the SYK model with global symmetries. We develop the chord diagram techniques to compute the moments of the Hamiltonian and the two point function in the presence of arbitrary chemical potential. We also derive a transfer matrix acting on an auxiliary hilbert space which can capture the chord diagram contributions. We present explicit results for the case of classical group symmetries namely orthogonal, unitary and symplectic groups. We also find the partition functions at fixed charges. [ABSTRACT FROM AUTHOR]

We answer a nonnegativity problem of G. E. Andrews and D. Newman by the q-unimodality of Gaussian polynomials. Some new considerations of the q-log-concavity and q-unimodality of Gaussian polynomials from a purely partition-theoretic perspective will also be presented. [ABSTRACT FROM AUTHOR]

Reaction rates are a complicated function of molecular interactions, which can be selected from vast chemical design spaces. Seeking the design that optimizes a rate is a particularly challenging problem since the rate calculation for any one design is itself a difficult computation. Toward this end, we demonstrate a strategy based on transition path sampling to generate an ensemble of designs and reactive trajectories with a preference for fast reaction rates. Each step of the Monte Carlo procedure requires a measure of how a design constrains molecular configurations, expressed via the reciprocal of the partition function for the design. Although the reciprocal of the partition function would be prohibitively expensive to compute, we apply Booth's method for generating unbiased estimates of a reciprocal of an integral to sample designs without bias. A generalization with multiple trajectories introduces a stronger preference for fast rates, pushing the sampled designs closer to the optimal design. We illustrate the methodology on two toy models of increasing complexity: escape of a single particle from a Lennard-Jones potential well of tunable depth and escape from a metastable tetrahedral cluster with tunable pair potentials. [ABSTRACT FROM AUTHOR]

Seaweeds or seaweed Lie algebras are subalgebras of the full-matrix algebra Mat (n) introduced by Dergachev and Kirillov to give an example of algebras for which it is possible to compute the Dixmier index via combinatorial methods. It is worth noting that finding such an index for general Lie algebras is a cumbersome problem. On the other hand, Brauer configuration algebras are multiserial and symmetric algebras whose representation theory can be described using combinatorial data. It is worth pointing out that the set of integer partitions and compositions of a fixed positive integer give rise to Brauer configuration algebras. However, giving a closed formula for the dimension of these kinds of algebras or their centers for all positive integer is also a tricky problem. This paper gives formulas for the dimension of Brauer configuration algebras (and their centers) induced by some restricted compositions. It is also proven that some of these algebras allow defining seaweeds of Dixmier index one. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) with v(G) vertices the partition function of the random cluster model is defined by Z G (q , w) = ∑ A ⊆ E (G) q k (A) w | A | , where k(A) denotes the number of connected components of the graph (V, A). Furthermore, let g(G) denote the girth of the graph G, that is, the length of the shortest cycle. In this paper we show that if (G n) n is a sequence of d-regular graphs such that the girth g (G n) → ∞ , then the limit lim n → ∞ 1 v (G n) ln Z G n (q , w) = ln Φ d , q , w exists if q ≥ 2 and w ≥ 0 . The quantity Φ d , q , w can be computed as follows. Let Φ d , q , w (t) : = 1 + w q cos (t) + (q - 1) w q sin (t) d + (q - 1) 1 + w q cos (t) - w q (q - 1) sin (t) d , then Φ d , q , w : = max t ∈ [ - π , π ] Φ d , q , w (t) , The same conclusion holds true for a sequence of random d-regular graphs with probability one. Our result extends the work of Dembo, Montanari, Sly and Sun for the Potts model (integer q), and we prove a conjecture of Helmuth, Jenssen and Perkins about the phase transition of the random cluster model with fixed q. [ABSTRACT FROM AUTHOR]

We investigate properties of attainable partitions of integers, where a partition (n 1 , n 2 , ⋯ , n r) of n is attainable if ∑ (3 - 2 i) n i ≥ 0 . Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian p-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes p. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions. [ABSTRACT FROM AUTHOR]

In this note, we consider the number of k's in all the partitions of n in order to provide a new proof of a classical identity involving Euler's partition function p(n) and the sum of the positive divisors function ρ(n). New relations connecting classical functions of multiplicative number theory with the partition function p(n) from additive number theory are introduced in this context. The fascinating feature of these relations is their common nature. A new identity for the number of 1's in all the partitions of n is derived in this context. [ABSTRACT FROM AUTHOR]

The present work describes the asymptotic local shape of a graph drawn uniformly at random from all connected simple planar graphs with n labelled vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched limit in the local topology as n → ∞. We also establish such limits for random 2-connected planar graphs and maps as their number of edges tends to infinity. Our approach encompasses a new probabilistic view on the Tutte decomposition. This allows us to follow the path along the decomposition of connectivity from planar maps to planar graphs in a uniform way, basing each step on condensation phenomena for random walks under subexponentiality and Gibbs partitions. Using large deviation results, we recover the asymptotic formula by Giménez and Noy (2009) for the number of planar graphs. [ABSTRACT FROM AUTHOR]

We investigate integer partitions λ of n that are nearly self-conjugate in the sense that there are n-1 overlapping cells among the Ferrers diagram of λ and its transpose, by establishing a correspondence, through the method of combinatorial telescoping, to partitions of n in which (i). there exists at least one even part; (ii). any even part is of size 2; (iii). the odd parts are distinct; and (iv). no odd part is of size 1. In particular, this correspondence confirms a conjecture that had been given in the OEIS. [ABSTRACT FROM AUTHOR]

We prove that the symmetric group Sn has a unique minimal cover M by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the size of M. In addition, we prove that the size of M is equal to the size of a maximal non-nilpotent subset of Sn. This cover M has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of n into distinct positive integers. We show that these results contrast with those for the alternating group An. In particular, we prove that, for all but finitely many values of n, no minimal cover of An by maximal nilpotent subgroups is a normal cover and the size of a minimal cover of An by maximal nilpotent subgroups is strictly greater than the size of a maximal non-nilpotent subset of An. [ABSTRACT FROM AUTHOR]

A partition into distinct parts is refinable if one of its parts a can be replaced by two different integers which do not belong to the partition and whose sum is a, and it is unrefinable otherwise. Clearly, the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distributions of the parts. We prove a O (n 1 / 2) -upper bound for the largest part in an unrefinable partition of n, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions for triangular numbers, proving that if n is even there exists only one maximal unrefinable partition of n (n + 1) / 2 , and that if n is odd the number of such partitions equals the number of partitions of ⌈ n / 2 ⌉ into distinct parts. In the second case, an explicit bijection is provided. [ABSTRACT FROM AUTHOR]

We study a bijective map from integer partitions to the prime factorizations of integers that we call the "supernorm" of a partition, in which the multiplicities of the parts of partitions are mapped to the multiplicities of prime factors of natural numbers. The supernorm is connected to a family of maps we define, which suggests the potential to apply techniques from partition theory to identify and prove multiplicative properties of integers. As an application of "supernormal" mappings (i.e., pertaining to the supernorm statistic), we prove an analogue of a formula of Kural-McDonald-Sah to give arithmetic densities of subsets of N instead of relative densities of subsets of P like previous formulas of this type; this builds on works of Alladi, Ono, Wagner, and the first and third authors. We then make a brief study of pertinent analytic aspects of the supernorm. Finally, using a table of "supernormal" additive-multiplicative correspondences, we conjecture Abelian-type formulas that specialize to our main theorem and other known results. [ABSTRACT FROM AUTHOR]

The recently introduced concept of a bent partition of a $2m$ -dimensional vector space $\mathbb {V}_{2m}^{(p)}$ over a prime field $\mathbb {F}_{p}$ exhibits similar properties as a partition from a spread. In particular, it gives rise to a large family of bent functions obtained in the same manner as spread bent functions. We show that the first non-spread construction of bent partitions introduced by Pirsic and the third author ($p=2$), respectively, the first and the third author ($p$ odd), gives rise to a large variety of different bent partitions. Especially, we show that the sets of bent functions obtained with any two such bent partitions do not intersect. We then show that every union of sets from one of these bent partitions always forms a partial difference set. This generalizes some known results on partial difference sets from spreads. Some general results on partial difference sets from bent partitions of $\mathbb {V}_{2m}^{(2)}$ are given in the last section. [ABSTRACT FROM AUTHOR]

The average size of the "smallest gap" of a partition was studied by Grabner and Knopfmacher in 2006. Recently, Andrews and Newman, motivated by the work of Fraenkel and Peled, studied the concept of the "smallest gap" under the name "minimal excludant" of a partition and rediscovered a result of Grabner and Knopfmacher. In this paper, we study the sum of the minimal excludants over partitions into distinct parts, and interestingly we observe that it has a nice connection with Ramanujan's function σ (q). As an application, we derive a stronger version of a result of Uncu. [ABSTRACT FROM AUTHOR]

Analogous to the definition of self-conjugate n-color partitions, we introduce here self-conjugate split n-color partitions. The self-conjugate split n-color partitions arise as a modification of another, existing class of partitions. In addition to the generating function and recurrence relation of self-conjugate split n-color partitions, we find several combinatorial identities which associate these partitions with other combinatorial structures. We give a bijection from the set of split n-color partitions of a positive integer ν onto that of partitions of ν with ' ' n + 1 2 copies of n". Moreover, an explicit bijection between the set of restricted self-conjugate split n-color partitions of ν and the set of restricted n-color partitions of ν has been constructed. Some results involving new restricted split n-color partition functions are also obtained. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the mathematics of set partitions (which specify indefiniteness and definiteness) linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac's Complete Sets of Commuting Operators. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM—as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being "definite all the way down". [ABSTRACT FROM AUTHOR]

We construct weight-preserving bijections between column strict shifted plane partitions with one row and alternating sign trapezoids with exactly one column in the left half that sums to 1. Amongst other things, they relate the number of - 1 s in the alternating sign trapezoids to certain elements in the column strict shifted plane partitions that generalise the notion of special parts in descending plane partitions. The advantage of these bijections is that they include configurations with - 1 s, which is a feature that many of the bijections in the realm of alternating sign arrays lack. [ABSTRACT FROM AUTHOR]

Motivated by recent works on the connection between 2D quantum gravity and timelike Liouville theory, we revisit the latter and clarify some aspects of the computation of its partition function: we present a detailed computation of the Liouville partition function on the fluctuating sphere at finite values of the central charge. The results for both the spacelike theory and the timelike theory are given, and their properties analyzed. We discuss the derivation of the partition function from the DOZZ formula, its derivation using the Coulomb gas approach, a semiclassical computation of it using the fixed area saddle point, and, finally, we arrive to an exact expression for the timelike partition function whose expansion can be compared with the 3-loop perturbative calculations reported in the literature. We also discuss the connection to the 2D black hole and other related topics. [ABSTRACT FROM AUTHOR]

In this paper, we prove an enhanced version of the Erdős‐Lovász Tihany Conjecture for line graphs of multigraphs. That is, for every line graph G $G$ whose chromatic number χ(G) $\chi (G)$ is more than its clique number ω(G) $\omega (G)$ and for any nonnegative integer ℓ $\ell $, any two integers s,t≥3.5ℓ+2 $s,t\ge 3.5\ell +2$ with s+t=χ(G)+1 $s+t=\chi (G)+1$, there is a partition (S,T) $(S,T)$ of the vertex set V(G) $V(G)$ such that χ(G[S])≥s $\chi (G[S])\ge s$ and χ(G[T])≥t+ℓ $\chi (G[T])\ge t+\ell $. In particular, when ℓ=1 $\ell =1$, we can obtain the same result just for any s,t≥4 $s,t\ge 4$. The Erdős‐Lovász Tihany conjecture for line graphs is a special case when ℓ=0 $\ell =0$. [ABSTRACT FROM AUTHOR]

In 2011, Santos, Ribeiro and Mondek have obtained a method, using two-line arrays, to representing partitions. Using this method we provide two formulas for the evaluation of the number of integer partitions of n for classes related to the first and second Rogers-Ramanujan identities, into k ≤ n parts. [ABSTRACT FROM AUTHOR]

For an edge e = uv in a graph G, MGu (e) is introduced as the set all edges of G that are at shorter distance to u than to v. We say that G is an edge quasi-distance-balanced graph whenever for every arbitrary edge e = uv, there exists a constant λ > 1 such that mGu(e) = λ+1 mGv (e). We investigate that edge quasi-distance-balanced garphs are complete bipartite graphs Km, n with m = n. The aim of this paper is to investigate the notion of cycles in edge quasi-distance-balanced graphs, and expand some techniques generalizing new outcome that every edge quasi-distance-balanced graph is complete bipartite graph. As well as, it is demontrated that connected quasi-distance-balanced graph admitting a bridge is not edge quasi-distance-balanced graph. [ABSTRACT FROM AUTHOR]

In the 1970s Nicolas proved that the coefficients p d (n) defined by the generating function ∑ n = 0 ∞ p d (n) q n = ∏ n = 1 ∞ 1 - q n - n d - 1 are log-concave for d = 1 . Recently, Ono, Pujahari, and Rolen have extended the result to d = 2 . Note that p 1 (n) = p (n) is the partition function and p 2 (n) = pp n is the number of plane partitions. In this paper, we invest in properties for p d (n) for general d. Let n ≥ 6 . Then p d (n) is almost log-concave for n divisible by 3 and almost strictly log-convex otherwise. [ABSTRACT FROM AUTHOR]

In this paper, we take the view that the area of a black hole's event horizon is quantized, A = l P 2 (4 ln 2) N , and the associated degrees of freedom are finite in number and of fermionic nature. We then investigate general aspects of the entropy, S BH , our main focus being black hole self-similarity. We first find a two-to-one map between the black hole's configurations and the ordered partitions of the integer N. Hence, we construct from there a composition law between the subparts making the whole configuration space. This gives meaning to black hole self-similarity, entirely within a single description, as a phenomenon stemming from the well-known self-similarity of the ordered partitions of N. Finally, we compare the above to the well-known results on the subleading (quantum) corrections, which necessarily require different (quantum) statistical weights for the various configurations. [ABSTRACT FROM AUTHOR]

For integers 0 < r ≤ t , let the function D r , t (n) denote the number of parts among all partitions of n into distinct parts that are congruent to r modulo t. We prove the asymptotic formula D r , t (n) ∼ 3 1 4 e π n 3 2 π t n 1 4 log (2) + 3 log (2) 8 π - π 4 3 r - t 2 n - 1 2 as n → ∞ . A corollary of this result is that for 0 < r < s ≤ t , the inequality D r , t (n) ≥ D s , t (n) holds for all sufficiently large n. We make this effective, showing that for 2 ≤ t ≤ 10 the inequality D r , t (n) ≥ D s , t (n) holds for all n > 8 . [ABSTRACT FROM AUTHOR]

We show that while the number of coprime compositions of a positive integer n into k parts can be expressed as a Q -linear combination of the Jordan totient functions, this is never possible for the coprime partitions of n into k parts. We also show that the number p k ′ (n) of coprime partitions of n into k parts can be expressed as a C -linear combination of the Jordan totient functions, for n sufficiently large, if and only if k ∈ { 2 , 3 } and in a unique way. Finally we introduce some generalizations of the Jordan totient functions and we show that p k ′ (n) can be always expressed as a C -linear combination of them. [ABSTRACT FROM AUTHOR]

We study the problem of preclustering a set B of imprecise points in R d : we wish to cluster the regions specifying the potential locations of the points such that, no matter where the points are located within their regions, the resulting clustering approximates the optimal clustering for those locations. We consider k-center, k-median, and k-means clustering, and obtain the following results. Let B : = { b 1 , ... , b n } be a collection of disjoint balls in R d , where each ball b i specifies the possible locations of an input point p i . A partition C of B into subsets is called an (f (k) , α) -preclustering (with respect to the specific k-clustering variant under consideration) if (i) C consists of f(k) preclusters, and (ii) for any realization P of the points p i inside their respective balls, the cost of the clustering on P induced by C is at most α times the cost of an optimal k-clustering on P. We call f(k) the size of the preclustering and we call α its approximation ratio. We prove that, even in R 1 , one may need at least 3 k - 3 preclusters to obtain a bounded approximation ratio—this holds for the k-center, the k-median, and the k-means problem—and we present a (3k, 1) preclustering for the k-center problem in R 1 . We also present various preclusterings for balls in R d with d ⩾ 2 , including a (3 k , α) -preclustering with α ≈ 13.9 for the k-center and the k-median problem, and α ≈ 193.9 for the k-means problem. [ABSTRACT FROM AUTHOR]

We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica–Speicher [5] and Weber [42]. For the infinite‐dimensional Temperley–Lieb algebra (corresponding to the quantum group ON+$O^+_N$) and the Motzkin algebra (BN+$B^+_N$), the classification of extremal traces implies a classification result for well‐known types of central random lattice paths. For the 2‐Fuss–Catalan algebra (HN+$H_N^+$), we solve the classification problem by computing the minimal or exit boundary (also known as the absolute) for central random walks on the Fibonacci tree, thereby solving a probabilistic problem of independent interest, and to our knowledge the first such result for a nonhomogeneous tree. In the course of this article, we also discuss the branching graphs for all seven examples of free partition quantum groups, compute those that were not already known, and provide new formulae for the dimensions of their irreducible representations. [ABSTRACT FROM AUTHOR]

Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. We also correct a small mistake in a result of Merca and Yee. [ABSTRACT FROM AUTHOR]

The goal of this article provides the common formula for Minimal Domination Cardinality for all Hamiltonian Path or Hamiltonian Circuit. It is explained the way of structures and characteristics of General Hamiltonian Path or Hamiltonian Circuit. It invents the result that all the Hamiltonian Circuit from the given graph should be Cycle Graph. These derived formulae are common for all graphs those have Hamiltonian path or Hamiltonian Circuit. This formula is used to find Maximal Dominating Cardinality in a simple manner. Even though complicated graph also which has Hamiltonian Circuit is evaluating the Cardinality of Maximal Dominating Set is very easy way. [ABSTRACT FROM AUTHOR]

We introduce a statistic on overpartitions called the k ¯ -rank. When there are no overlined parts, this coincides with the k -rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when k = 2. The generating function for the k ¯ -rank of overpartitions is given. We also establish a relation between the generating function of self-3-conjugate overpartitions and the tenth-order mock theta functions X(q) and χ (q). [ABSTRACT FROM AUTHOR]