Pairs Of Partitions Research Articles

New Congruences for 9-Regular Cubic Partition Pairs

New Congruences for 9-Regular Cubic Partition Pairs

The Squish Map and the $SL_2$ Double Dimer Model

A plane partition, whose 3D Young diagram is made of unit cubes, can be approximated by a "coarser" plane partition, made of cubes of side length 2. Indeed, there are two such approximations obtained by "rounding up" or "rounding down" to the nearest cube. We relate this coarsening (or downsampling) operation to the squish map introduced by the second author in earlier work. We exhibit a related measure-preserving map between the dimer model on the honeycomb graph, and the SL2 double dimer model on a coarser honeycomb graph; we compute the most interesting special case of this map, related to plane partition q-enumeration with 2-periodic weights. As an application, we specialize the weights to be certain roots of unity, obtain novel generating functions (some known, some new, and some conjectural) that (-1)-enumerate certain classes of pairs of plane partitions according to how their dimer configurations interact.

AROUND THE ERDÖS–GALLAI CRITERION

By an (integer) partition we mean a non-increasing sequence \(\lambda=(\lambda_1, \lambda_2, \dots)\) of non-negative integers that contains a finite number of non-zero components. A partition \(\lambda\) is said to be graphic if there exists a graph \(G\) such that \(\lambda = \mathrm{dpt}\,G\), where we denote by \(\mathrm{dpt}\,G\) the degree partition of \(G\) composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper.

ON A CONJECTURE FOR A REFINEMENT OF THE SUM OF MINIMAL EXCLUDANTS

AbstractIn 2019, Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb.23(2) (2019), 249–254] introduced the arithmetic function $\sigma \textrm {mex}(n)$ , which denotes the sum of minimal excludants over all the partitions of n. Baruah et al. [‘A refinement of a result of Andrews and Newman on the sum of minimal excludants’, Ramanujan J., to appear] showed that the sum of minimal excludants over all the partitions of n is the same as the number of partition pairs of n into distinct parts. They proved three congruences modulo $4$ and $8$ for two functions appearing in this refinement and conjectured two further congruences modulo $8$ and $16$ . We confirm these two conjectures by using q-series manipulations and modular forms.

Validation and calibration of partitioned integral ocean wave parameters from co-polarized synthetic aperture radar data

The inversion of wave spectrum is the embodiment of the fine observation of ocean swells by synthetic aperture radar. Due to the nonlinear distortion from velocity bunching, the operational Sentinel-1 Wave Mode (WV) level-2 (L2) Ocean (OCN) wave products retrieved by quasi-linear inversion are still quite different from the real wave spectrum, requiring further calibration. Limited by the azimuth cut-off, the quasi-linear inversed wave spectrum mostly concentrates on swells with a wavelength longer than 150 m. 2D spectral partitioning separates different swell systems, denoted as partitions in the Sentinel-1 WV L2 OCN product. We compared and calibrated the Sentinel-1 WV L2 OCN products' swell partitions (More specifically partitioned integral ocean wave parameters: partitioned wave height, partitioned wavelength, and partitioned wave direction) with WAVEWATCH III (WW3) in different polarization modes (HH and VV polarization) and different incidence angles (∼23°-WV1 and ∼ 36°-WV2). The accuracy of swell partitions in the OCN products varies with peak wave direction relative to the Sentinel-1 flight direction, and it is difficult to calibrate the partitions derived from quasi-linear inversion unless deleting partitions severely distorted by nonlinearity and large-scale features. A machine learning method is proposed to calibrate the wave partitions. Spectral distances of collocated Sentinel-1 and WW3 partition pairs larger than two are considered “bad” partitions to be filtered. Inputting high-dimensional features (twenty-one features) derived from Sentinel-1 WV L2 OCN products into the extreme gradient boosting (XGBoost) model can effectively reduce the bias (root-mean-square error, RMSE) of significant wave height (SWH) of all the partitions from ∼0.10 m (∼0.70 m) to ∼0.00 m (∼0.46 m) and the bias (RMSE) of wavelengths of all the partitions from ∼20 m (∼35 m) to ∼0.20 m (∼26 m) compared with WW3 swell partitions. This method improves the precision of the wave partition and retains more partitions in the Sentinel-1 WV L2 OCN products than previous studies, especially for wave partitions with moderate wave height (< 4 m), potentially improving the understanding of swell dynamics globally.

ON THE CLASSIFICATION OF PAIRS OF DISCRETE MEASURABLE PARTITIONS OF TYPE II

In this paper, we consider the problem of classifying maximal approximatively finite pairs of discrete measurable partitions of type II Lebesgue space and construct a complete invariant for such pairs of partitions.

Minimal Partitions with a Given s-Core and t-Core

Suppose s and t are coprime positive integers, and let sigma be an s-core partition and tau a t-core partition. In this paper, we consider the set {mathcal {P}}_{sigma ,tau }(n) of partitions of n with s-core sigma and t-core tau . We find the smallest n for which this set is non-empty, and show that for this value of n the partitions in {mathcal {P}}_{sigma ,tau }(n) (which we call (sigma ,tau )-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 {mathcal {P}}_{sigma ,tau }(n) consists of a conjugate pair of partitions, and when {mathcal {P}}_{sigma ,tau }(n) contains a unique self-conjugate partition.

On partitions with [formula omitted] corners not containing the staircase with one more corner

We give three proofs of the following result conjectured by Carriegos, De Castro-García and Muñoz Castañeda in their work on enumeration of control systems: when k+12≤n<k+22, there are as many partitions of n with k corners as pairs of partitions (α,β) such that k+12+|α|+|β|=n.

Arithmetic properties of (2, 3)-regular overcubic bipartitions

PurposeLet b¯2,3(n), which enumerates the number of (2, 3)-regular overcubic bipartition of n. The purpose of the paper is to describe some congruences modulo 8 for b¯2,3(n). For example, for each α ≥ 0 and n ≥ 1, b¯2,3(8n+5)≡0(mod8), b¯2,3(2⋅3α+3n+4⋅3α+2)≡0(mod8).Design/methodology/approachH.C. Chan has studied the congruence properties of cubic partition function a(n), which is defined by ∑n=0∞a(n)qn=1(q;q)∞(q2;q2)∞.FindingsTo establish several congruence modulo 8 for b¯2,3(n), here the author keeps to the classical spirit of q-series techniques in the proofs.Originality/valueThe results established in the work are extension to those proved in ℓ-regular cubic partition pairs.

Unimodality of the Andrews-Garvan-Dyson cranks of partitions

The main objective of this paper is to investigate the distribution of the Andrews-Garvan-Dyson cranks of partitions. Let M(m,n) denote the number of partitions of n with the Andrews-Garvan-Dyson crank m, we show that the sequence {M(m,n)}|m|≤n−1 is unimodal for n≥44. It turns out that the unimodality of {M(m,n)}|m|≤n−1 is related to the monotonicity properties of two partition functions pr(n) and ppr(n). Let pr(n) denote the number of partitions of n with parts taken from {2,3,…,r} and let ppr(n) denote the number of pairs (α,β) of partitions, where α is a partition counted by pr(i) and β is a partition counted by pr+1(n−i) for 0≤i≤n. We show that pr(n)≥pr(n−1) for r≥5 and n≥14 and ppr(n)≥ppr(n−1) for r≥3 and n≥8. With the aid of the monotonicity properties on pr(n) and ppr(n), we show that M(m,n)≥M(m,n−1) for n≥14 and 0≤m≤n−2 and M(m−1,n)≥M(m,n) for n≥44 and 1≤m≤n−1. By means of the symmetry M(m,n)=M(−m,n), we find that M(m−1,n)≥M(m,n) for n≥44 and 1≤m≤n−1 implies that the sequence {M(m,n)}|m|≤n−1 is unimodal for n≥44. We also give a proof of an upper bound for ospt(n) conjectured by Chan and Mao in light of M(m−1,n)≥M(m,n) for n≥44 and 0≤m≤n−1.

A crank for bipartitions with designated summands

Andrews, Lewis, and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. A bipartition of n is an ordered pair of partitions \((\pi _1, \pi _2)\) with the sum of all of the parts being n. In this paper, we introduce a generalized crank named the pd-crank for bipartitions with designated summands and give some inequalities for the pd-crank of bipartitions with designated summands modulo 2 and 3. We also define the pd-crank moments weighted by the parity of pd-cranks \(\mu _{2k,bd}(-1,n)\) and show the positivity of \((-1)^n\mu _{2k,bd}(-1,n)\). Let \(M_{bd}(m,n)\) denote the number of bipartitions of n with designated summands with pd-crank m. We prove a monotonicity property of pd-cranks of bipartitions with designated summands and find that the sequence \(\{M_{bd}(m,n)\}_{|m|\le n}\) is unimodal for \(n\not = 1,5,7\).

Chiral differential operators on the upper half plane and modular forms

In this paper we study the vertex operator algebra Dch(H,Γ) constructed from the fixed points of the chiral differential operators on the upper half plane which is holomorphic at all the cusps, under the action of a congruence subgroup Γ. To this end, we introduce an SL(2,R)-invariant filtration labeled by partition pairs and study its successive quotient. We show that the successive quotient under the cuspidal condition is isomorphic to the space of modular forms. We also give a description of the structure of Dch(H,Γ) and compute its character.

Modular Irreducible Representations of the FpW4-Submodules ,()pFNof the Modules ,()pFMas Linear Codes, where W4is the Weyl Group of Type B4

The modular representations of the FpWn-Specht modules( , )KSas linear codes is given in our paper [6], and the modular irreducible representations of the FpW4-submodules( , )pFNof the Specht modules pFS ( , )as linear codes where W4is the Weyl group of type B4is given in our paper [5]. In this paper we are concerning of finding the linear codes of the representations of the irreducible FpW4-submodules( , )pFNof the FpW4-modules( , )pFMfor each pair of partitions( , )of a positive integer n4, where FpGF(p) is the Galois field (finite field) of order p, and pis a prime number greater than or equal to 3. We will find in this paper a generator matrix of a subspace((2,1),(1))()pU representing the irreducible FpW4-submodules((2,1),(1))pFNof the FpW4-modules((2,1),(1))pF Mand give the linear code of ((2,1),(1))()pU for each prime number p greater than or equal to 3. Then we will give the linear codes of all the subspaces( , )()pUfor all pair of partitions( , )of a positive integer n4, and for each prime number p greater than or equal to 3.We mention that some of the ideas of this work in this paper have been influenced by that of Adalbert Kerber and Axel Kohnert [13], even though that their paper is about the symmetric group and this paper is about the Weyl groups of type Bn

Revealing Consensus and Dissensus between Network Partitions

Community detection methods attempt to divide a network into groups of nodes that share similar properties, thus revealing its large-scale structure. A major challenge when employing such methods is that they are often degenerate, typically yielding a complex landscape of competing answers. As an attempt to extract understanding from a population of alternative solutions, many methods exist to establish a consensus among them in the form of a single partition “point estimate” that summarizes the whole distribution. Here, we show that it is, in general, not possible to obtain a consistent answer from such point estimates when the underlying distribution is too heterogeneous. As an alternative, we provide a comprehensive set of methods designed to characterize and summarize complex populations of partitions in a manner that captures not only the existing consensus but also the dissensus between elements of the population. Our approach is able to model mixed populations of partitions, where multiple consensuses can coexist, representing different competing hypotheses for the network structure. We also show how our methods can be used to compare pairs of partitions, how they can be generalized to hierarchical divisions, and how they can be used to perform statistical model selection between competing hypotheses.9 MoreReceived 23 June 2020Revised 22 December 2020Accepted 12 January 2021DOI:https://doi.org/10.1103/PhysRevX.11.021003Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCommunity structureNP-hard problemsStochastic inferenceTechniquesBlock modelsMonte Carlo methodsStatistical PhysicsNetworks

Optimization over Young diagrams

We consider the problem of finding a Young diagram minimizing the sum of evaluations of a given pair of functions on the parts of the associated pair of conjugate partitions. While there are exponentially many diagrams, we show it is polynomial time solvable.

Twin Nicomachean q-identities and conjectures for the associated discriminants, polynomials, and inequalities

We insert additional variables into Warnaar’s [Formula: see text]-analogue of Nicomachus’ identity and other related identities, and compute discriminants with respect to [Formula: see text]. Factorization of these discriminants reveals pairs of partitions that conjecturally relate in the manner of Wheatstone. The factorization also yields, conjecturally, families of polynomials with relations to various Molien series, remarkable rational generating functions, and notable root distributions. For a [Formula: see text]-analogue of Nicomachus’ identity produced by Cigler, we provide proofs of the partition properties. We also state and in part prove tight inequalities for the elements of two interlacing sequences that led us to the “twin” of Warnaar’s [Formula: see text]-analogue.

Gluing affine Yangians with bi-fundamentals

The affine Yangian of {mathfrak{gl}}_1 is isomorphic to the universal enveloping algebra of {mathcal{W}}_{1+infty } and can serve as a building block in the construction of new vertex operator algebras. In [1], a two-parameter family generalization of mathcal{N} = 2 supersymmetric {mathcal{W}}_{infty } algebra was constructed by “gluing” two affine Yangians of {mathfrak{gl}}_1 using operators that transform as (□, overline{square} ) and ( overline{square} , □) w.r.t. the two affine Yangians. In this paper we realize a similar (but non-isomorphic) two-parameter gluing construction where the gluing operators transform as (□, □) and ( overline{square} , overline{square} ) w.r.t. the two affine Yangians. The corresponding representation space consists of pairs of plane partitions connected by a common leg whose cross-section takes the shape of Young diagrams, offering a more transparent geometric picture than the previous construction.

Enhanced PEEC Model Based on Automatic Voronoi Decomposition of Triangular Meshes

We present an enhanced partial element equivalent circuit (PEEC) model based on Voronoi decomposition of triangular meshes. A complex geometry is accurately represented by many surface triangles, while keeping the number of elements in the corresponding equivalent circuit far less than the number of triangles. To do this, the triangular geometry is automatically decomposed into surface partitions using Voronoi's algorithm. These partitions describe capacitance cells, while inductance elements are set between all neighboring partition pairs. The inductance ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> ) and capacitance ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ) matrix calculations are done independently using solvers based on quasi-static electric and magnetic full-wave method of moments (MoM). To get the current and charge distribution over the triangles, the MoM results during <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</i> evaluation are combined with the results of modified nodal analysis. We validated our method against the standard PEEC and MoM results as well as measured data. Then, we used our method to simulate conducted emission for a real automotive LED brake-light module.

Speciation in North American Junonia from a genomic perspective.

Delineating species boundaries in phylogenetic groups undergoing recent radiation is a daunting challenge akin to discretizing continuity. Here, we propose a general approach exemplified by American butterflies from the genus Junonia Hübner notorious for the variety of similar phenotypes, ease of hybridization, and the lack of consensus about their classification. We obtain whole-genome shotgun sequences of about 200 specimens. We reason that discreteness emerges from continuity by means of a small number of key players, and search for the proteins that diverged markedly between sympatric populations of different species, while keeping low polymorphism within these species. Being 0.25% of the total number, these three dozen 'speciation' proteins indeed partition pairs of Junonia populations into two clusters with a prominent break in between, while all proteins taken together fail to reveal this discontinuity. Populations with larger divergence from each other, comparable to that between two sympatric species, form the first cluster and correspond to different species. The other cluster is characterized by smaller divergence, similar to that between allopatric populations of the same species and comprise conspecific pairs. Using this method, we conclude that J. genoveva (Cramer), J. litoralis Brévignon, J. evarete (Cramer), and J. divaricata C. & R. Felder are restricted to South America. We find that six species of Junonia are present in the United States, one of which is new: Junonia stemosa Grishin, sp.n. (i), found in south Texas and phenotypically closest to J. nigrosuffusa W. Barnes & McDunnough (ii) in its dark appearance. In the pale nudum of the antennal club, these two species resemble J. zonalis C. & R. Felder (iii) from Florida and the Caribbean Islands. The pair of sister species, J. grisea Austin & J. Emmel (iv) and J. coenia Hübner (v), represent the classic west/east U.S.A. split. The mangrove feeder (as caterpillar), dark nudum J. neildi Brévignon (vi) enters south Texas as a new subspecies Junonia neildi varia Grishin ssp.n. characterized by more extensive hybridization with and introgression from J. coenia, and, as a consequence, more variable wing patterns compared with the nominal J. n. neildi in Florida. Furthermore, a new mangrove-feeding species from the Pacific Coast of Mexico is described as Junonia pacoma Grishin sp.n. Finally, genomic analysis suggests that J. nigrosuffusa may be a hybrid species formed by the ancestors of J. grisea and J. stemosa sp.n.

The Use of Collaboration Distance in Scheduling Conference Talks

Several bibliographic databases offer a free tool that enables one to determine the collaboration distance or co-authorship distance between researchers. This paper addresses a real-life application of the collaboration distance. It concerns somewhat unusual clustering; namely clustering in which the the average distances in each cluster need to be maximised. We briefly consider a pair of clusterings in which two cluster partitions are uniform and orthogonal in the sense that in each partition all clusters are of the same size and that no pair of elements belongs to the same cluster in both partitions. We consider different objective functions when calculating the score of the pair of orthogonal partitions. The one used on the Wiener index (a graph invariant, known in chemical graph theory) is used. The main application of our work is an algorithm for scheduling a series of parallel talks in a major conference.