scholarly journals Lattice Points and Simultaneous Core Partitions

10.37236/5734 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Paul Johnson
Keyword(s):  
Lattice Point ◽  
Lattice Points ◽  
Ehrhart Theory ◽  
Average Size

We apply lattice point techniques to the study of simultaneous core partitions. Our central observation is that for $a$ and $b$ relatively prime, the abacus construction identifies the set of simultaneous $(a,b)$-core partitions with lattice points in a rational simplex. We apply this result in two main ways: using Ehrhart theory, we reprove Anderson's theorem that there are $(a+b-1)!/a!b!$ simultaneous $(a,b)$-cores; and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an $(a,b)$-core is $(a+b+1)(a-1)(b-1)/24$. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate $(a,b)$-cores.

scholarly journals Bounds on the Lattice Point Enumerator via Slices and Projections

2021 ◽  
Author(s):  
Ansgar Freyer ◽  
Martin Henk
Keyword(s):  
Convex Body ◽  
Lattice Point ◽  
Geometric Mean ◽  
Discrete Analogue ◽  
Lattice Points ◽  
General Context ◽  
General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

scholarly journals GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

2019 ◽  
Vol 7 ◽  
Author(s):  
SPENCER BACKMAN ◽  
MATTHEW BAKER ◽  
CHI HO YUEN
Keyword(s):  
Spanning Trees ◽  
Finite Abelian Group ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Lattice Points ◽  
Combinatorial Interpretation ◽  
Ehrhart Polynomial ◽  
Ehrhart Theory ◽  
Regular Matroid ◽  
Definition Of

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$ ). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$ , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$ , and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$ . (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$ ) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$ .) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$ ; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$ .

Two classical lattice point problems

1961 ◽  
Vol 57 (4) ◽  
pp. 699-721 ◽  
Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

Theoretical Investigation of Alloy Phase Equilibria by Continuous Displacement Cluster Variation Method

2011 ◽  
Vol 172-174 ◽  
pp. 1119-1127
Author(s):  
Tetsuo Mohri
Keyword(s):  
Phase Equilibria ◽  
Lattice Point ◽  
Cluster Variation Method ◽  
Lattice Points ◽  
Variation Method ◽  
Bravais Lattice ◽  
Diffuse Intensity ◽  
Wide Range ◽  
Cluster Variation ◽  
Continuous Displacement

Continuous Displacement Cluster Variation Method is employed to study binary phase equilibria on the two dimensional square lattice with Lennard-Jones type pair potentials. It is confirmed that the transition temperature decreases significantly as compared with the one obtained by conventional Cluster Variation Method. This is ascribed to the distribution of atomic pairs in a wide range of atomic distance, which enables the system to attain the lower free energy. The spatial distribution of atomic species around a Bravais lattice point is visualized. Although the average position of an atom is centred at the Bravais lattice point, the maximum pair probability is not necessarily attained for the pairs located at the neighboring Bravais lattice points. In addition to the real space information, k-space information are calculated in the present study. Among them, the diffuse intensity spectra due to short range ordering and atomic displacement are discussed.

scholarly journals On the maximal circumradius of a planar convex set containing one lattice point

1995 ◽  
Vol 52 (1) ◽  
pp. 137-151 ◽  
Author(s):  
Poh W. Awyong ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Set ◽  
Compact Convex ◽  
Lattice Points ◽  
Compact Convex Set ◽  
Planar Convex

We obtain a result about the maximal circumradius of a planar compact convex set having circumcentre O and containing no non-zero lattice points in its interior. In addition, we show that under certain conditions, the set with maximal circumradius is a triangle with an edge containing two lattice points.

scholarly journals Partial structure factors derived from coarse-grained phase-separation dynamics on a disordered lattice with density fluctuations

2021 ◽  
Author(s):  
Puwadet Sutipanya ◽  
Takashi Arai
Keyword(s):  
Phase Separation ◽  
Structure Factor ◽  
Lattice Point ◽  
Local Density ◽  
Lattice Points ◽  
Concentration Fluctuations ◽  
Coarse Grained ◽  
Partial Structure ◽  
Disordered Lattice ◽  
Structure Factors

Abstract The simplest and most time-efficient phase-separation dynamics simulations are carried out on a disordered lattice to calculate the partial structure factors of coarse-grained A-B binary mixtures. The typical coarse-grained phase-separation models use regular lattices and can describe the local concentrations but cannot describe both local density and concentration fluctuations. To introduce fluctuation for local density in the model, the particle positions from a hard sphere fluid model are determined as disordered lattice points for the model. Then we place the local order parameter as the difference of the concentrations of A and B components on each lattice point. The concentration at each lattice point is time-evolved by discrete equations derived from the Cahn-Hilliard equation. From both fluctuations, Bhatia and Thornton’s structure factor can be accurately calculated. The structure factor for concentration fluctuations at the large wavenumber region gives us the correct mean concentrations of the components. Using the mean concentrations, partial structure factors can be converted from three of Bhatia and Thornton’s structure factors. The present model and procedures can provide a means of analysing the structural properties of many materials that exhibit complex morphological changes.

scholarly journals Uniform distribution and lattice point counting

1992 ◽  
Vol 53 (1) ◽  
pp. 39-50 ◽  
Author(s):  
G. R. Everest
Keyword(s):  
Asymptotic Formula ◽  
Uniform Distribution ◽  
Lattice Point ◽  
Lattice Points ◽  
Analytic Method ◽  
Point Counting

AbstractA well-known theorem of Hardy and Littlewood gives a three-term asymptotic formula, counting the lattice points inside an expanding, right triangle. In this paper a generalisation of their theorem is presented. Also an analytic method is developed which enables one to interpret the coefficients in the formula. These methods are combined to give a generalisation of a “heightcounting” formula of Györy and Pethö which itself was a generalisation of a theorem of Lang.

scholarly journals Circumradius-diameter and width-inradius relations for lattice constrained convex sets

1999 ◽  
Vol 59 (1) ◽  
pp. 147-152 ◽  
Author(s):  
Poh Wah Awyong ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Integer Lattice ◽  
Lattice Points ◽  
Compact Convex Set ◽  
Planar Convex ◽  
Rectangular Lattices

Let K be a planar, compact, convex set with circumradius R, diameter d, width w and inradius r, and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2R − d) and (w − 2r) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

scholarly journals The mean lattice point discrepancy

1995 ◽  
Vol 38 (3) ◽  
pp. 523-531 ◽  
Author(s):  
M. N. Huxley
Keyword(s):  
Root Mean Square ◽  
Lattice Point ◽  
Plane Curve ◽  
Lattice Points ◽  
Mean Square ◽  
Convex Plane ◽  
The Mean

Consider a sufficiently smooth simple closed convex plane curve enclosing the origin, expanding linearly with time. The root mean square of the discrepancy (number of lattice points minus area) from time t = M to t = M + 1 is almost as small as the root mean square discrepancy from time t = 0 to t = M, so the discrepancy has no memory.

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