Determinant method

Abstract: The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri?Iwaniec?Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves. This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri?Iwaniec?Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.

Abstract: In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser. A 66 (1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the project of proving all ten formulas. We enumerate cyclically symmetric, self-complementary plane partitions. We first convert plane partitions to tilings of a hexagon in the plane by rhombuses, or equivalently to matchings in a certain planar graph. We can then use the permanent-determinant method or a variant, the Hafnian-Pfaffian method, to obtain the answer as the determinant or Pfaffian of a matrix in each of the ten cases. We row-reduce the resulting matrix in the case under consideration to prove the formula. A similar row-reduction process can be carried out in many of the other cases, and we analyze three other symmetry classes of plane partitions for comparison.

Abstract: The permanent-determinant method and its generalization, the Hafnian-Pfaffian method, are methods to enumerate perfect matchings of plane graphs that was discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanent-determinant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a bipartite graph on the sphere with 4n vertices is invariant under the antipodal map, the number of matchings is the square of the number of matchings of the quotient graph. 2. The number of matchings of the edge graph of a graph with vertices of degree at most 3 is a power of 2. 3. The three Carlitz matrices whose determinants count a x b x c plane partitions all have the same cokernel. 4. Two symmetry classes of plane partitions can be enumerated with almost no calculation.