A364781 Triangular array read by rows: T(n, k) is the number of zero-energy states from the partition function in the Ising model for a finite n*k square lattice with periodic boundary conditions.

0, 2, 12, 0, 26, 0, 2, 100, 1346, 20524, 0, 322, 0, 272682, 0, 2, 1188, 72824, 3961300, 226137622, 13172279424, 0, 4258, 0, 58674450, 0, 777714553240, 0, 2, 15876, 3968690, 876428620, 199376325322, 46463664513012, 10990445640557042, 2627978003957146636, 0, 59138, 0, 13184352554, 0, 2799323243348702, 0, 633566123999182005386, 0

Offset: 1

Author

Thomas Scheuerle, Aug 07 2023

Keywords

• nonn
• tabl

Comments

Imagine an n X k square tiling on a 2D surface with torus topology. T(n, k) is the number of ways two colors can be assigned to all tiles such that the overall length of the boundary between the colored regions is n*k.

The number of solutions with the additional constrain that exactly k tiles must have the lesser represented color is given for tilings with size 2 X 2*k by A241023(k). In the case 2 X 2*k is k also the minimum count of tiles with the same color in all solutions.

Examples

			Triangle begins:
  0;
  2,    12;
  0,    26,       0;
  2,   100,    1346,     20524;
  0,   322,       0,    272682,            0;
  2,  1188,   72824,   3961300,    226137622,    13172279424;
  0,  4258,       0,  58674450,            0,   777714553240,                 0;
  2, 15876, 3968690, 876428620, 199376325322, 46463664513012, 10990445640557042, 2627978003957146636;
  ...
		

Links

• Manuel Kauers, Triangular array flattened. Table of n, a(n) for n = 1..120
• Roland Häggkvist, Anders Rosengren, Daniel Andrén, Petras Kundrotas, Per Håkan Lundow, and Klas Markström, Computation of the Ising partition function for 2-dimensional square grids, Phys. Rev. E 69, 046104 (April 16 2004).
• Manuel Kauers, Onsager's solution of the Ising model could have been guessed, presentation slides (2018).
• Thomas Scheuerle, Some values for T(k, k) from Klas Markström and R. Häggkvist et al., extracted from calculation results provided with their work. (See link.)

Crossrefs

Cf. A001411, A002931, A010566, A241023.

Programs

MATLAB

function a = A364781( n, k )
    a = 0;
    for m = 1:2^(n*k)-2
        if isingSum( reshape(1-2*bitget(m,1:n*k),n ,k)) == 0
            a = a + 1;
        end
    end
end
function e = isingSum( config )
    e = 0; si = size(config);
    for j = 1:si(2)
        for k = 1:si(1)
            S = config(k, j);
            nb = config(1+mod(k , si(1)), j) + config(k, 1+mod(j , si(2)));
            e = e + (-nb)*S;
        end
    end
end

Formula

T(n, k) = 0 if n*k is odd.

Extensions

a(27) - a(45) from Manuel Kauers, Sep 07 2023