login
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.
1

%I #54 Feb 01 2024 02:51:58

%S 0,2,12,0,26,0,2,100,1346,20524,0,322,0,272682,0,2,1188,72824,3961300,

%T 226137622,13172279424,0,4258,0,58674450,0,777714553240,0,2,15876,

%U 3968690,876428620,199376325322,46463664513012,10990445640557042,2627978003957146636,0,59138,0,13184352554,0,2799323243348702,0,633566123999182005386,0

%N 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.

%C 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.

%C 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.

%H Manuel Kauers, <a href="/A364781/b364781.txt">Triangular array flattened. Table of n, a(n) for n = 1..120</a>

%H Roland Häggkvist, Anders Rosengren, Daniel Andrén, Petras Kundrotas, Per Håkan Lundow, and Klas Markström, <a href="http://abel.math.umu.se/~klasm/Uppsatser/plancheb.pdf">Computation of the Ising partition function for 2-dimensional square grids</a>, Phys. Rev. E 69, 046104 (April 16 2004).

%H Manuel Kauers, <a href="http://www.algebra.uni-linz.ac.at/people/mkauers/publications/kauers18k.pdf">Onsager's solution of the Ising model could have been guessed</a>, presentation slides (2018).

%H Thomas Scheuerle, <a href="/A364781/a364781.txt">Some values for T(k, k) from Klas Markström and R. Häggkvist et al.</a>, extracted from calculation results provided with their work. (See link.)

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

%e Triangle begins:

%e 0;

%e 2, 12;

%e 0, 26, 0;

%e 2, 100, 1346, 20524;

%e 0, 322, 0, 272682, 0;

%e 2, 1188, 72824, 3961300, 226137622, 13172279424;

%e 0, 4258, 0, 58674450, 0, 777714553240, 0;

%e 2, 15876, 3968690, 876428620, 199376325322, 46463664513012, 10990445640557042, 2627978003957146636;

%e ...

%o (MATLAB)

%o function a = A364781( n, k )

%o a = 0;

%o for m = 1:2^(n*k)-2

%o if isingSum( reshape(1-2*bitget(m,1:n*k),n ,k)) == 0

%o a = a + 1;

%o end

%o end

%o end

%o function e = isingSum( config )

%o e = 0; si = size(config);

%o for j = 1:si(2)

%o for k = 1:si(1)

%o S = config(k, j);

%o nb = config(1+mod(k , si(1)), j) + config(k, 1+mod(j , si(2)));

%o e = e + (-nb)*S;

%o end

%o end

%o end

%Y Cf. A001411, A002931, A010566, A241023.

%K nonn,tabl

%O 1,2

%A _Thomas Scheuerle_, Aug 07 2023

%E a(27) - a(45) from _Manuel Kauers_, Sep 07 2023