parameter. This quotient is unfortunately rather sensitive to noise, making it difficult to determine the parameters p and q for sampled data. Section IV gives detailed results for the special case when p = q . This case corresponds exactly to the complete graph and is the only case where we can give asymptotically exact expressions for the sum of the coefficients. In section V we provide some useful tools for working with p, q -binomial coefficients in the case when p = q . After this build-up of tools we are, at long last, ready to give some general results on the distribution of p, q -binomial coefficients in section VI. Using the parameterization p = 1 + y/n and q = 1 + z/n we find the asymptotic value of y , given z , where the distribution is flat in the middle. We also allow for a small change in y , using a higher order parameter a , so that we can follow properly how the distribution changes from unimodal to bimodal. However, the computations that we rely on involve some rather complicated series expansions that were made using Mathematica. These are much too long to fit into this paper. We have prepared a simplified Mathematica notebook that performs all the necessary computations. The interested reader can obtain it by contacting the first author. Section VII looks into the case of moving y and z along a line with any given slope as opposed to the previous section where z stays fixed. In section VIII we give exact scaling formulae for the moments of the distributions depending on the parameters a and z . For a given moment of these distributions we always obtain the same exponent on n , regardless of a and z . In section IX we try to remedy this by letting the previously fixed parameter z depend ever so slightly (at most logarithmically) on n . This is based on the assumption that the previous formulae in section VI still hold. However, we can now adjust the exponent of n though this comes at the cost of an extremely slow convergence. Section X defines the Ising model, laying the ground for studying distributions of mag- netisations, the intended application of our endeavour. In section XI we apply our tools to the d -dimensional lattice graphs for d = 1 , 2 , 3 , 4 , 5 fitting p, q -distributions to simulated distributions and comparing them. A condensed reading, more suitable to the reader who is pressed for time, should include a look at (1), (14), (27), (28), (29), (49), (50), (51) for the necessary definitions and results concerning the basics. After that, the most important results are stated in equations (62), (65), (96), (109) and (112). After looking up the basic definitions regarding the Ising model the reader can skip to (121). In Section XI the reader can now pick and choose his favourite 3