
COMMENTS

a(d) is the number of parts into which ddimensional space (x_1,...,x_d) is split by a set of (2^d  1) hyperplanes c_1 x_1 + c_2 x_2 + ...+ c_d x_d =0 where c_j are 0 or +1 and we exclude the case with all c=0.
Also, a(d) is the number of independent realtime Green functions of Quantum Field Theory produced when analytically continuing from euclidean time/energy (d+1 = number of energy/time variables). These are also known as Generalized Retarded Functions.
The numbers up to d=6 were first produced by T. S. Evans using a Pascal program, strictly as upper bounds only. M. van Eijck wrote a C program using a direct enumeration of hyperplanes which confirmed these and produced the value for d=7. Kamiya et al. showed how to find these numbers and some associated polynomials using more sophisticated methods, giving results up to d=7. T. S. Evans added the last number on Aug 01 2011 using an updated version of van Eijck's program, which took 7 days on a standard desktop computer.


REFERENCES

L. J. Billera, J. T. Moore, C. D. Moraites, Y. Wang and K. Williams, Maximal unbalanced families, arXiv preprint arXiv:1209.2309, 2012.  From N. J. A. Sloane, Dec 26 2012
T. S. Evans, Npoint finite temperature expectation values at real times, Nuclear Physics B 374 (1992) 340370.
T. S. Evans, What is being calculated with thermal field theory?, in "Particle Physics and Cosmology: Proceedings of the Ninth Lake Louise Winter School", World Scientific, 1995 (ISBN 9810221002), preprint arXiv:hepph/9404262.
H. Kamiya, A. Takemura and H. Terao, Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47 (2011) 379  400.
M. van Eijck, Thermal Field Theory and FiniteTemperature Renormalisation Group, PhD thesis, Univ. Amsterdam, 4th Dec. 1995.
