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%I #80 Jun 04 2022 01:57:00
%S 2,6,32,370,11292,1066044,347326352,419172756930,1955230985997140
%N Number of generalized retarded functions in quantum field theory.
%C a(d) is the number of parts into which d-dimensional 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.
%C Also, a(d) is the number of independent real-time 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".
%C 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 a(8) on Aug 01 2011 using an updated version of van Eijck's program, which took 7 days on a standard desktop computer.
%D Björner, Anders. "Positive Sum Systems", in Bruno Benedetti, Emanuele Delucchi, and Luca Moci, editors, Combinatorial Methods in Topology and Algebra. Springer International Publishing, 2015. 157-171.
%D M. van Eijck, Thermal Field Theory and Finite-Temperature Renormalisation Group, PhD thesis, Univ. Amsterdam, 4th Dec. 1995.
%H Louis J. Billera, Sara C. Billey, and Vasu Tewari, <a href="https://arxiv.org/abs/1806.02943">Boolean product polynomials and Schur-positivity</a>, arXiv:1806.02943 [math.CO], 2018.
%H L. J. Billera, J. T. Moore, C. D. Moraites, Y. Wang and K. Williams, <a href="http://arxiv.org/abs/1209.2309">Maximal unbalanced families</a>, arXiv preprint arXiv:1209.2309 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 26 2012
%H Taylor Brysiewicz, Holger Eble, and Lukas Kühne, <a href="https://arxiv.org/abs/2105.14542">Enumerating chambers of hyperplane arrangements with symmetry</a>, arXiv:2105.14542 [math.CO], 2021.
%H Antoine Deza, Mingfei Hao, and Lionel Pournin, <a href="https://arxiv.org/abs/2205.13309">Sizing the White Whale</a>, arXiv:2205.13309 [math.CO], 2022.
%H Antoine Deza, George Manoussakis, and Shmuel Onn, <a href="https://dx.doi.org/10.1007/s00454-017-9873-z">Primitive Zonotopes</a>, Discrete & Computational Geometry, 2017, p. 1-13. (See p. 5.)
%H T. S. Evans, <a href="http://plato.tp.ph.ic.ac.uk/~time/TSEpaper/nptr.pdf">N-point finite temperature expectation values at real times</a>, Nuclear Physics B 374 (1992) 340-370.
%H T. S. Evans, <a href="http://arXiv.org/abs/hep-ph/9404262">What is being calculated with Thermal Field Theory?</a>, arXiv:hep-ph/9404262, 1994-2011 and in "Particle Physics and Cosmology: Proceedings of the Ninth Lake Louise Winter School", World Scientific, 1995 (ISBN 9810221002).
%H Samuel C. Gutekunst, Karola Mészáros, and T. Kyle Petersen, <a href="https://arxiv.org/abs/1903.06595">Root Cones and the Resonance Arrangement</a>, arXiv:1903.06595 [math.CO], 2019.
%H Lukas Kühne, <a href="https://arxiv.org/abs/2008.10553">The Universality of the Resonance Arrangement and its Betti Numbers</a>, arXiv:2008.10553 [math.CO], 2020.
%H H. Kamiya, A. Takemura and H. Terao, <a href="https://doi.org/10.1016/j.aam.2010.11.002">Ranking patterns of unfolding models of codimension one</a>, Advances in Applied Mathematics 47 (2011) 379 - 400.
%H Lars Kastner and Marta Panizzut, <a href="https://arxiv.org/abs/2003.13548">Hyperplane arrangements in polymake</a>, arXiv:2003.13548 [math.CO], 2020.
%H Zhengwei Liu, William Norledge, and Adrian Ocneanu, <a href="https://arxiv.org/abs/1901.03243">The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations</a>, arXiv:1901.03243 [math.CO], 2019.
%H William Norledge and Adrian Ocneanu, <a href="https://arxiv.org/abs/1911.11736">Hopf monoids, permutohedral tangent cones, and generalized retarded functions</a>, arXiv:1911.11736 [math.CO], 2019.
%e a(1)=2 because the point x=0 splits the real line into two parts, the positive and negative reals.
%e a(2)=6 because we can split two-dimensional space into 6 parts using lines x=0, y=0 and x+y=0.
%K nonn,more
%O 1,1
%A _Tim S. Evans_
%E a(9) from _Zachary Chroman_, Feb 19 2021
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