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 A344494 Triangle read by rows: The d-th row contains the Betti numbers of the d-dimensional resonance arrangement. 0
 1, 3, 2, 7, 15, 9, 15, 80, 170, 104, 31, 375, 2130, 5270, 3485, 63, 1652, 22435, 159460, 510524, 371909, 127, 7035, 215439, 3831835, 37769977, 169824305, 135677633, 255, 29360, 1957200, 81029004, 2076831708, 30623870732, 207507589302, 178881449368, 511, 120975, 17153460, 1582492380, 96834110730, 3829831100340, 89702833260450, 973784079284874, 887815808473419 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(d,i) is the i-th Betti number of the d-dimensional resonance arrangement (for 1 <= i <= d). The d-dimensional resonance arrangement is the hyperplane arrangement in the d-dimensional space (x_1,...,x_d) consisting 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. This arrangement is also called the all-subset arrangement. The Betti numbers are also called Whitney numbers of the second kind and they are also the absolute values of the coefficients of the characteristic polynomial of the arrangement. The sum of the Betti numbers equals the number of chambers of this arrangement. The Betti numbers for the 8- and 9-dimensional resonance arrangement were computed with the julia package CountingChambers.jl. LINKS Table of n, a(n) for n=1..45. T. Brysiewicz, H. Eble, and L. Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021 (provides the Betti numbers for the d-dimensional resonance arrangement for 1<=d<=9). Z. Chroman and M. Singhal, Computations associated with the resonance arrangemnt, arXiv:2106.09940 [math.CO], 2021 (provides a formula for the fourth Betti number of this arrangement and the Betti numbers for the d-dimensional resonance arrangement for 1<=d<=9). H. Kamiya, A. Takemura, and H. Terao, Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47 (2011) 379 - 400 (provides the Betti numbers for the d-dimensional resonance arrangement for 1<=d<=7). Lukas Kühne, The Universality of the Resonance Arrangement and its Betti Numbers, arXiv:2008.10553 [math.CO], 2020 (provides formula for the second and third Betti number of this arrangement). EXAMPLE Triangle begins 1; 3, 2; 7, 15, 9; 15, 80, 170, 104; 31, 375, 2130, 5270, 3485; CROSSREFS A034997 is the sum of each row (Number of generalized retarded functions in quantum field theory). A000225 is the first column (2^d - 1). A036239 is the second column (1/2) * (4^n - 3^n - 2^n + 1). Sequence in context: A363584 A365279 A260141 * A286940 A049968 A363399 Adjacent sequences: A344491 A344492 A344493 * A344495 A344496 A344497 KEYWORD nonn,tabl,hard AUTHOR Lukas Kühne, May 21 2021 STATUS approved

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Last modified August 11 04:26 EDT 2024. Contains 375059 sequences. (Running on oeis4.)