A344494 - OEIS

%I #25 Jun 25 2021 11:46:54

%S 1,3,2,7,15,9,15,80,170,104,31,375,2130,5270,3485,63,1652,22435,

%T 159460,510524,371909,127,7035,215439,3831835,37769977,169824305,

%U 135677633,255,29360,1957200,81029004,2076831708,30623870732,207507589302,178881449368,511,120975,17153460,1582492380,96834110730,3829831100340,89702833260450,973784079284874,887815808473419

%N Triangle read by rows: The d-th row contains the Betti numbers of the d-dimensional resonance arrangement.

%C a(d,i) is the i-th Betti number of the d-dimensional resonance arrangement (for 1 <= i <= d).

%C 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.

%C 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.

%C The sum of the Betti numbers equals the number of chambers of this arrangement.

%C The Betti numbers for the 8- and 9-dimensional resonance arrangement were computed with the julia package CountingChambers.jl.

%H T. Brysiewicz, H. Eble, and L. Kühne, <a href="https://arxiv.org/abs/2105.14542">Enumerating chambers of hyperplane arrangements with symmetry</a>, arXiv:2105.14542 [math.CO], 2021 (provides the Betti numbers for the d-dimensional resonance arrangement for 1<=d<=9).

%H Z. Chroman and M. Singhal, <a href="https://arxiv.org/abs/2106.09940">Computations associated with the resonance arrangemnt</a>, 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 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 (provides the Betti numbers for the d-dimensional resonance arrangement for 1<=d<=7).

%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 (provides formula for the second and third Betti number of this arrangement).

%e Triangle begins

%e    1;

%e    3,   2;

%e    7,  15,    9;

%e   15,  80,  170,  104;

%e   31, 375, 2130, 5270, 3485;

%Y A034997 is the sum of each row (Number of generalized retarded functions in quantum field theory).

%Y A000225 is the first column (2^d - 1).

%Y A036239 is the second column (1/2) * (4^n - 3^n - 2^n + 1).

%K nonn,tabl,hard

%O 1,2

%A _Lukas Kühne_, May 21 2021