A034997 -id:A034997 - cp's OEIS frontend

A344494 Triangle read by rows: The d-th row contains the Betti numbers of the d-dimensional resonance arrangement.

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

Offset: 1

Lukas Kühne, May 21 2021

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.

			Triangle begins
   1;
   3,   2;
   7,  15,    9;
  15,  80,  170,  104;
  31, 375, 2130, 5270, 3485;

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).

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).

cp's OEIS Frontend

A045990 a(n) = A034997(n)/2.

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A344494 Triangle read by rows: The d-th row contains the Betti numbers of the d-dimensional resonance arrangement.

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