

A344494


Triangle read by rows: The dth row contains the Betti numbers of the ddimensional 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
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OFFSET

1,2


COMMENTS

a(d,i) is the ith Betti number of the ddimensional resonance arrangement (for 1 <= i <= d).
The ddimensional resonance arrangement is the hyperplane arrangement in the ddimensional 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 allsubset 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 9dimensional resonance arrangement were computed with the julia package CountingChambers.jl.


LINKS



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).


KEYWORD



AUTHOR



STATUS

approved



