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A177821 a(n) gives the number of nonisomorphic connected compact Lie groups of dimension n which are simple products. 0
1, 1, 1, 3, 3, 3, 6, 6, 8, 12, 14, 18, 23, 27, 34, 43, 52, 62, 79, 93, 109, 138, 159, 187, 236, 270, 316, 385, 443, 518, 620, 719, 836, 983, 1138, 1314, 1541, 1770, 2041, 2388, 2726, 3122, 3628, 4124, 4720, 5459, 6204, 7063, 8116, 9203, 10440, 11940, 13525, 15306, 17436, 19690, 22231, 25208, 28388, 32013, 36217, 40673, 45729, 51575, 57808, 64817 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
By the structure theorem for compact Lie groups, every compact connected Lie group is a finite central quotient of a product of copies of the circle group U(1) and compact simple Lie groups which are all known due to Cartan's classification. This sequence counts only those which are direct products of such groups.
LINKS
FORMULA
G.f.: 1/((1-x)*(1-x^3)^2*(1-x^8)^2*(1-x^10)^2*(1-x^14)*...) = (1/(1-x)) * Product_{k>=0} (1-x^k)^A178176(k) with (1-x^k)^0 taken to be 1.
EXAMPLE
For n=0, the trivial group is the only such group.
For n=8, the 8 Lie groups are U(1)^8, U(1)^5 x SU(2), U(1)^5 x SO(3), U(1)^2 x SU(2)^2, U(1)^2 x SU(2) x SO(3), U(1)^2 x SO(3)^2, SU(3) and SU(3)/3.
CROSSREFS
See also A178176 for enumeration of the simple factors giving these counts.
Sequence in context: A013322 A211534 A219816 * A166273 A244482 A219299
KEYWORD
nonn
AUTHOR
Andrew Rupinski, Dec 18 2010
STATUS
approved

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Last modified February 24 01:16 EST 2024. Contains 370288 sequences. (Running on oeis4.)