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A232164
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Number of Weyl group elements, not containing an s_r factor, which contribute nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the adjoint representation for the Lie algebra of type C and rank n.
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3
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0, 1, 1, 2, 6, 12, 25, 57, 124, 268, 588, 1285, 2801, 6118, 13362, 29168, 63685, 139057, 303608, 662888, 1447352, 3160121, 6899745, 15064810, 32892270, 71816436, 156802881, 342360937, 747505396, 1632091412, 3563482500, 7780451037, 16987713169, 37090703118
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OFFSET
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0,4
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COMMENTS
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Apart from the initial 0, number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the last element. - Sergey Kitaev, Dec 08 2020
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REFERENCES
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P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
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LINKS
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FORMULA
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a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
G.f.: -x/(x^4 + 3*x^3 + x^2 + x - 1). (End)
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EXAMPLE
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MAPLE
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a:=proc(n::nonnegint)
if n=0 then return 0:
elif n=1 then return 1:
elif n=2 then return 1:
elif n=3 then return 2:
else return
a(n-1)+a(n-2)+3*a(n-3)+a(n-4):
end if;
end proc:
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MATHEMATICA
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CoefficientList[Series[x/(1 - x - x^2 -3 x^3- x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 31 2013 *)
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PROG
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(PARI) Vec(-x/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 31 2013
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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