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%I #52 May 19 2023 10:58:25
%S 0,1,1,2,6,12,25,57,124,268,588,1285,2801,6118,13362,29168,63685,
%T 139057,303608,662888,1447352,3160121,6899745,15064810,32892270,
%U 71816436,156802881,342360937,747505396,1632091412,3563482500,7780451037,16987713169,37090703118
%N 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.
%C Apart from the offset the same as A214663. - _R. J. Mathar_, Nov 27 2013
%C 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
%D P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
%H Vincenzo Librandi, <a href="/A232164/b232164.txt">Table of n, a(n) for n = 0..1000</a>
%H Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%H P. E. Harris, E. Insko, and L. K. Williams, <a href="http://arxiv.org/abs/1401.0055">The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula</a>, arXiv preprint arXiv:1401.0055, 2013
%H B. Kostant, <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528626/">A Formula for the Multiplicity of a Weight</a>, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588-589.
%H László Németh and Dragan Stevanović, <a href="https://www.researchgate.net/publication/370765824_Graph_solution_of_system_of_recurrence_equations">Graph solution of system of recurrence equations</a>, Research Gate, 2023. See Table 1 at p. 3.
%H Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, <a href="https://arxiv.org/abs/2101.12061">Permutations Avoiding Certain Partially-ordered Patterns</a>, arXiv:2101.12061 [math.CO], 2021.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,3,1).
%F a(n) = A232164(n-1) + A232164(n-2) + 3*A232164(n-3) + A232164(n-4).
%F From _Colin Barker_, Dec 31 2013: (Start)
%F a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
%F G.f.: -x/(x^4 + 3*x^3 + x^2 + x - 1). (End)
%e For n=4, a(4)= A232164(3) + A232164(2) + 3*A232164(1) + A232164(0) = 2+1+3*1+0=6.
%p a:=proc(n::nonnegint)
%p if n=0 then return 0:
%p elif n=1 then return 1:
%p elif n=2 then return 1:
%p elif n=3 then return 2:
%p else return
%p a(n-1)+a(n-2)+3*a(n-3)+a(n-4):
%p end if;
%p end proc:
%t CoefficientList[Series[x/(1 - x - x^2 -3 x^3- x^4),{x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 31 2013 *)
%o (PARI) Vec(-x/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ _Colin Barker_, Dec 31 2013
%K nonn,easy
%O 0,4
%A _Pamela E Harris_, Nov 19 2013
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