

A232164


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 zeroweight in the adjoint representation for the Lie algebra of type C and rank n.


3



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

0,4


COMMENTS

Apart from the offset the same as A214663.  R. J. Mathar, Nov 27 2013
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


REFERENCES

P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of WisconsinMilwaukee, 2012.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
P. E. Harris, E. Insko, L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055, 2013
B. Kostant, A Formula for the Multiplicity of a Weight, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588589.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partiallyordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).


FORMULA

a(n) = A232164(n1) + A232164(n2) + 3*A232164(n3) + A232164(n4).
a(n) = a(n1)+a(n2)+3*a(n3)+a(n4). G.f.: x / (x^4+3*x^3+x^2+x1).  Colin Barker, Dec 31 2013


EXAMPLE

For n=4, a(4)= A232164(3) + A232164(2) + 3*A232164(1) + A232164(0) = 2+1+3*1+0=6.


MAPLE

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(n1)+a(n2)+3*a(n3)+a(n4):
end if;
end proc:


MATHEMATICA

CoefficientList[Series[x/(1  x  x^2 3 x^3 x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 31 2013 *)


PROG

(PARI) Vec(x/(x^4+3*x^3+x^2+x1) + O(x^100)) \\ Colin Barker, Dec 31 2013


CROSSREFS

Sequence in context: A140659 A238462 A099495 * A214663 A151385 A034875
Adjacent sequences: A232161 A232162 A232163 * A232165 A232166 A232167


KEYWORD

nonn,easy


AUTHOR

Pamela E Harris, Nov 19 2013


STATUS

approved



