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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 zero-weight 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 (list; graph; refs; listen; history; text; internal format)
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 Wisconsin-Milwaukee, 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): 588-589.

Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.

Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).

FORMULA

a(n) = A232164(n-1) + A232164(n-2) + 3*A232164(n-3) + A232164(n-4).

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). - 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(n-1)+a(n-2)+3*a(n-3)+a(n-4):

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+x-1) + 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

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Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)