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.

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

Offset: 0

Author

Pamela E Harris, Nov 19 2013

Keywords

• nonn
• easy

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

Examples

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

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 and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
• Alice L. L. Gao and 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, and 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.
• László Németh and Dragan Stevanović, Graph solution of system of recurrence equations, Research Gate, 2023. See Table 1 at p. 3.
• 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).

Programs

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 *)

PARI

Vec(-x/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 31 2013

Formula

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

From Colin Barker, Dec 31 2013: (Start)

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)