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 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).
FORMULA
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)
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
KEYWORD
nonn,easy
AUTHOR
Pamela E Harris, Nov 19 2013
STATUS
approved