OFFSET
0,3
COMMENTS
For n>=4, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2, x_3, x_4 in {1,2,...,n} and fixed y_1, y_2, y_3, y_ 4 in {1,2,3,4} we have f(x_i)<>y_i, (i=1,2,3,4). - Milan Janjic, May 13 2007
Also the number of monic polynomials of degree n over GF(4) without any linear factors. - Greyson C. Wesley, Jul 05 2022
REFERENCES
C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1661
Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, and Mike Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082, 2005. See also Canad. J. Math. 60 (2008), no. 2, 266-296.
Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 21.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (4).
FORMULA
G.f.: (1-q)^4/(1-4q).
a(n) = Sum_{k=0..min(n,4)} (-1)^k*C(4,k)*4^(n-k).
a(n) = 81*4^(n-4) for n>3. - Jean-François Alcover, Dec 10 2018
EXAMPLE
The enveloping algebra of the derived free Lie algebra is characterized as the intersection of the kernels of all partial derivative operators in the space of non-commutative polynomials, a(0) = 1 since all constants are killed by derivatives, a(1) = 0 since no polys of degree 1 are killed, a(2) = 6 since all Lie brackets [x1,x2], [x1,x3], [x1, x4], [x2,x3], [x2,x4], [x3,x4] are killed by all derivative operators.
MAPLE
f:=n->add((-1)^k*C(4, k)*4^(n-k), k=0..min(n, 4)); seq(f(i), i=0..15);
MATHEMATICA
a[n_] := If[n<4, {1, 0, 6, 20}[[n+1]], 81*4^(n-4)];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Dec 10 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Mike Zabrocki, Apr 20 2006
STATUS
approved