A090327 Number of rules of a context-free grammar in Chomsky normal form that generates all permutations of n symbols.

1, 4, 11, 30, 83, 234, 671, 1950, 5723, 16914, 50231, 149670, 446963, 1336794, 4002191, 11990190, 35937803, 107747874, 323112551, 969075510, 2906702243, 8719058154, 26155077311, 78461037630, 235374724283, 706107395634, 2118288632471, 6354798788550

Offset: 1

Author

Peter R. J. Asveld, Jan 27 2004

Keywords

• nonn
• easy

Examples

			S -> AD | DA | BE | EB, D -> BC | CB, E -> AC | CA, A -> a, B -> b, C-> c; so a(3)=11.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• P. R. J. Asveld, Generating all permutations by context-free grammars in Chomsky normal form, Theoretical Computer Science 354 (2006) 118-130.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Programs

Maple

f:= gfun:-rectoproc({a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3),a(1)=1,a(2)=4,a(3)=11,a(4)=30},a(n),'remember'):
seq(f(n),n=1..100); # Robert Israel, Jan 15 2015

Mathematica

f[n_] := Ceiling[5/2*3^(n - 2) + 2^(n - 1) - 1/2]; Table[ f[n], {n, 2, 27}] (* Robert G. Wilson v, Jan 30 2004 *)

PARI

Vec(-x*(2*x^3-2*x^2-2*x+1)/((x-1)*(2*x-1)*(3*x-1)) + O(x^100)) \\ Colin Barker, Jan 15 2015

Formula

a(n) = ceiling[ (5*3^(n-2))/2 + 2^(n-1) - 1/2 ] for n > 1.

G.f.: -x*(2*x^3-2*x^2-2*x+1) / ((x-1)*(2*x-1)*(3*x-1)). - Colin Barker, Jan 15 2015

a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3) for n >= 5. - Robert Israel, Jan 15 2015

Extensions

More terms from Robert G. Wilson v, Jan 30 2004