OFFSET
1,2
COMMENTS
A gap-free composition contains all the parts between its smallest and largest part. a(5)=11 because we have: 5, 3+2, 2+3, 2+2+1, 2+1+2, 1+2+2, 2+1+1+1, 1+2+1+1, 1+1+2+1, 1+1+1+2, 1+1+1+1+1. - Geoffrey Critzer, Apr 13 2014
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
Alois P. Heinz, Plot of (a(n)-2^(n-2))/2^(n-2) for n = 42..1000
P. Hitczenko and A. Knopfmacher, Gap-free compositions and gap-free samples of geometric random variables, Discrete Math., 294 (2005), 225-239.
FORMULA
a(n) ~ 2^(n-2). - Alois P. Heinz, Dec 07 2014
G.f.: Sum_{j>0} Sum_{k>=j} C({j..k},x) where C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)) is the g.f. for compositions such that the set of parts equals {s} with C({},x) = 1. - John Tyler Rascoe, Jun 01 2024
EXAMPLE
From Gus Wiseman, Oct 04 2022: (Start)
The a(0) = 1 through a(5) = 11 gap-free compositions:
() (1) (2) (3) (4) (5)
(11) (12) (22) (23)
(21) (112) (32)
(111) (121) (122)
(211) (212)
(1111) (221)
(1112)
(1121)
(1211)
(2111)
(11111)
(End)
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, t!,
`if`(i<1 or n<i, 0, add(b(n-i*j, i-1, t+j)/j!, j=1..n/i)))
end:
a:= n-> add(b(n, i, 0), i=1..n):
seq(a(n), n=1..40); # Alois P. Heinz, Apr 14 2014
MATHEMATICA
Table[Length[Select[Level[Map[Permutations, IntegerPartitions[n]], {2}], Length[Union[#]]==Max[#]-Min[#]+1&]], {n, 1, 20}] (* Geoffrey Critzer, Apr 13 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, t!, If[i < 1 || n < i, 0, Sum[b[n - i*j, i - 1, t + j]/j!, {j, 1, n/i}]]]; a[n_] := Sum[b[n, i, 0], {i, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, May 26 2005
EXTENSIONS
More terms from Vladeta Jovovic, May 26 2005
STATUS
approved