A107428 Number of gap-free compositions of n.

1, 2, 4, 6, 11, 21, 39, 71, 141, 276, 542, 1070, 2110, 4189, 8351, 16618, 33134, 66129, 131937, 263483, 526453, 1051984, 2102582, 4203177, 8403116, 16800894, 33593742, 67174863, 134328816, 268624026, 537192064, 1074288649, 2148414285, 4296543181, 8592585289

Offset: 1

Author

N. J. A. Sloane, May 26 2005

Keywords

• nonn

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

Examples

			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)
		

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.

Crossrefs

The unordered version (partitions) is A034296, ranked by A073491.

The initial case is A107429, unordered A000009, ranked by A333217.

The unordered complement is counted by A239955, ranked by A073492.

These compositions are ranked by A356841.

The complement is counted by A356846, ranked by A356842

A356230 ranks gapless factorization lengths, firsts A356603.

A356233 counts factorizations into gapless numbers.

Cf. A055932, A073493, A137921, A251729, A356224, A356843, A356845.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t!,
      `if`(i<1 or 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 *)

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

Extensions

More terms from Vladeta Jovovic, May 26 2005