A356842 -id:A356842 - cp's OEIS frontend

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

N. J. A. Sloane, May 26 2005

nonn

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

			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)

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.

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.

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

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 *)

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

More terms from Vladeta Jovovic, May 26 2005

A356841 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 18, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 58, 59, 61, 62, 63, 64, 68, 72, 74, 75, 77, 78, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101

Offset: 1

Gus Wiseman, Aug 31 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms and their corresponding standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
  18: (3,2)
  20: (2,3)
  21: (2,2,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
An unordered version is A073491, complement A073492.
These compositions are counted by A107428.
The complement is A356842.
The non-initial case is A356843, unordered A356845.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224/A356225.

nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],nogapQ[stc[#]]&]

A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

Original entry on oeis.org

2, 4, 8, 10, 16, 18, 20, 32, 36, 42, 64, 68, 72, 74, 82, 84, 128, 136, 146, 148, 164, 170, 256, 264, 272, 274, 276, 290, 292, 296, 298, 324, 328, 330, 338, 340, 512, 528, 548, 580, 584, 586, 594, 596, 658, 660, 676, 682, 1024, 1040, 1056, 1092, 1096, 1098

Offset: 1

Gus Wiseman, Sep 01 2022

nonn

			The terms together with their corresponding standard compositions begin:
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   18: (3,2)
   20: (2,3)
   32: (6)
   36: (3,3)
   42: (2,2,2)
   64: (7)
   68: (4,3)
   72: (3,4)
   74: (3,2,2)
   82: (2,3,2)
   84: (2,2,3)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
A subset of A022340.
These compositions are counted by A251729.
The unordered version (using Heinz numbers of partitions) is A356845.
A333217 ranks complete compositions.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356841 ranks gapless compositions, counted by A107428.
A356842 ranks non-gapless compositions, counted by A356846.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073491, A073492, A073493, A137921, A356224/A356225.

nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],!MemberQ[stc[#],1]&&nogapQ[stc[#]]&]

Complement of A333217 in A356841.

A356846 Number of integer compositions of n into parts not covering an interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 11, 25, 57, 115, 236, 482, 978, 1986, 4003, 8033, 16150, 32402, 64943, 130207, 260805, 522123, 1045168, 2091722, 4185431, 8374100, 16753538, 33515122, 67042865, 134106640, 268246886, 536549760, 1073194999, 2146553011, 4293391411, 8587283895

Offset: 0

Gus Wiseman, Sep 03 2022

nonn

			The a(0) = 0 through a(6) = 8 compositions:
  .  .  .  .  (13)  (14)   (15)
              (31)  (41)   (24)
                    (113)  (42)
                    (131)  (51)
                    (311)  (114)
                           (141)
                           (411)
                           (1113)
                           (1131)
                           (1311)
                           (3111)

The complement is counted by A107428, initial A107429.
The case of partitions is A239955, ranked by A073492, initial A053251, complement A034296.
These compositions are ranked by A356842, complement A356841.
A000041 counts partitions, compositions A011782.
A066208 lists numbers with all odd prime indices, counted by A000009.
A073491 lists numbers with gapless prime indices, initial A055932.
Cf. A001227, A060142, A080259, A188575, A239327, A356604, A356605.

gappyQ[m_]:=And[m!={},Union[m]!=Range[Min[m],Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],gappyQ]],{n,0,15}]

a(n) = A011782(n) - A107428(n).

A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 16, 20, 32, 52, 64, 72, 128, 256, 272, 328, 512, 840, 1024, 1056, 2048, 2320, 4096, 4160, 8192, 10512, 16384, 16512, 17440, 26896, 32768, 65536, 65792, 131072, 135232, 148512, 262144, 262656, 524288, 672800, 1048576, 1049600, 1065088, 1721376

Offset: 1

Gus Wiseman, Sep 24 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

			The terms and corresponding intervals begin:
        0: ()
        1: (1)
        2: (2)
        4: (3)
        6: (1,2)
        8: (4)
       16: (5)
       20: (2,3)
       32: (6)
       52: (1,2,3)
       64: (7)
       72: (3,4)
      128: (8)
      256: (9)
      272: (4,5)
      328: (2,3,4)
      512: (10)
      840: (1,2,3,4)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
These compositions are counted by A001227.
An unordered version is A073485, non-strict A073491 (complement A073492).
The initial version is A164894, non-strict A356843 (unordered A356845).
The non-strict version is A356841, initial A333217, counted by A107428.
A066311 lists gapless numbers.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224, A356842.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Select[Range[0,1000],chQ[stc[#]]&]

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A107428 Number of gap-free compositions of n.

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A356841 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).

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A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

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A356846 Number of integer compositions of n into parts not covering an interval of positive integers.

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A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

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