A178470 -id:A178470 - cp's OEIS frontend

A328172 Number of integer partitions of n with all pairs of consecutive parts relatively prime.

1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 24, 28, 36, 43, 51, 62, 74, 87, 104, 122, 143, 169, 195, 227, 260, 302, 346, 397, 455, 521, 599, 686, 780, 889, 1001, 1138, 1286, 1454, 1638, 1846, 2076, 2330, 2614, 2929, 3280, 3666, 4093, 4565, 5085, 5667, 6300, 7002

Offset: 0

Gus Wiseman, Oct 12 2019

nonn

Except for any number of 1's, these partitions must be strict. The fully strict case is A328188.

Partitions with no consecutive pair of parts relatively prime are A328187, with strict case A328220.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (52)       (71)
                    (1111)  (311)    (411)     (61)       (431)
                            (2111)   (3111)    (511)      (521)
                            (11111)  (21111)   (3211)     (611)
                                     (111111)  (4111)     (5111)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The case of compositions is A167606.
The strict case is A328188.
The Heinz numbers of these partitions are given by A328335.
Cf. A000837, A018783, A178470, A328028, A328170, A328171, A328187, A328188 A328220.

b:= proc(n, i, s) option remember; `if`(n=0 or i=1, 1,
      `if`(andmap(j-> igcd(i, j)=1, s), b(n-i, min(n-i, i-1),
           numtheory[factorset](i)), 0)+b(n, i-1, s))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 13 2019

Table[Length[Select[IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]>1]&]],{n,0,30}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[n == 0 || i == 1, 1,
     If[AllTrue[s, GCD[i, #] == 1&], b[n - i, Min[n - i, i - 1],
     FactorInteger[i][[All, 1]]], 0] + b[n, i - 1, s]];
a[n_] := b[n, n, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A128695 Number of compositions of n with parts in N which avoid the adjacent pattern 111.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 24, 46, 89, 170, 324, 618, 1183, 2260, 4318, 8249, 15765, 30123, 57556, 109973, 210137, 401525, 767216, 1465963, 2801115, 5352275, 10226930, 19541236, 37338699, 71345449, 136324309, 260483548, 497722578, 951030367

Offset: 0

Ralf Stephan, May 08 2007

nonn

			From _Gus Wiseman_, Jul 06 2020: (Start)
The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)    (3)    (4)      (5)
           (1,1)  (1,2)  (1,3)    (1,4)
                  (2,1)  (2,2)    (2,3)
                         (3,1)    (3,2)
                         (1,1,2)  (4,1)
                         (1,2,1)  (1,1,3)
                         (2,1,1)  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,2,1)
                                  (1,2,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. Heubach and T. Mansour, Enumeration of 3-letter patterns in compositions, arXiv:math/0603285 [math.CO], 2006
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Column k=0 of A232435.
Cf. A091616, A232432.
The matching version is A335464.
Contiguously (1,1)-avoiding compositions is A003242.
Contiguously (1,1)-matching compositions are A261983.
Compositions with some part > 2 are A008466
Compositions by number of adjacent equal parts are A106356.
Compositions where each part is adjacent to an equal part are A114901.
Compositions with adjacent parts coprime are A167606.
Compositions with equal parts contiguous are A274174.
Patterns contiguously matched by compositions are A335457.
Patterns contiguously matched by a given partition are A335516.
Cf. A005251, A032020, A131044, A178470, A242882, A333175, A335455.

b:= proc(n, t) option remember; `if`(n=0, 1, add(`if`(abs(t)<>j,
       b(n-j, j), `if`(t=-j, 0, b(n-j, -j))), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 23 2013

nn=33;CoefficientList[Series[1/(1-Sum[(x^i+x^(2i))/(1+x^i+x^(2i)),{i,1,nn}]),{x,0,nn}],x] (* Geoffrey Critzer, Nov 23 2013 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,x_,x_,_}]&]],{n,13}] (* Gus Wiseman, Jul 06 2020 *)

G.f.: 1/(1-Sum(i>=1, x^i*(1+x^i)/(1+x^i*(1+x^i)) ) ).
a(n) ~ c * d^n, where d is the root of the equation Sum_{k>=1} 1/(d^k + 1/(1 + d^k)) = 1, d=1.9107639262818041675000243699745706859615884029961947632387839..., c=0.4993008137128378086219448701860326113802027003939127932922782... - Vaclav Kotesovec, May 01 2014, updated Jul 07 2020
For n>=2, a(n) = A091616(n) + A003242(n). - Vaclav Kotesovec, Jul 07 2020

A328187 Number of integer partitions of n with no pair of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 14, 1, 16, 9, 22, 3, 38, 4, 46, 19, 58, 9, 94, 18, 106, 41, 144, 28, 221, 37, 246, 92, 318, 87, 465, 95, 530, 198, 693, 169, 963, 220, 1108, 424, 1383, 381, 1899, 492, 2216, 815, 2732, 799, 3644, 1041, 4231, 1585, 5194, 1608

Offset: 0

Gus Wiseman, Oct 12 2019

nonn

			The a(1) = 1 through a(15) = 9 partitions (A..F = 10..15):
  1  2  3  4   5  6    7  8     9    A      B  C       D  E        F
           22     33      44    63   55        66         77       96
                  42      62    333  64        84         86       A5
                  222     422        82        93         A4       C3
                          2222       442       A2         C2       555
                                     622       444        644      663
                                     4222      633        662      933
                                     22222     642        842      6333
                                               822        A22      33333
                                               3333       4442
                                               4422       6422
                                               6222       8222
                                               42222      44222
                                               222222     62222
                                                          422222
                                                          2222222

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

The Heinz numbers of these partitions are given by A328336.
The case of compositions is A178470.
The strict case is A328220.
Partitions with all pairs of consecutive parts relatively prime are A328172.
Cf. A000837, A018783, A328028, A328170, A328171, A328188.

Table[Length[Select[IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]==1]&]],{n,0,30}]

A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

Original entry on oeis.org

0, 1, 9, 19, 22, 28, 34, 69, 74, 84, 104, 132, 135, 141, 153, 177, 225, 265, 271, 274, 283, 286, 292, 307, 310, 316, 328, 355, 358, 364, 376, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 553, 562, 593, 610, 673, 706, 833, 898, 1041, 1047, 1053, 1058

Offset: 1

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. 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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The sequence together with the corresponding compositions begins:
    0: ()
    1: (1)
    9: (3,1)
   19: (3,1,1)
   22: (2,1,2)
   28: (1,1,3)
   34: (4,2)
   69: (4,2,1)
   74: (3,2,2)
   84: (2,2,3)
  104: (1,2,4)
  132: (5,3)
  135: (5,1,1,1)
  141: (4,1,2,1)
  153: (3,1,3,1)
  177: (2,1,4,1)
  225: (1,1,5,1)

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

See link for sequences related to standard compositions.
For product equal to sum we have A335404, counted by A335405.
For sum equal to twice alternating sum we have A348614, counted by A262977.
These compositions are counted by A357182.
For absolute value we have A357184, counted by A357183.
The case of partitions is counted by A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.
Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],Length[stc[#]]==ats[stc[#]]&]

A345195 Number of non-alternating anti-run compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 4, 10, 23, 49, 96, 192, 368, 692, 1299, 2403, 4400, 8029, 14556, 26253, 47206, 84574, 151066, 269244, 478826, 849921, 1506309, 2665829, 4711971, 8319763, 14675786, 25865400, 45552678, 80171353, 141015313, 247905305, 435614270, 765132824

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.

			The a(9) = 23 anti-runs:
  (1,2,6)  (1,2,4,2)  (1,2,1,2,3)
  (1,3,5)  (1,2,5,1)  (1,2,3,1,2)
  (2,3,4)  (1,3,4,1)  (1,2,3,2,1)
  (4,3,2)  (1,4,3,1)  (1,3,2,1,2)
  (5,3,1)  (1,5,2,1)  (2,1,2,3,1)
  (6,2,1)  (2,1,2,4)  (2,1,3,2,1)
           (2,4,2,1)  (3,2,1,2,1)
           (3,1,2,3)
           (3,2,1,3)
           (4,2,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Non-anti-run compositions are counted by A261983.
A version counting partitions is A345166, ranked by A345173.
These compositions are ranked by A345169.
Non-alternating compositions are counted by A345192, ranked by A345168.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A032020 counts strict compositions.
A106356 counts compositions by number of maximal anti-runs.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A345194 counts alternating patterns (with twins: A344605).
Cf. A005649, A008965, A114901, A178470, A333755, A344604, A344614, A344654, A344740, A345162, A345163, A348380, A348612, A348613.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], sepQ[#]&&!wigQ[#]&]],{n,0,15}]

a(n) = A003242(n) - A025047(n).

a(21) onwards from Andrew Howroyd, Jan 31 2024

A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 2, 3, 2, 5, 12, 22, 26, 58, 100, 203, 282, 616, 962, 2045, 2982, 6518, 9858, 21416, 31680, 69623, 104158, 228930, 339978, 751430, 1119668, 2478787, 3684082, 8182469, 12171900, 27082870, 40247978, 89748642, 133394708, 297933185, 442628598, 990210110

Offset: 0

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(1) = 1 through a(8) = 12 compositions:
  (1)  (13)  (113)  (24)  (124)  (35)
       (31)  (212)  (42)  (151)  (53)
             (311)        (223)  (1115)
                          (322)  (1151)
                          (421)  (1214)
                                 (1313)
                                 (1412)
                                 (1511)
                                 (2141)
                                 (3131)
                                 (4121)
                                 (5111)

For product instead of length we have A114220.
For sum equal to twice alternating sum we have A262977, ranked by A348614.
For product equal to sum we have A335405, ranked by A335404.
This is the absolute value version of A357182.
These compositions are ranked by A357185.
The case of partitions is A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A261983 counts non-anti-run compositions.
A357136 counts compositions by alternating sum.
Cf. A000120, A032020, A070939, A106356, A114901, A131044, A178470, A233564, A242882, A262046, A301987.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==Abs[ats[#]]&]],{n,0,15}]

a(21)-a(39) from Alois P. Heinz, Sep 29 2022

A348377 Number of non-alternating compositions of n, excluding twins (x,x).

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 19, 45, 98, 208, 436, 906, 1861, 3803, 7731, 15659, 31628, 63747, 128257, 257722, 517338, 1037652, 2079983, 4167325, 8346203, 16710572, 33449694, 66944254, 133959020, 268028868, 536231902, 1072737537, 2145905284, 4292486690, 8586035992

Offset: 0

Gus Wiseman, Oct 26 2021

nonn

First differs from A348382 at a(6) = 19, A348382(6) = 17. The two non-alternating non-twin compositions of 6 that are not an anti-run are (1,2,3) and (3,2,1).

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			The a(3) = 1 through a(6) = 19 compositions:
  (1,1,1)  (1,1,2)    (1,1,3)      (1,1,4)
           (2,1,1)    (1,2,2)      (1,2,3)
           (1,1,1,1)  (2,2,1)      (2,2,2)
                      (3,1,1)      (3,2,1)
                      (1,1,1,2)    (4,1,1)
                      (1,1,2,1)    (1,1,1,3)
                      (1,2,1,1)    (1,1,2,2)
                      (2,1,1,1)    (1,1,3,1)
                      (1,1,1,1,1)  (1,2,2,1)
                                   (1,3,1,1)
                                   (2,1,1,2)
                                   (2,2,1,1)
                                   (3,1,1,1)
                                   (1,1,1,1,2)
                                   (1,1,1,2,1)
                                   (1,1,2,1,1)
                                   (1,2,1,1,1)
                                   (2,1,1,1,1)
                                   (1,1,1,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Alternating permutation

The version for patterns is A000670(n) - A344605(n).
Non-twin compositions are counted by A051049.
The complement is counted by A344604.
An unordered version is A344654.
The complement is ranked by A345167 \/ A007582.
These compositions are ranked by A345168 \ A007582.
Including twins gives A345192, complement A025047.
The version for factorizations is A347706, or A348380 with twins.
The non-anti-run case is A348382.
A001250 counts alternating permutations.
A011782 counts compositions, strict A032020.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A261983 counts non-anti-run compositions, complement A003242.
A325535 counts inseparable partitions, ranked by A335448.
A344614 counts compositions avoiding (1,2,3) and (3,2,1) adjacent.
A345165 = partitions with no alternating permutations, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
Cf. A005649, A178470, A238279, A333755, A335126, A344653, A344740, A345166, A345169, A345173, A348379, A348381.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,15}]

For n > 0, a(n) = A345192(n) - 1 if n is even; otherwise A345192(n).

a(26) onwards from Andrew Howroyd, Jan 31 2024

A328460 Number of compositions of n with no part divisible by the next.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 8, 11, 16, 26, 35, 53, 76, 115, 168, 244, 363, 528, 782, 1144, 1685, 2474, 3633, 5347, 7844, 11539, 16946, 24919, 36605, 53782, 79053, 116142, 170700, 250800, 368585, 541610, 795884, 1169572, 1718593, 2525522, 3711134, 5453542, 8013798, 11776138

Offset: 0

Gus Wiseman, Oct 17 2019

nonn

			The a(1) = 1 through a(9) = 16 compositions:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)     (9)
            (21)  (31)  (23)  (42)   (25)   (35)    (27)
                        (32)  (51)   (34)   (53)    (45)
                        (41)  (231)  (43)   (62)    (54)
                              (321)  (52)   (71)    (63)
                                     (61)   (251)   (72)
                                     (232)  (323)   (81)
                                     (421)  (341)   (234)
                                            (431)   (252)
                                            (521)   (342)
                                            (2321)  (351)
                                                    (423)
                                                    (432)
                                                    (531)
                                                    (621)
                                                    (3231)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The case of partitions is A328171.
If we also require no part to be divisible by the prior, we get A328508.
Compositions with each part relatively prime to the next are A167606.
Compositions with no part relatively prime to the next are A178470.
Cf. A328026, A328028, A328161, A328172, A328189.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;Divisible[y,x]]&]],{n,0,10}]

seq(n)={my(r=matid(n)); for(k=1, n, for(i=1, k-1, r[i,k]=sum(j=1, k-i, if(j%i, r[j, k-i])))); concat([1], vecsum(Col(r)))} \\ Andrew Howroyd, Oct 19 2019

Terms a(26) and beyond from Andrew Howroyd, Oct 19 2019

A328598 Number of compositions of n with no part circularly followed by a divisor.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 0, 4, 2, 7, 12, 11, 22, 26, 55, 63, 99, 149, 215, 324, 458, 699, 1006, 1492, 2185, 3202, 4734, 6928, 10242, 14951, 22023, 32365, 47557, 69905, 102633, 150983, 221712, 325918, 478841, 703647, 1034103, 1519431, 2233061, 3281003, 4821790, 7085358

Offset: 0

Gus Wiseman, Oct 24 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

Circularity means the last part is followed by the first.

			The a(5) = 2 through a(12) = 22 compositions (empty column not shown):
  (2,3)  (2,5)  (3,5)  (2,7)    (3,7)      (2,9)    (5,7)
  (3,2)  (3,4)  (5,3)  (4,5)    (4,6)      (3,8)    (7,5)
         (4,3)         (5,4)    (6,4)      (4,7)    (2,3,7)
         (5,2)         (7,2)    (7,3)      (5,6)    (2,7,3)
                       (2,4,3)  (2,3,5)    (6,5)    (3,2,7)
                       (3,2,4)  (2,5,3)    (7,4)    (3,4,5)
                       (4,3,2)  (3,2,5)    (8,3)    (3,5,4)
                                (3,5,2)    (9,2)    (3,7,2)
                                (5,2,3)    (2,4,5)  (4,3,5)
                                (5,3,2)    (4,5,2)  (4,5,3)
                                (2,3,2,3)  (5,2,4)  (5,3,4)
                                (3,2,3,2)           (5,4,3)
                                                    (7,2,3)
                                                    (7,3,2)
                                                    (2,3,2,5)
                                                    (2,3,4,3)
                                                    (2,5,2,3)
                                                    (3,2,3,4)
                                                    (3,2,5,2)
                                                    (3,4,3,2)
                                                    (4,3,2,3)
                                                    (5,2,3,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The necklace version is A328600, or A318729 without singletons.
The version with singletons is A318726.
The non-circular version is A328460.
Also forbidding parts circularly followed by a multiple gives A328599.
Partitions with no part followed by a divisor are A328171.
Cf. A000740, A008965, A167606, A178470, A318748, A328187, A328508, A328593, A328597, A328601, A328603, A328609.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Not/@Divisible@@@Partition[#,2,1,1]&]],{n,0,10}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i,j)->i%j<>0)))} \\ Andrew Howroyd, Oct 26 2019

a(n > 0) = A318726(n) - 1.

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

A335464 Number of compositions of n with a run of length > 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 8, 18, 39, 86, 188, 406, 865, 1836, 3874, 8135, 17003, 35413, 73516, 152171, 314151, 647051, 1329936, 2728341, 5587493, 11424941, 23327502, 47567628, 96879029, 197090007, 400546603, 813258276, 1649761070, 3343936929, 6772740076, 13707639491

Offset: 0

Gus Wiseman, Jul 06 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

Also compositions contiguously matching the pattern (1,1,1).

			The a(3) = 1 through a(7) = 18 compositions:
  (111)  (1111)  (1112)   (222)     (1114)
                 (2111)   (1113)    (1222)
                 (11111)  (3111)    (2221)
                          (11112)   (4111)
                          (11121)   (11113)
                          (12111)   (11122)
                          (21111)   (11131)
                          (111111)  (13111)
                                    (21112)
                                    (22111)
                                    (31111)
                                    (111112)
                                    (111121)
                                    (111211)
                                    (112111)
                                    (121111)
                                    (211111)
                                    (1111111)

Compositions contiguously avoiding (1,1) are A003242.
Compositions with some part > 2 are A008466.
Compositions by number of adjacent equal parts are A106356.
Compositions where each part is adjacent to an equal part are A114901.
Compositions contiguously avoiding (1,1,1) are A128695.
Compositions with adjacent parts coprime are A167606.
Compositions contiguously matching (1,1) are A261983.
Compositions with all equal parts contiguous are A274174.
Patterns contiguously matched by compositions are A335457.
Cf. A005251, A011782, A056986, A032020, A131044, A178470, A242882, A335455, A335458.

b:= proc(n, t) option remember; `if`(n=0, 1, add(`if`(abs(t)<>j,
       b(n-j, j), `if`(t=-j, 0, b(n-j, -j))), j=1..n))
    end:
a:= n-> ceil(2^(n-1))-b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,x_,x_,_}]&]],{n,0,10}]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[If[Abs[t] != j,
     b[n - j, j], If[t == -j, 0, b[n - j, -j]]], {j, 1, n}]];
a[n_] := Ceiling[2^(n-1)] - b[n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

a(n) = A011782(n) - A128695(n). - Alois P. Heinz, Jul 06 2020

a(23)-a(35) from Alois P. Heinz, Jul 06 2020

cp's OEIS Frontend

A328172 Number of integer partitions of n with all pairs of consecutive parts relatively prime.

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A128695 Number of compositions of n with parts in N which avoid the adjacent pattern 111.

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A328187 Number of integer partitions of n with no pair of consecutive parts relatively prime.

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A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

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A345195 Number of non-alternating anti-run compositions of n.

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A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

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A348377 Number of non-alternating compositions of n, excluding twins (x,x).

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A328460 Number of compositions of n with no part divisible by the next.

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