A032020 -id:A032020 - cp's OEIS frontend

A102726 Number of compositions of the integer n into positive parts that avoid a fixed pattern of three letters.

1, 1, 2, 4, 8, 16, 31, 60, 114, 214, 398, 732, 1334, 2410, 4321, 7688, 13590, 23869, 41686, 72405, 125144, 215286, 368778, 629156, 1069396, 1811336, 3058130, 5147484, 8639976, 14463901, 24154348, 40244877, 66911558, 111026746, 183886685, 304034456, 501877227

Offset: 0

Herbert S. Wilf, Feb 07 2005

The sequence is the same no matter which of the six patterns of three letters is chosen as the one to be avoided.

			a(6) = 31 because there are 32 compositions of 6 into positive parts and only one of these, namely 6 = 1+2+3, contains the pattern (123), the other 31 compositions of 6 avoid that pattern.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..900 (first 400 terms from Alois P. Heinz)
M. Elder and V. Vatter, Problems and conjectures presented at the third international conference on permutation petterns
Carla D. Savage and Herbert S. Wilf, Pattern avoidance in compositions and multiset permutations, Advances in Applied Mathematics 36 (2006), pp.194-201.
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for patterns is A226316.
These compositions are ranked by the complement of A335479.
The matching version is A335514.
The version for prime indices is A335521.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Compositions are counted by A011782.
Strict compositions are counted by A032020 and ranked by A233564.
Patterns matched by compositions are counted by A335456.
Minimal patterns avoided by a given composition are counted by A335465.
Cf. A056986, A106356, A232464, A269134, A333755.

b:= proc(n, m, t) option remember; `if`(n=0, 1,
      add(b(n-i, min(m, i, n-i), min(t, n-i,
      `if`(i>m, i, t))), i=1..min(n, t)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 18 2014

b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[b[n - i, Min[m, i, n - i], Min[t, n - i, If[i > m, i, t]]], {i, 1, Min[n, t]}]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union[mstype/@Subsets[#]],{1,2,3}]&]],{n,0,10}] (* Gus Wiseman, Jun 22 2020 *)

seq(n)={Vec(sum(i=1, n, prod(j=1, n, if(i==j, 1, (1-x^i)/((1-x^(j-i))*(1-x^i-x^j))) + O(x*x^n))/(1-x^i)))} \\ Andrew Howroyd, Dec 31 2020

G.f.: Sum_{i>=1} (1/(1-x^i))*Product_{j>=1, j<>i} (1-x^i)/((1-x^(j-i))*(1-x^i-x^j)).

Asymptotics (Savage and Wilf, 2005): a(n) ~ c * ((1+sqrt(5))/2)^n, where c = r/(r-1)/(r-s) * (r * Product_{j>=3} (1-1/r)/(1-r^(1-j))/(1-1/r-r^(-j)) - Product_{j>=3} (1-1/r^2)/(1-r^(2-j))/(1-1/r^2-r^(-j)) ) = 18.9399867283479198666671671745270505487677312850521421513193261105... and r = (1+sqrt(5))/2, s = (1-sqrt(5))/2. - Vaclav Kotesovec, May 02 2014

More terms from Ralf Stephan, May 27 2005

A329739 Number of compositions of n whose run-lengths are all different.

Original entry on oeis.org

1, 1, 2, 2, 5, 8, 10, 20, 28, 41, 62, 102, 124, 208, 278, 426, 571, 872, 1158, 1718, 2306, 3304, 4402, 6286, 8446, 11725, 15644, 21642, 28636, 38956, 52296, 70106, 93224, 124758, 165266, 218916, 290583, 381706, 503174, 659160, 865020, 1124458, 1473912, 1907298

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(7) = 20 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (113)    (33)      (115)
                    (112)   (122)    (114)     (133)
                    (211)   (221)    (222)     (223)
                    (1111)  (311)    (411)     (322)
                            (1112)   (1113)    (331)
                            (2111)   (3111)    (511)
                            (11111)  (11112)   (1114)
                                     (21111)   (1222)
                                     (111111)  (2221)
                                               (4111)
                                               (11113)
                                               (11122)
                                               (22111)
                                               (31111)
                                               (111112)
                                               (111211)
                                               (112111)
                                               (211111)
                                               (1111111)

The normal case is A329740.
The case of partitions is A098859.
Strict compositions are A032020.
Compositions with relatively prime run-lengths are A000740.
Compositions with distinct multiplicities are A242882.
Compositions with distinct differences are A325545.
Compositions with equal run-lengths are A329738.
Compositions with normal run-lengths are A329766.
Cf. A003242, A008965, A059966, A098504, A098859, A107429, A175342, A238130, A328592, A329745.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Length/@Split[#]&]],{n,0,10}]

a(21)-a(26) from Giovanni Resta, Nov 22 2019

a(27)-a(43) from Alois P. Heinz, Jul 06 2020

A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85

Offset: 1

Gus Wiseman, Jul 13 2024

nonn

These are compositions avoiding the patterns (1,2,1) and (2,1,2).

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 together with their 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)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  12: (1,3)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
See A374253 for the complement: 13, 22, 25, 27, 29, ...

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

The strict (also anti-run) case is A233564, counted by A032020.
Compositions of this type are counted by A274174.
Permutations of prime indices of this type are counted by A333175.
The complement is A374253 (anti-run A374254), counted by A335548.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A066099 lists compositions in standard order.
A124767 counts runs in standard compositions, anti-runs A333381.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
- A335470 counts (1,2,1)-matching compositions, ranks A335466.
- A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
- A335472 counts (2,1,2)-matching compositions, ranks A335468.
- A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
Cf. A106356, A124762, A238130, A238279, A261982, A272919, A333382, A335450, A335460, A335524, A335525.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@First/@Split[stc[#]]&]

Equals A335467 /\ A335469.

A167606 Number of compositions of n where each pair of adjacent parts is relatively prime.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 25, 48, 90, 168, 316, 594, 1116, 2096, 3935, 7388, 13877, 26061, 48944, 91919, 172623, 324188, 608827, 1143390, 2147309, 4032677, 7573426, 14223008, 26711028, 50163722, 94208254, 176924559, 332267039, 624002605, 1171886500, 2200820905

Offset: 0

Franklin T. Adams-Watters, Nov 07 2009

nonn

			For n = 4, there are 8 compositions: [4], [3,1], [2,2], [2,1,1], [1,3], [1,2,1], [1,1,2], and [1,1,1,1]. Of these, only [2,2] has adjacent terms that are not relatively prime, so a(4) = 7.

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

Cf. A066099, A003242, A032020.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(`if`(igcd(i, j)=1, b(n-j, j), 0), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 27 2014

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[If[GCD[i, j]==1, b[n-j, j], 0], {j, n}]];
a[n_] := b[n, 1];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz *)

am(n)={local(r);r=matrix(n,n);
for(k=1,n,
for(i=1,k-1,r[k,i]=sum(j=1,k-i,if(gcd(i,j)==1,r[k-i,j],0)));r[k,k]=1);
r}
al(n)=local(m);m=am(n);vector(n,k,sum(i=1,k,m[k,i]))
a(left,last=1)={local(r);if(left==0,return(1));
for(k=1,left,if(gcd(k,last)==1,r+=a(left-k,k)));r}

a(n) ~ c * d^n, where d=1.8780154065731862176678940156530410192010138618103068156064519919669849911..., c=0.5795813856338135589080831265343299561832275012313700387790334792220408848... - Vaclav Kotesovec, May 01 2014

A345192 Number of non-alternating compositions of n.

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 20, 45, 99, 208, 437, 906, 1862, 3803, 7732, 15659, 31629, 63747, 128258, 257722, 517339, 1037652, 2079984, 4167325, 8346204, 16710572, 33449695, 66944254, 133959021, 268028868, 536231903, 1072737537, 2145905285, 4292486690, 8586035993, 17173742032, 34350108745, 68704342523, 137415168084

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

First differs from A261983 at a(6) = 20, A261983(6) = 18.

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).

			The a(2) = 1 through a(6) = 20 compositions:
  (11)  (111)  (22)    (113)    (33)
               (112)   (122)    (114)
               (211)   (221)    (123)
               (1111)  (311)    (222)
                       (1112)   (321)
                       (1121)   (411)
                       (1211)   (1113)
                       (2111)   (1122)
                       (11111)  (1131)
                                (1221)
                                (1311)
                                (2112)
                                (2211)
                                (3111)
                                (11112)
                                (11121)
                                (11211)
                                (12111)
                                (21111)
                                (111111)

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

The complement is counted by A025047 (ascend: A025048, descend: A025049).
Dominates A261983 (non-anti-run compositions), ranked by A348612.
These compositions are ranked by A345168, complement A345167.
The case without twins is A348377.
The version for factorizations is A348613.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions.
A032020 counts strict compositions.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A274174 counts compositions with equal parts contiguous.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344605 counts alternating patterns with twins.
A344654 counts non-twin partitions with no alternating permutation.
A345162 counts normal partitions with no alternating permutation.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions w/ alternating permutation, ranked by A345172.
A345165 counts partitions w/o alternating permutation, ranked by A345171.
Patterns:
- A128761 avoiding (1,2,3) adjacent.
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000070, A008965, A178470, A238279, A333755, A335126, A344606, A344653, A344740, A345163, A345166, A345169, A345173, A348380.

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

a(n) = A011782(n) - A025047(n).

A261982 Number of compositions of n with some part repeated.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 21, 51, 109, 229, 455, 959, 1947, 3963, 7999, 16033, 32333, 64919, 130221, 260967, 522733, 1045825, 2093855, 4189547, 8382315, 16768455, 33543127, 67093261, 134193413, 268404995, 536829045, 1073686083, 2147408773, 4294869253, 8589803783

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020

			a(2) = 1: 11.
a(3) = 1: 111.
a(4) = 5: 22, 211, 121, 112, 1111.

Alois P. Heinz, Table of n, a(n) for n = 0..3322
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Row sums of A261981 and of A262191.
Cf. A262047.
The version for patterns is A019472.
The (1,1)-avoiding version is A032020.
The case of partitions is A047967.
(1,1,1)-matching compositions are counted by A335455.
Patterns matched by compositions are counted by A335456.
(1,1)-matching compositions are ranked by A335488.
Cf. A011782, A056986, A106356, A181796, A238279, A269134, A333755.

b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
    end:
a:= n-> ceil(2^(n-1))-add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);

b[n_, k_] := b[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], b[n-k, k] + k*b[n-k, k-1]]]; a[n_] := Ceiling[2^(n-1)]-Sum[b[n, k], {k, 0, Floor[ (Sqrt[8n+1]-1)/2]}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>Length[Split[#]]&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = A011782(n) - A032020(n).

G.f.: (1 - x) / (1 - 2*x) - Sum_{k>=0} k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 30 2020

A351014 Number of distinct runs in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 1, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 3

Offset: 0

Gus Wiseman, Feb 07 2022

nonn

The n-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 n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The number 3310 has binary expansion 110011101110 and standard composition (1,3,1,1,2,1,1,2), with runs (1), (3), (1,1), (2), (1,1), (2), of which 4 are distinct, so a(3310) = 4.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Counting not necessarily distinct runs gives A124767.
Using binary expansions instead of standard compositions gives A297770.
Positions of first appearances are A351015.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351201.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Split[stc[n]]]],{n,0,100}]

A169942 Number of Golomb rulers of length n.

Original entry on oeis.org

1, 1, 3, 3, 5, 7, 13, 15, 27, 25, 45, 59, 89, 103, 163, 187, 281, 313, 469, 533, 835, 873, 1319, 1551, 2093, 2347, 3477, 3881, 5363, 5871, 8267, 9443, 12887, 14069, 19229, 22113, 29359, 32229, 44127, 48659, 64789, 71167, 94625, 105699, 139119, 151145, 199657

Offset: 1

N. J. A. Sloane, Aug 01 2010

nonn

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?
Leading entry in row n of triangle in A169940. Also the number of Sidon sets A with min(A) = 0 and max(A) = n. Odd for all n since {0,n} is the only symmetric Golomb ruler, and reversal preserves the Golomb property. Bounded from above by A032020 since the ruler {0 < r_1 < ... < r_t < n} gives rise to a composition of n: (r_1 - 0, r_2 - r_1, ... , n - r_t) with distinct parts. - Tomas Boothby, May 15 2012
Also the number of compositions of n such that every restriction to a subinterval has a different sum. This is a stronger condition than all distinct consecutive subsequences having a different sum (cf. A325676). - Gus Wiseman, May 16 2019

			For n=2, there is one Golomb Ruler: {0,2}.  For n=3, there are three: {0,3}, {0,1,3}, {0,2,3}. - _Tomas Boothby_, May 15 2012
From _Gus Wiseman_, May 16 2019: (Start)
The a(1) = 1 through a(8) = 15 compositions such that every restriction to a subinterval has a different sum:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)
            (12)  (13)  (14)  (15)   (16)   (17)
            (21)  (31)  (23)  (24)   (25)   (26)
                        (32)  (42)   (34)   (35)
                        (41)  (51)   (43)   (53)
                              (132)  (52)   (62)
                              (231)  (61)   (71)
                                     (124)  (125)
                                     (142)  (143)
                                     (214)  (152)
                                     (241)  (215)
                                     (412)  (251)
                                     (421)  (341)
                                            (512)
                                            (521)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..99
T. Pham, Enumeration of Golomb Rulers (Master's thesis), San Francisco State U., 2011.
Eric Weisstein's World of Mathematics, Golomb Ruler.
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to Golomb rulers

Related to thickness: A169940-A169954, A061909.
Related to Golomb rulers: A036501, A054578, A143823.
Row sums of A325677.
Cf. A000079, A103295, A103300, A108917, A143824, A325466, A325545, A325676, A325678, A325679, A325683, A325686.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]],{n,15}] (* Gus Wiseman, May 16 2019 *)

def A169942(n):
    R = QQ['x']
    return sum(1 for c in cartesian_product([[0, 1]]*n) if max(R([1] + list(c) + [1])^2) == 2)
[A169942(n) for n in range(1,8)]
# Tomas Boothby, May 15 2012

a(n) = A169952(n) - A169952(n-1) for n>1. - Andrew Howroyd, Jul 09 2017

a(15)-a(30) from Nathaniel Johnston, Nov 12 2011

a(31)-a(50) from Tomas Boothby, May 15 2012

A261983 Number of compositions of n such that at least two adjacent parts are equal.

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 18, 41, 89, 185, 388, 810, 1670, 3435, 7040, 14360, 29226, 59347, 120229, 243166, 491086, 990446, 1995410, 4016259, 8076960, 16231746, 32599774, 65437945, 131293192, 263316897, 527912140, 1058061751, 2120039885, 4246934012, 8505864640

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

			a(5) = 9: 311, 113, 221, 122, 2111, 1211, 1121, 1112, 11111.
From _Gus Wiseman_, Jul 07 2020: (Start)
The a(2) = 1 through a(6) = 18 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,2,2,1)
                             (1,1,1,1,1)  (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)
(End)

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

Column k=1 of A261981.
Cf. A011782, A262046.
The complement A003242 counts anti-runs.
Sum of positive-indexed terms of row n of A106356.
Row sums of A131044.
The (1,1,1) matching case is A335464.
Strict compositions are A032020.
Compositions with adjacent parts coprime are A167606.
Compositions with equal parts contiguous are A274174.
Cf. A114901, A178470, A242882, A244164, A325534, A335448, A335452.

b:= proc(n, i) option remember; `if`(n=0, 0, add(
      `if`(i=j, ceil(2^(n-j-1)), b(n-j, j)), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}] (* Gus Wiseman, Jul 06 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 0, Sum[If[i == j, Ceiling[2^(n-j-1)], b[n-j, j]], {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz's Maple code *)

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 08 2015

a(n) = A011782(n) - A003242(n). - Emeric Deutsch, Jul 03 2020

A353850 Number of integer compositions of n with all distinct run-sums.

Original entry on oeis.org

1, 1, 2, 4, 5, 12, 24, 38, 52, 111, 218, 286, 520, 792, 1358, 2628, 4155, 5508, 9246, 13182, 23480, 45150, 54540, 94986, 146016, 213725, 301104, 478586, 851506, 1302234, 1775482, 2696942, 3746894, 6077784, 8194466, 12638334, 21763463, 28423976, 45309850, 62955524, 94345474

Offset: 0

Gus Wiseman, May 31 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (1111)  (41)
                                (113)
                                (122)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)
For n=4, (211) is invalid because the two runs (2) and (11) have the same sum. - _Joseph Likar_, Aug 04 2023

Joseph Likar, Table of n, a(n) for n = 0..120

For distinct parts instead of run-sums we have A032020.
For distinct multiplicities instead of run-sums we have A242882.
For distinct run-lengths instead of run-sums we have A329739, ptns A098859.
For runs instead of run-sums we have A351013.
For partitions we have A353837, ranked by A353838 (complement A353839).
For equal instead of distinct run-sums we have A353851, ptns A304442.
These compositions are ranked by A353852.
The weak version (rucksack compositions) is A354580, ranked by A354581.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A175413 lists numbers whose binary expansion has all distinct runs.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353847 gives composition run-sum transformation.
A353929 counts distinct runs in binary expansion, firsts A353930.
Cf. A238279, A333755, A351016, A351017, A353832, A353848, A353849, A353853-A353859, A353860, A353863, A353932.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@Total/@Split[#]&]],{n,0,15}]

Terms a(21) and onwards from Joseph Likar, Aug 04 2023

cp's OEIS Frontend

A102726 Number of compositions of the integer n into positive parts that avoid a fixed pattern of three letters.

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A329739 Number of compositions of n whose run-lengths are all different.

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A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

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A167606 Number of compositions of n where each pair of adjacent parts is relatively prime.

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A345192 Number of non-alternating compositions of n.

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A261982 Number of compositions of n with some part repeated.

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A351014 Number of distinct runs in the n-th composition in standard order.

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