A329766 -id:A329766 - cp's OEIS frontend

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

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

A329738 Number of compositions of n whose run-lengths are all equal.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 19, 24, 45, 75, 133, 215, 401, 662, 1177, 2035, 3587, 6190, 10933, 18979, 33339, 58157, 101958, 178046, 312088, 545478, 955321, 1670994, 2925717, 5118560, 8960946, 15680074, 27447350, 48033502, 84076143, 147142496, 257546243, 450748484, 788937192

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(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (1111)  (131)    (51)
                            (212)    (123)
                            (11111)  (132)
                                     (141)
                                     (213)
                                     (222)
                                     (231)
                                     (312)
                                     (321)
                                     (1122)
                                     (1212)
                                     (2121)
                                     (2211)
                                     (111111)

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

Compositions with relatively prime run-lengths are A000740.
Compositions with equal multiplicities are A098504.
Compositions with equal differences are A175342.
Compositions with distinct run-lengths are A329739.
Cf. A003242, A008965, A107429, A164707, A238130, A242882, A274174, A329745, A329766.

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

seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); concat([1], vector(n, k, sumdiv(k, d, b[d])))} \\ Andrew Howroyd, Dec 30 2020

a(n) = Sum_{d|n} A003242(d).

a(n) = A329745(n) + A000005(n).

A227038 Number of (weakly) unimodal compositions of n where all parts 1, 2, ..., m appear where m is the largest part.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 13, 19, 30, 44, 71, 98, 147, 205, 294, 412, 575, 783, 1077, 1456, 1957, 2634, 3492, 4627, 6082, 7980, 10374, 13498, 17430, 22451, 28767, 36806, 46803, 59467, 75172, 94839, 119285, 149599, 187031, 233355, 290340, 360327, 446222, 551251, 679524, 835964, 1026210

Offset: 0

Joerg Arndt, Jun 28 2013

nonn

			There are a(8) = 30 such compositions of 8:
01:  [ 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 2 2 ]
06:  [ 1 1 1 2 1 1 1 ]
07:  [ 1 1 1 2 2 1 ]
08:  [ 1 1 1 2 3 ]
09:  [ 1 1 1 3 2 ]
10:  [ 1 1 2 1 1 1 1 ]
11:  [ 1 1 2 2 1 1 ]
12:  [ 1 1 2 2 2 ]
13:  [ 1 1 2 3 1 ]
14:  [ 1 1 3 2 1 ]
15:  [ 1 2 1 1 1 1 1 ]
16:  [ 1 2 2 1 1 1 ]
17:  [ 1 2 2 2 1 ]
18:  [ 1 2 2 3 ]
19:  [ 1 2 3 1 1 ]
20:  [ 1 2 3 2 ]
21:  [ 1 3 2 1 1 ]
22:  [ 1 3 2 2 ]
23:  [ 2 1 1 1 1 1 1 ]
24:  [ 2 2 1 1 1 1 ]
25:  [ 2 2 2 1 1 ]
26:  [ 2 2 3 1 ]
27:  [ 2 3 1 1 1 ]
28:  [ 2 3 2 1 ]
29:  [ 3 2 1 1 1 ]
30:  [ 3 2 2 1 ]
From _Gus Wiseman_, Mar 05 2020: (Start)
The a(1) = 1 through a(6) = 13 compositions:
  (1)  (11)  (12)   (112)   (122)    (123)
             (21)   (121)   (221)    (132)
             (111)  (211)   (1112)   (231)
                    (1111)  (1121)   (321)
                            (1211)   (1122)
                            (2111)   (1221)
                            (11111)  (2211)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)
(End)

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Composition (combinatorics)
Eric Weisstein's World of Mathematics, Unimodal Sequence
Wikipedia, Unimodality

Cf. A001523 (unimodal compositions), A001522 (smooth unimodal compositions with first and last part 1), A001524 (unimodal compositions such that each up-step is by at most 1 and first part is 1).
Organizing by length rather than sum gives A007052.
The complement is counted by A332743.
The case of run-lengths of partitions is A332577, with complement A332579.
Compositions covering an initial interval are A107429.
Non-unimodal compositions are A115981.
Cf. A000009, A055932, A072704, A317086, A329766, A332578, A332669, A332670.

b:= proc(n,i) option remember;
      `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+
      add(b(n-i*j, i+1)*(j+1), j=1..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i] == 0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 1, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&unimodQ[#]&]],{n,0,10}] (* Gus Wiseman, Mar 05 2020 *)

a(n) ~ c * exp(Pi*sqrt(r*n)) / n, where r = 0.9409240878664458093345791978063..., c = 0.05518035191234679423222212249... - Vaclav Kotesovec, Mar 04 2020

a(n) + A332743(n) = 2^(n - 1). - Gus Wiseman, Mar 05 2020

A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 8, 27, 75, 185, 441, 1025, 2276, 4985, 10753, 22863, 48142, 100583, 208663, 430563, 884407, 1809546, 3690632, 7506774, 15233198, 30851271, 62377004, 125934437, 253936064, 511491634, 1029318958, 2069728850, 4158873540, 8351730223, 16762945432

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

			The a(6) = 3 and a(7) = 8 compositions:
  (1221)   (2113)
  (2112)   (3112)
  (11211)  (11311)
           (12112)
           (21112)
           (21121)
           (111211)
           (112111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence.

The case of partitions is A332641.
The version for unsorted prime signature is A332831.
The version for the compositions themselves (not run-lengths) is A332834.
The complement is counted by A332835.
Unimodal compositions are A001523.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Compositions whose run-lengths are not unimodal are A332727.
Partitions with weakly increasing or weakly decreasing run-lengths: A332745.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A072704, A072706, A107429, A181819, A329398, A329744, A329746, A329766, A332273, A332640, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,10}]

a(n) = 2^(n - 1) - 2 * A332836(n) + A329738(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 29, 56, 101, 181, 327, 583, 1023, 1820, 3207, 5631, 9905, 17394, 30489, 53481, 93725, 164169, 287606, 503672, 881834, 1544018, 2703161, 4731860, 8283291, 14499392, 25379278, 44422866, 77754798, 136093756, 238204369, 416923752, 729728031

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

			The a(6) = 29 compositions:
  (6)    (141)  (213)   (1113)  (21111)
  (51)   (114)  (132)   (222)   (12111)
  (15)   (33)   (123)   (2211)  (11121)
  (42)   (321)  (3111)  (2121)  (11112)
  (24)   (312)  (1311)  (1212)  (111111)
  (411)  (231)  (1131)  (1122)
Missing are: (2112), (1221), (11211).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence

The version for the compositions themselves (not run-lengths) is A329398.
Compositions with equal run-lengths are A329738.
The case of partitions is A332745.
The version for unsorted prime signature is the complement of A332831.
The complement is counted by A332833.
Unimodal compositions are A001523.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Neither weakly increasing nor weakly decreasing compositions are A332834.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A181819, A329744, A329766, A332273, A332640, A332641, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,20}]

a(n) = 2 * A332836(n) - A329738(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A332726 Number of compositions of n whose run-lengths are unimodal.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 61, 120, 228, 438, 836, 1580, 2976, 5596, 10440, 19444, 36099, 66784, 123215, 226846, 416502, 763255, 1395952, 2548444, 4644578, 8452200, 15358445, 27871024, 50514295, 91446810, 165365589, 298730375, 539127705, 972099072, 1751284617, 3152475368

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

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

			The only composition of 6 whose run-lengths are not unimodal is (1,1,2,1,1).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not run-lengths) gives A001523.
The case of partitions is A332280, with complement counted by A332281.
The complement is counted by A332727.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negated run-lengths are unimodal are A332578.
Compositions whose negated run-lengths are not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332642, A332670, A332741, A332833, A332835.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],unimodQ[Length/@Split[#]]&]],{n,0,10}]

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
desc(M, m)={my(n=matsize(M)[1]); while(m>1, m--; M=step(M,m)); vector(n, i, vecsum(M[i,]))/(#M-1)}
seq(n)={my(M=matrix(n+1, n+1, i, j, i==1), S=M[,1]~); for(m=1, n, my(D=M); M=step(M, m); D=(M-D)[m+1..n+1,1..n-m+2]; S+=concat(vector(m), desc(D,m))); S} \\ Andrew Howroyd, Dec 31 2020

a(n) + A332727(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A332340 Number of widely alternately co-strongly normal compositions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 9, 11, 13, 23, 53, 78, 120, 207, 357, 707, 1183, 2030, 3558, 6229, 10868

Offset: 0

Gus Wiseman, Feb 17 2020

An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-length (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (11)  (12)   (121)   (122)    (123)     (1213)     (1232)
             (21)   (211)   (212)    (132)     (1231)     (1322)
             (111)  (1111)  (1211)   (213)     (1312)     (2123)
                            (11111)  (231)     (1321)     (2132)
                                     (312)     (2122)     (2312)
                                     (321)     (2131)     (2321)
                                     (1212)    (2311)     (3122)
                                     (2121)    (3121)     (3212)
                                     (111111)  (3211)     (12131)
                                               (12121)    (13121)
                                               (1111111)  (21212)
                                                          (122111)
                                                          (11111111)
For example, starting with the composition y = (122111) and repeatedly taking run-lengths and reversing gives (122111) -> (321) -> (111). All of these are normal with weakly increasing run-lengths and the last is all 1's, so y is counted under a(8).

Normal compositions are A107429.
Compositions with normal run-lengths are A329766.
The Heinz numbers of the case of partitions are A332290.
The case of partitions is A332289.
The total (instead of alternating) version is A332337.
Not requiring normality gives A332338.
The strong version is this same sequence.
The narrow version is a(n) + 1 for n > 1.
Cf. A181819, A317245, A317491, A329744, A329741, A329746, A332278, A332279, A332292.

totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],totnQ]],{n,0,10}]

A332836 Number of compositions of n whose run-lengths are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 24, 40, 73, 128, 230, 399, 712, 1241, 2192, 3833, 6746, 11792, 20711, 36230, 63532, 111163, 194782, 340859, 596961, 1044748, 1829241, 3201427, 5604504, 9808976, 17170112, 30051470, 52601074, 92063629, 161140256, 282033124, 493637137, 863982135, 1512197655

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

Also compositions whose run-lengths are weakly decreasing.

			The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (121)   (41)
                        (211)   (122)
                        (1111)  (131)
                                (212)
                                (311)
                                (1211)
                                (2111)
                                (11111)
For example, the composition (2,3,2,2,1,1,2,2,2) has run-lengths (1,1,2,2,3) so is counted under a(17).

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

The version for the compositions themselves (not run-lengths) is A000041.
The case of partitions is A100883.
The case of unsorted prime signature is A304678, with dual A242031.
Permitting the run-lengths to be weakly decreasing also gives A332835.
The complement is counted by A332871.
Unimodal compositions are A001523.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Cf. A001462, A072704, A072706, A100882, A181819, A329744, A329766, A332641, A332727, A332745, A332833, A332834.

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

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
seq(n)={my(M=matrix(n+1, n, i, j, i==1)); for(m=1, n, M=step(M, m)); M[1,n]=0; vector(n+1, i, vecsum(M[i,]))/(n-1)} \\ Andrew Howroyd, Dec 31 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A329740 Number of compositions of n whose multiplicities are distinct and cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 4, 10, 10, 10, 73, 196, 133, 379, 319, 379, 502, 805, 562, 1108, 13648, 51448, 51691, 115174, 140011, 178597, 203617, 329737, 292300, 456703, 456160, 608386, 633466, 898186, 823009, 39014392, 190352269, 266293795, 493345615, 834326995, 947714938

Offset: 0

Gus Wiseman, Nov 21 2019

nonn

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

			The a(1) = 1 through a(9) = 10 compositions:
  (1)  (2)  (3)  (4)      (5)      (6)      (7)      (8)      (9)
                 (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
                 (1,2,1)  (1,2,2)  (1,4,1)  (1,3,3)  (1,6,1)  (1,4,4)
                 (2,1,1)  (1,3,1)  (4,1,1)  (1,5,1)  (2,2,4)  (1,7,1)
                          (2,1,2)           (2,2,3)  (2,3,3)  (2,2,5)
                          (2,2,1)           (2,3,2)  (2,4,2)  (2,5,2)
                          (3,1,1)           (3,1,3)  (3,2,3)  (4,1,4)
                                            (3,2,2)  (3,3,2)  (4,4,1)
                                            (3,3,1)  (4,2,2)  (5,2,2)
                                            (5,1,1)  (6,1,1)  (7,1,1)

The version allowing repeated multiplicities is A329741.
Complete compositions are A107429.
Compositions whose multiplicities are distinct are A242882.
Cf. A059966, A098504, A244164, A274174, A329739, A329766, A329748.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Range[Length[Union[#]]]==Sort[Length/@Split[Sort[#]]]&]],{n,0,10}]

a(21)-a(40) from Alois P. Heinz, Nov 21 2019

A332727 Number of compositions of n whose run-lengths are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 8, 28, 74, 188, 468, 1120, 2596, 5944, 13324, 29437, 64288, 138929, 297442, 632074, 1333897, 2798352, 5840164, 12132638, 25102232, 51750419, 106346704, 217921161, 445424102, 908376235, 1848753273, 3755839591, 7617835520, 15428584567, 31207263000

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

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

			The a(6) = 1 through a(8) = 8 compositions:
  (11211)  (11311)   (11411)
           (111211)  (111311)
           (112111)  (112112)
                     (113111)
                     (211211)
                     (1111211)
                     (1112111)
                     (1121111)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not its run-lengths) gives A115981.
The case of partitions is A332281, with complement counted by A332280.
The complement is counted by A332726.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negation is not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A007052, A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332578, A332638, A332639, A332670, A332741, A332833.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[Length/@Split[#]]&]],{n,0,10}]

a(n) + A332726(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

cp's OEIS Frontend

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

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A329738 Number of compositions of n whose run-lengths are all equal.

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A227038 Number of (weakly) unimodal compositions of n where all parts 1, 2, ..., m appear where m is the largest part.

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A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

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A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

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A332726 Number of compositions of n whose run-lengths are unimodal.

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A332340 Number of widely alternately co-strongly normal compositions of n.

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