A072706 -id:A072706 - cp's OEIS frontend

A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

1, 1, 2, 4, 12, 30, 66, 164, 419, 1049, 2625, 6372, 15451, 37335, 89855, 216523, 518714, 1235897, 2930050, 6911149, 16217817, 37914515, 88304358, 204971388, 474172899, 1093547574, 2513959446, 5761735383, 13165908506, 29998936859, 68164839887, 154478212575

Offset: 0

Gus Wiseman, Jul 01 2022

nonn

If a composition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of compositions.

			The initial terms count the following containments:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)
                (11)(1)  (21)(21)  (31)(31)
                         (12)(12)  (13)(13)
                         (111)(1)  (22)(2)
                                   (211)(11)
                                   (211)(21)
                                   (121)(11)
                                   (121)(12)
                                   (121)(21)
                                   (112)(11)
                                   (112)(12)
                                   (1111)(1)

Christian Sievers, Table of n, a(n) for n = 0..59

The homog. case is A032020, w/o containment A355388 (partitions A355385).
For partitions we have A355383, homog. A000009, w/ multiplicity A339006.
A000244 counts splittings of compositions, for partitions A323583.
Cf. A001970, A022811, A063834, A070933, A072706, A133494, A336139.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,5}]

a(21) and beyond from Christian Sievers, May 08 2025

A218004 Number of equivalence classes of compositions of n where two compositions a,b are considered equivalent if the summands of a can be permuted into the summands of b with an even number of transpositions.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 14, 19, 27, 37, 51, 67, 91, 118, 156, 202, 262, 334, 430, 543, 690, 867, 1090, 1358, 1696, 2099, 2600, 3201, 3939, 4820, 5899, 7181, 8738, 10590, 12821, 15467, 18644, 22396, 26878, 32166, 38450, 45842, 54599, 64870, 76990, 91181, 107861, 127343, 150182, 176788, 207883

Offset: 0

Geoffrey Critzer, Oct 17 2012

nonn

a(n) = A000041(n) + A000009(n) - 1  where A000041 is the partition numbers and A000009 is the number of partitions into distinct parts.
From Gus Wiseman, Oct 14 2020: (Start)
Also the number of compositions of n that are either strictly increasing or weakly decreasing. For example, the a(1) = 1 through a(6) = 14 compositions are:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (211)   (41)     (42)
                    (1111)  (221)    (51)
                            (311)    (123)
                            (2111)   (222)
                            (11111)  (321)
                                     (411)
                                     (2211)
                                     (3111)
                                     (21111)
                                     (111111)
A007997 counts only compositions of length 3.
A329398 appears to be the weakly increasing version.
A333147 is the strictly decreasing version.
A333255 union A114994 ranks these compositions using standard compositions (A066099).
A337482 counts the complement.
(End)

			a(4) = 6 because the 6 classes can be represented by: 4, 3+1, 1+3, 2+2, 2+1+1, 1+1+1+1.

A000009 counts strictly increasing compositions, ranked by A333255.
A000041 counts weakly decreasing compositions, ranked by A114994.
A001523 counts unimodal compositions (strict: A072706).
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A332834 counts compositions not increasing nor decreasing (strict: A333149).
Cf. A115981, A225620, A332578, A332833, A332874, A333150, A333190, A333191, A333256, A337483, A337484.

nn=50;p=CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x];d= CoefficientList[Series[Sum[Product[x^i/(1-x^i),{i,1,k}],{k,0,nn}],{x,0,nn}],x];p+d-1
(* second program *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@#||GreaterEqual@@#&]],{n,0,15}] (* Gus Wiseman, Oct 14 2020 *)

A072707 Number of non-unimodal compositions of n into distinct terms.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 24, 26, 46, 64, 100, 224, 276, 416, 590, 850, 1144, 2214, 2644, 3938, 5282, 7504, 9776, 13704, 21984, 27632, 38426, 51562, 69844, 91950, 123504, 159658, 246830, 303400, 416068, 540480, 730268, 933176, 1248110

Offset: 0

Henry Bottomley, Jul 04 2002

Also the number of compositions of n into distinct terms whose negation is not unimodal. - Gus Wiseman, Mar 05 2020

			a(6)=2 since 6 can be written as 2+1+3 or 3+1+2.
From _Gus Wiseman_, Mar 05 2020: (Start)
The a(6) = 2 through a(9) = 6 strict compositions:
  (2,1,3)  (2,1,4)  (2,1,5)  (2,1,6)
  (3,1,2)  (4,1,2)  (3,1,4)  (3,1,5)
                    (4,1,3)  (3,2,4)
                    (5,1,2)  (4,2,3)
                             (5,1,3)
                             (6,1,2)
(End)

Eric Weisstein's World of Mathematics, Unimodal Sequence

The complement is counted by A072706.
The non-strict version is A115981.
The case where the negation is not unimodal either is A332874.
Unimodal compositions are A001523.
Strict compositions are A032020.
Non-unimodal permutations are A059204.
A triangle for strict unimodal compositions is A072705.
Non-unimodal sequences covering an initial interval are A328509.
Numbers whose prime signature is not unimodal are A332282.
Strict partitions whose 0-appended differences are not unimodal are A332286.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Non-unimodal compositions covering an initial interval are A332743.
Cf. A007052, A072704, A107429, A227038, A329398, A332281, A332284, A332285, A332579, A332870.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!unimodQ[#]&]],{n,0,16}] (* Gus Wiseman, Mar 05 2020 *)

a(n) = A032020(n) - A072706(n) = Sum_{k} A059204(k) * A060016(n, k).

A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 2, 0, 0, 0, 0, 4, 5, 1, 0, 0, 0, 0, 5, 6, 1, 0, 0, 0, 0, 0, 6, 9, 2, 0, 0, 0, 0, 0, 0, 8, 13, 3, 0, 0, 0, 0, 0, 0, 0, 10, 23, 10, 1, 0, 0, 0, 0, 0, 0, 0, 12, 27, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 15, 40, 19, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 18 2019

			Triangle begins:
  1
  0  1
  0  1  0
  0  2  1  0
  0  2  1  0  0
  0  3  2  0  0  0
  0  4  5  1  0  0  0
  0  5  6  1  0  0  0  0
  0  6  9  2  0  0  0  0  0
  0  8 13  3  0  0  0  0  0  0
  0 10 23 10  1  0  0  0  0  0  0
  0 12 27 11  1  0  0  0  0  0  0  0
  0 15 40 19  2  0  0  0  0  0  0  0  0
Row n = 8 counts the following set partitions:
  {{8}}      {{1},{7}}    {{1},{2},{5}}
  {{3,5}}    {{2},{6}}    {{1},{3},{4}}
  {{2,6}}    {{3},{5}}
  {{1,7}}    {{1},{3,4}}
  {{1,3,4}}  {{1},{2,5}}
  {{1,2,5}}  {{2},{1,5}}
             {{3},{1,4}}
             {{4},{1,3}}
             {{5},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n=0..50)

Row sums are A294617.
Column k = 1 is A000009 (n > 0).
Cf. A000110, A008277, A050342, A060016, A072706, A270995, A271619, A279375, A279785, A326701, A330459, A330462, A330463, A330759.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 k*
         b(n-i, t, k)+b(n-i, t, k+1))(min(n-i, i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..15);  # Alois P. Heinz, Dec 29 2019

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,2],Length[#]==k&&And[UnsameQ@@#,UnsameQ@@Join@@#]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, x^k, b[n, i-1, k] + Function[t, k*b[n-i, t, k] + b[n-i, t, k + 1]][Min[n-i, i-1]]]];
T[n_] := PadRight[CoefficientList[b[n, n, 0], x], n + 1];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n)))); vector(#v, n, my(p=v[n]); vector(n, k, sum(i=k, n, polcoef(p,i-1)*stirling(i-1, k-1, 2))))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

T(n,k) = Sum_{k <= i <= n} A060016(n,i) * A008277(i,k).

For n > 0, T(n,2) = Sum_{k = 1..n} (2^(k - 1) -1) * A060016(n,k).

A333147 Number of compositions of n that are either strictly increasing or strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 19, 23, 29, 35, 43, 53, 63, 75, 91, 107, 127, 151, 177, 207, 243, 283, 329, 383, 443, 511, 591, 679, 779, 895, 1023, 1169, 1335, 1519, 1727, 1963, 2225, 2519, 2851, 3219, 3631, 4095, 4607, 5179, 5819, 6527, 7315, 8193, 9163

Offset: 0

Gus Wiseman, May 16 2020

nonn

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

			The a(1) = 1 through a(9) = 15 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)    (1,8)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)    (2,7)
                          (3,2)  (4,2)    (3,4)    (3,5)    (3,6)
                          (4,1)  (5,1)    (4,3)    (5,3)    (4,5)
                                 (1,2,3)  (5,2)    (6,2)    (5,4)
                                 (3,2,1)  (6,1)    (7,1)    (6,3)
                                          (1,2,4)  (1,2,5)  (7,2)
                                          (4,2,1)  (1,3,4)  (8,1)
                                                   (4,3,1)  (1,2,6)
                                                   (5,2,1)  (1,3,5)
                                                            (2,3,4)
                                                            (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)

Eric Weisstein's World of Mathematics, Unimodal Sequence

Strict partitions are A000009.
Unimodal compositions are A001523 (strict: A072706).
Strict compositions are A032020.
The non-strict version appears to be A329398.
Partitions with incr. or decr. run-lengths are A332745 (strict: A333190).
Compositions with incr. or decr. run-lengths are A332835 (strict: A333191).
The complement is counted by A333149 (non-strict: A332834).
Cf. A059204, A072705, A072707, A115981, A332285, A332578, A332746, A332831, A332833, A332874, A333150.

Mathematica
```
Table[2*PartitionsQ[n]-1,{n,0,30}]
```

a(n) = 2*A000009(n) - 1.

A333150 Number of strict compositions of n whose non-adjacent parts are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 13, 18, 26, 31, 42, 52, 68, 89, 110, 136, 173, 212, 262, 330, 398, 487, 592, 720, 864, 1050, 1262, 1508, 1804, 2152, 2550, 3037, 3584, 4236, 5011, 5880, 6901, 8095, 9472, 11048, 12899, 14996, 17436, 20261, 23460, 27128, 31385, 36189

Offset: 0

Gus Wiseman, May 16 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. It is strict if there are no repeated parts.

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)
                          (3,2)  (4,2)    (3,4)    (3,5)
                          (4,1)  (5,1)    (4,3)    (5,3)
                                 (2,3,1)  (5,2)    (6,2)
                                 (3,1,2)  (6,1)    (7,1)
                                 (3,2,1)  (2,4,1)  (2,5,1)
                                          (4,1,2)  (3,4,1)
                                          (4,2,1)  (4,1,3)
                                                   (4,3,1)
                                                   (5,1,2)
                                                   (5,2,1)
For example, (3,5,1,2) is such a composition, because the non-adjacent pairs of parts are (3,1), (3,2), (5,2), all of which are strictly decreasing.

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

The case of permutations appears to be A000045(n + 1).
Unimodal strict compositions are A072706.
A version for ordered set partitions is A332872.
The non-strict version is A333148.
Cf. A001523, A028859, A056242, A059204, A107429, A115981, A329398, A332578, A332669, A332673, A332724, A332834, A333193.

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

seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=0, n, fibonacci(k+1) * polcoef(p,k,y)))} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=0} Fibonacci(k+1) * [y^k](Product_{j>=1} 1 + y*x^j). - Andrew Howroyd, Apr 16 2021

A333191 Number of compositions of n whose run-lengths are either strictly increasing or strictly decreasing.

Original entry on oeis.org

1, 1, 2, 2, 5, 8, 10, 18, 24, 29, 44, 60, 68, 100, 130, 148, 201, 256, 310, 396, 478, 582, 736, 898, 1068, 1301, 1594, 1902, 2288, 2750, 3262, 3910, 4638, 5510, 6538, 7686, 9069, 10670, 12560, 14728, 17170, 20090, 23462, 27292, 31710, 36878, 42704, 49430

Offset: 0

Gus Wiseman, May 17 2020

nonn

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

			The a(1) = 1 through a(7) = 18 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)
                                               (211111)
                                               (1111111)

Giovanni Resta, Table of n, a(n) for n = 0..1000

The non-strict version is A332835.
The case of partitions is A333190.
Unimodal compositions are A001523.
Strict compositions are A032020.
Partitions with distinct run-lengths are A098859.
Partitions with strictly increasing run-lengths are A100471.
Partitions with strictly decreasing run-lengths are A100881.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Compositions whose run-lengths are unimodal or co-unimodal are A332746.
Compositions whose run-lengths are neither incr. nor decr. are A332833.
Compositions that are neither increasing nor decreasing are A332834.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are strictly incr. or strictly decr. are A333147.
Compositions with strictly increasing run-lengths are A333192.
Cf. A059204, A072706, A329398, A329744, A329766, A332641, A332726, A332831, A332745, A333149.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Less@@Length/@Split[#],Greater@@Length/@Split[#]]&]],{n,0,15}]

a(n > 0) = 2*A333192(n) - A000005(n).

Terms a(26) and beyond from Giovanni Resta, May 19 2020

A072705 Triangle of number of unimodal compositions of n into exactly k distinct terms.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 4, 0, 0, 0, 1, 6, 4, 0, 0, 0, 0, 1, 6, 8, 0, 0, 0, 0, 0, 1, 8, 12, 0, 0, 0, 0, 0, 0, 1, 8, 16, 8, 0, 0, 0, 0, 0, 0, 1, 10, 20, 8, 0, 0, 0, 0, 0, 0, 0, 1, 10, 28, 16, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 32, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 40, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Jul 04 2002

Also the number of compositions of n into exactly k distinct terms whose negation is unimodal. - Gus Wiseman, Mar 06 2020

			Rows start: 1; 1,0; 1,2,0; 1,2,0,0; 1,4,0,0,0; 1,4,4,0,0,0; 1,6,4,0,0,0,0; 1,6,8,0,0,0,0,0; etc. T(6,3)=4 since 6 can be written as 1+2+3, 1+3+2, 2+3+1, or 3+2+1 but not 2+1+3 or 3+1+2.
From _Gus Wiseman_, Mar 06 2020: (Start)
Triangle begins:
  1
  1  0
  1  2  0
  1  2  0  0
  1  4  0  0  0
  1  4  4  0  0  0
  1  6  4  0  0  0  0
  1  6  8  0  0  0  0  0
  1  8 12  0  0  0  0  0  0
  1  8 16  8  0  0  0  0  0  0
  1 10 20  8  0  0  0  0  0  0  0
  1 10 28 16  0  0  0  0  0  0  0  0
  1 12 32 24  0  0  0  0  0  0  0  0  0
  1 12 40 40  0  0  0  0  0  0  0  0  0  0
  1 14 48 48 16  0  0  0  0  0  0  0  0  0  0
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A060016, A072574, A072704. Row sums are A072706.
Column k = 2 is A052928.
Unimodal compositions are A001523.
Unimodal sequences covering an initial interval are A007052.
Strict compositions are A032020.
Non-unimodal strict compositions are A072707.
Unimodal compositions covering an initial interval are A227038.
Numbers whose prime signature is not unimodal are A332282.
Cf. A059204, A107429, A115981, A329398, A332285, A332286, A332578, A332870, A332874.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i)*ceil(2^(i-1)), i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[n > i*(i+1)/2, 0, If[n == 0, 1, Expand[b[n, i-1] + If[i > n, 0, x*b[n-i, i-1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i]* Ceiling[2^(i-1)], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@#&&unimodQ[#]&]],{n,12},{k,n}] (* Gus Wiseman, Mar 06 2020 *)

T(n,k) = 2^(k-1)*A060016(n,k) = T(n-k,k)+2*T(n-k,k-1) [starting with T(0,0)=0, T(0,1)=0 and T(n,1)=1 for n>0].

A333149 Number of strict compositions of n that are neither increasing nor decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 4, 8, 12, 38, 42, 72, 98, 150, 298, 372, 542, 760, 1070, 1428, 2600, 3120, 4550, 6050, 8478, 10976, 15220, 23872, 29950, 41276, 55062, 74096, 97148, 129786, 167256, 256070, 314454, 429338, 556364, 749266, 955746, 1275016, 1618054

Offset: 0

Gus Wiseman, May 16 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. It is strict if there are no repeated parts.

			The a(6) = 4 through a(9) = 12 compositions:
  (1,3,2)  (1,4,2)  (1,4,3)  (1,5,3)
  (2,1,3)  (2,1,4)  (1,5,2)  (1,6,2)
  (2,3,1)  (2,4,1)  (2,1,5)  (2,1,6)
  (3,1,2)  (4,1,2)  (2,5,1)  (2,4,3)
                    (3,1,4)  (2,6,1)
                    (3,4,1)  (3,1,5)
                    (4,1,3)  (3,2,4)
                    (5,1,2)  (3,4,2)
                             (3,5,1)
                             (4,2,3)
                             (5,1,3)
                             (6,1,2)

Eric Weisstein's World of Mathematics, Unimodal Sequence

The non-strict case is A332834.
The complement is counted by A333147.
Strict partitions are A000009.
Strict compositions are A032020.
Non-unimodal strict compositions are A072707.
Strict partitions with increasing or decreasing run-lengths are A333190.
Strict compositions with increasing or decreasing run-lengths are A333191.
Unimodal compositions are A001523, with strict case A072706.
Cf. A059204, A115981, A227038, A329398, A332745, A332746, A332831, A332833, A332835, A332874, A333150, A333192.

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

a(n) = A032020(n) - 2*A000009(n) + 1.

A333190 Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 26, 29, 39, 49, 50, 68, 80, 92, 109, 129, 142, 181, 201, 227, 262, 317, 343, 404, 456, 516, 589, 677, 742, 870, 949, 1077, 1207, 1385, 1510, 1704, 1895, 2123, 2352, 2649, 2877, 3261, 3571, 3966, 4363, 4873, 5300, 5914, 6466

Offset: 0

Gus Wiseman, May 17 2020

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The non-strict version is A332745.
The generalization to compositions is A333191.
Partitions with distinct run-lengths are A098859.
Partitions with strictly increasing run-lengths are A100471.
Partitions with strictly decreasing run-lengths are A100881.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Partitions with unimodal run-lengths are A332280.
Partitions whose run-lengths are not increasing nor decreasing are A332641.
Compositions whose run-lengths are unimodal or co-unimodal are A332746.
Compositions that are neither increasing nor decreasing are A332834.
Strictly increasing or strictly decreasing compositions are A333147.
Compositions with strictly increasing run-lengths are A333192.
Numbers with strictly increasing prime multiplicities are A334965.
Cf. A032020, A059204, A072706, A332726, A332831, A332833, A332835, A333149.

Table[Length[Select[IntegerPartitions[n],Or[Less@@Length/@Split[#],Greater@@Length/@Split[#]]&]],{n,0,30}]

cp's OEIS Frontend

A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

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A218004 Number of equivalence classes of compositions of n where two compositions a,b are considered equivalent if the summands of a can be permuted into the summands of b with an even number of transpositions.

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A072707 Number of non-unimodal compositions of n into distinct terms.

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A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

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A333147 Number of compositions of n that are either strictly increasing or strictly decreasing.

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A333150 Number of strict compositions of n whose non-adjacent parts are strictly decreasing.

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A333191 Number of compositions of n whose run-lengths are either strictly increasing or strictly decreasing.

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A072705 Triangle of number of unimodal compositions of n into exactly k distinct terms.