A001523 -id:A001523 - cp's OEIS frontend

A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 27, 8, 5, 3, 2, 1, 1, 47, 13, 7, 5, 3, 2, 1, 1, 79, 21, 11, 7, 5, 3, 2, 1, 1, 130, 33, 16, 11, 7, 5, 3, 2, 1, 1, 209, 52, 24, 15, 11, 7, 5, 3, 2, 1, 1, 330, 80, 35, 22, 15, 11, 7, 5, 3, 2, 1, 1, 512, 122, 52, 31, 22, 15, 11, 7, 5, 3

Offset: 1

Henry Bottomley, Feb 01 2001

Weakly unimodal means nondecreasing then nonincreasing.

			Rows are {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc.
As an upper right triangle:
  1,  1,  2,  4,  8, 15, ...,
      1,  1,  2,  3,  5,  8, ...,
          1,  1,  2,  3,  5,  7, ...,
              ...
As a left downward triangle, it starts:
   1;
   1, 1;
   2, 1, 1;
   4, 2, 1, 1;
   8, 3, 2, 1, 1;
  15, 5, 3, 2, 1, 1;
  27, 8, 5, 3, 2, 1, 1;
  ...
T(9,3)=11 since 9 can be written as 3+6, 3+5+1, 3+4+2, 3+4+1+1, 3+3+3, 3+3+2+1, 3+3+1+1+1, 3+2+2+2, 3+2+2+1+1, 3+2+1+1+1+1 or 3+1+1+1+1+1.

Column sums give A001523. Cf. A008284, A026836, A008284, A059607, A059619.

T(n, k) = S(n, k) - S(n-k, k) + Sum_j[T(n-k, j)] for j >= k, where S(n, k) = A008284(n, k) = Sum_j[S(n-k, j)] for n>k >= j [note reversal] with S[n, n] = 1.

A226541 Number of unimodal compositions of n where the maximal part appears three times.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 3, 5, 7, 11, 16, 24, 34, 51, 71, 102, 143, 201, 276, 384, 522, 714, 964, 1301, 1739, 2328, 3084, 4085, 5377, 7064, 9226, 12036, 15616, 20228, 26092, 33584, 43067, 55125, 70308, 89502, 113598, 143889, 181755, 229160, 288186, 361750, 453046, 566346, 706464

Offset: 0

Joerg Arndt, Jun 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A006330 (max part appears once), A114921 (max part appears twice).
Cf. A188674 (max part m appears m times), A001522 (max part m appears at least m times).
Cf. A001523 (max part appears any number of times).
Cf. A000009 (symmetric, max part m appears once; also symmetric, max part appears an odd number of times).
Cf. A035363 (symmetric, max part m appears twice; also symmetric, max part appears an even number of times).
Cf. A087897 (symmetric, max part m appears 3 times).
Cf. A027349 (symmetric, max part m appears m times), A189357 (symmetric, max part m appears at least m times).
Column k=3 of A247255.

N=66; x='x+O('x^N); Vec(sum(n=0,N, x^(3*n) / prod(k=1,n-1, 1-x^k )^2 ))

G.f.: sum(n>=0, x^(3*n) / prod(k=1..n-1, 1-x^k )^2 ); replace 3 by m to obtain g.f. for "... max part appears m times".

a(n) ~ Pi^2 * exp(2*Pi*sqrt(n/3)) / (16 * 3^(7/4) * n^(9/4)). - Vaclav Kotesovec, Oct 24 2018

A332725 Heinz numbers of integer partitions whose negated first differences are not unimodal.

Original entry on oeis.org

90, 126, 180, 198, 234, 252, 270, 306, 342, 350, 360, 378, 396, 414, 450, 468, 504, 522, 525, 540, 550, 558, 594, 612, 630, 650, 666, 684, 700, 702, 720, 738, 756, 774, 792, 810, 825, 828, 846, 850, 882, 900, 910, 918, 936, 950, 954, 975, 990, 1008, 1026, 1044

Offset: 1

Gus Wiseman, Feb 26 2020

nonn

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

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
    90: {1,2,2,3}
   126: {1,2,2,4}
   180: {1,1,2,2,3}
   198: {1,2,2,5}
   234: {1,2,2,6}
   252: {1,1,2,2,4}
   270: {1,2,2,2,3}
   306: {1,2,2,7}
   342: {1,2,2,8}
   350: {1,3,3,4}
   360: {1,1,1,2,2,3}
   378: {1,2,2,2,4}
   396: {1,1,2,2,5}
   414: {1,2,2,9}
   450: {1,2,2,3,3}
   468: {1,1,2,2,6}
   504: {1,1,1,2,2,4}
   522: {1,2,2,10}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
For example, 350 is the Heinz number of (4,3,3,1), with negated first differences (1,0,2), which is not unimodal, so 350 is in the sequence.

MathWorld, Unimodal Sequence
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The complement is too full.
The enumeration of these partitions by sum is A332284.
The version where the last part is taken to be 0 is A332832.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Heinz numbers of partitions with weakly increasing differences are A325360.
Cf. A001523, A007052, A240026, A332280, A332283, A332285, A332286, A332288, A332294, A332579, A332639, A332642.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Select[Range[1000],!unimodQ[Differences[primeMS[#]]]&]

A333145 Number of unimodal negated permutations of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 09 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also permutations of the multiset of prime indices of n avoiding the patterns (1,2,1), (1,3,2), and (2,3,1).
Also the number divisors of n not divisible by the least prime factor of n. The other divisors are counted by A069157. - Gus Wiseman, Apr 12 2022

			The a(n) permutations for n = 2, 6, 18, 30, 90, 162, 210, 450:
  (1)  (12)  (122)  (123)  (1223)  (12222)  (1234)  (12233)
       (21)  (212)  (213)  (2123)  (21222)  (2134)  (21233)
             (221)  (312)  (2213)  (22122)  (3124)  (22133)
                    (321)  (3122)  (22212)  (3214)  (31223)
                           (3212)  (22221)  (4123)  (32123)
                           (3221)           (4213)  (32213)
                                            (4312)  (33122)
                                            (4321)  (33212)
                                                    (33221)

Eric Weisstein's World of Mathematics, Unimodal Sequence
Wikipedia, Permutation pattern

Dominated by A008480.
The complementary divisors are counted by A069157.
The non-negated version is A332288.
A more interesting version is A332741.
The complement is counted by A333146.
A001523 counts unimodal compositions.
A007052 counts unimodal normal sequences.
A028233 gives the highest power of the least prime factor, quotient A028234.
A332578 counts compositions whose negation is unimodal.
A332638 counts partitions with unimodal negated run-lengths.
A332642 lists numbers with non-unimodal negated unsorted prime signature.
Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332294, A332639, A332669, A332670, A332671.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[primeMS[n]],unimodQ[-#]&]],{n,30}]

a(n) + A333146(n) = A008480(n).
a(n) = A000005(A028234(n)). - Gus Wiseman, Apr 14 2022
a(n) = A000005(n) - A069157(n). - Gus Wiseman, Apr 14 2022

A059619 As upper right triangle, number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing) where initial part is k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 3, 1, 1, 1, 4, 2, 0, 1, 1, 6, 2, 1, 1, 1, 1, 10, 4, 2, 1, 1, 1, 1, 15, 6, 3, 1, 2, 1, 1, 1, 21, 9, 4, 2, 1, 2, 1, 1, 1, 30, 12, 6, 3, 2, 2, 2, 1, 1, 1, 43, 18, 8, 5, 3, 2, 2, 2, 1, 1, 1, 59, 25, 12, 6, 3, 3, 3, 2, 2, 1, 1, 1, 82, 34, 17, 9, 5, 4, 3, 3, 2, 2, 1, 1, 1, 111, 48, 22, 12

Offset: 0

Henry Bottomley, Jan 31 2001

			Rows start:
1,  1,  1,  3,  4,  6, 10, 15, 21, 30,  43,  59,  82, 111, ...
1,  0,  1,  2,  2,  4,  6,  9, 12, 18,  25,  34,  48, ...
1,  1,  0,  1,  2,  3,  4,  6,  8, 12,  17,  22, ...
1,  1,  1,  1,  1,  2,  3,  5,  6,  9,  12, ...
1,  1,  1,  2,  1,  2,  3,  3,  5, ...
1,  1,  1,  2,  2,  2,  3,  4, ...
1,  1,  1,  2,  2,  3,  3, ...
1,  1,  1,  2,  2,  3, ...
1,  1,  1,  2,  2, ...
1,  1,  1,  2, ...
1,  1,  1, ...
1,  1, ...
1, ...  etc.
T(16,6)=8 since 16 can be written as 6+10, 6+9+1, 6+8+2, 6+7+3, 6+7+2+1, 6+5+4+1, 6+5+3+2, or 6+4+3+2+1 (but for example neither 6+6+4 nor 6+8+1+1 which are only weakly unimodal).

R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms

Top row is A059618 and is sum of other rows (for n>0). Cf. A000009, A000041, A001523, A059607.

s[n_?Positive, k_] := s[n, k] = Sum[s[n-k, j], {j, 0, k-1}]; s[0, 0] = 1; s[0, ] = 0; s[?Negative, ] = 0; t[n, k_] := t[n, k] = s[n, k] + Sum[t[n-k, j], {j, k+1, n}]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 11 2012 *)

T(n, k)=S(n, k)+sum_j[T(n-k, j)] for j>k, where S(n, k)=A059607(n, k)=sum_j[S(n-k, j)] for k>j [note reversal] with S(0, 0)=1.

A229706 Triangular array read by rows: T(n,k) is the number of unimodal compositions of n with greatest part equal to k; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 5, 2, 1, 1, 9, 9, 5, 2, 1, 1, 12, 16, 10, 5, 2, 1, 1, 16, 25, 19, 10, 5, 2, 1, 1, 20, 39, 32, 20, 10, 5, 2, 1, 1, 25, 56, 54, 35, 20, 10, 5, 2, 1, 1, 30, 80, 84, 61, 36, 20, 10, 5, 2, 1, 1, 36, 109, 129, 99, 64, 36, 20, 10, 5, 2, 1

Offset: 1

Geoffrey Critzer, Sep 27 2013

A unimodal composition is a composition such that for some j, m, 1 <= x(1) <= x(2) <= ... <= x(j) >= x(j+1) >= ... >= x(m) >= 1.
Row sums are A001523.
T(2*n+1,n+1) = A000712(n) for n>=0. - Alois P. Heinz, Oct 03 2013

			1;
1,  1;
1,  2,  1;
1,  4,  2,  1;
1,  6,  5,  2,  1;
1,  9,  9,  5,  2,  1;
1, 12, 16, 10,  5,  2,  1;
1, 16, 25, 19, 10,  5,  2, 1;
1, 20, 39, 32, 20, 10,  5, 2, 1;
1, 25, 56, 54, 35, 20, 10, 5, 2, 1;
T(5,3) = 5 because we have: 3+2 = 2+3 = 3+1+1 = 1+3+1 = 1+1+3.

E. M. Wright, Stacks, Quart. J. Math. Oxford 19 (1968) 313-320, table s(r).

Alois P. Heinz, rows n = 1..141, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46

Cf. A229707.

b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(k>0, `if`(n b(n, 1, k):
seq(seq(T(n, k), k=1..n), n=1..16);  # Alois P. Heinz, Oct 03 2013

Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[Series[x^n/(1-x^n)/Product[1-x^i,{i,1,n-1}]^2,{x,0,nn}],x],{n,1,nn}]],1]]//Grid

O.g.f. for column k: x^k/prod(i=1..k-1, 1-x^i )^2.

A238871 Number of weakly unimodal compositions of n with absolute difference of successive parts <= 1.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 14, 21, 27, 40, 52, 70, 92, 124, 156, 206, 264, 335, 425, 539, 673, 847, 1052, 1300, 1611, 1990, 2433, 2977, 3638, 4420, 5367, 6496, 7829, 9439, 11341, 13590, 16270, 19425, 23135, 27525, 32697, 38745, 45844, 54168, 63875, 75247, 88493, 103892

Offset: 0

Joerg Arndt, Mar 21 2014

nonn

			The a(8) = 27 such compositions are:
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 2 1 1 1 1 ]
10:  [ 1 1 2 2 1 1 ]
11:  [ 1 1 2 2 2 ]
12:  [ 1 2 1 1 1 1 1 ]
13:  [ 1 2 2 1 1 1 ]
14:  [ 1 2 2 2 1 ]
15:  [ 1 2 2 3 ]
16:  [ 1 2 3 2 ]
17:  [ 2 1 1 1 1 1 1 ]
18:  [ 2 2 1 1 1 1 ]
19:  [ 2 2 2 1 1 ]
20:  [ 2 2 2 2 ]
21:  [ 2 3 2 1 ]
22:  [ 2 3 3 ]
23:  [ 3 2 1 1 1 ]
24:  [ 3 2 2 1 ]
25:  [ 3 3 2 ]
26:  [ 4 4 ]
27:  [ 8 ]

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

Cf. A001522, A001523, A005169, A034297, A238870, A238872.

A288578 q-Expansion of wedge character chi^(2)(q).

Original entry on oeis.org

1, 2, 5, 9, 18, 31, 55, 91, 151, 240, 381, 587, 900, 1352, 2018, 2966, 4332, 6250, 8962, 12725, 17962, 25147, 35015, 48414, 66603, 91071, 123945, 167786, 226154, 303375, 405337, 539249, 714740, 943659, 1241605, 1627812, 2127302, 2770966, 3598567

Offset: 0

N. J. A. Sloane, Jul 01 2017

nonn

Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.

Cf. A006330 (chi^(0)(q)), A001523 (chi^(1)(q)).

chi := proc(l,e)
    local gf,m,n,f;
    gf := 0 ;
    for m from 0 to e+1 do
        f := 1;
        for n from 1 to min(m+abs(l),e-m+1) do
            f := f/(1-q^n) ;
        end do:
        for n from 1 to min(m,e-m+1) do
            f := f/(1-q^n) ;
        end do:
        gf := gf+f*q^m ;
    end do:
    expand(gf) ;
    coeftayl(%,q=0,e) ;
end proc:
A288578 := proc(n)
    chi(2,n) ;
end proc:
for n from 0 do
    printf("%d,\n",A288578(n)) ;
end do: # R. J. Mathar, Jul 04 2017

A332871 Number of compositions of n whose run-lengths are not weakly increasing.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 8, 24, 55, 128, 282, 625, 1336, 2855, 6000, 12551, 26022, 53744, 110361, 225914, 460756, 937413, 1902370, 3853445, 7791647, 15732468, 31725191, 63907437, 128613224, 258626480, 519700800, 1043690354, 2094882574, 4202903667, 8428794336, 16897836060

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 not weakly decreasing.

			The a(4) = 1 through a(6) = 8 compositions:
  (112)  (113)   (114)
         (221)   (1113)
         (1112)  (1131)
         (1121)  (1221)
                 (2112)
                 (11112)
                 (11121)
                 (11211)
For example, the composition (2,1,1,2) has run-lengths (1,2,1), which are not weakly increasing, so (2,1,1,2) is counted under a(6).

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

The version for the compositions themselves (not run-lengths) is A056823.
The version for unsorted prime signature is A112769, with dual A071365.
The case without weakly decreasing run-lengths either is A332833.
The complement is counted by A332836.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are not unimodal are A332727.
Cf. A001523, A072704, A100883, A181819, A329744, A329766, A332641, A332669, A332726, A332745, A332746, A332834, A332835.

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

a(n) = 2^(n - 1) - A332836(n).

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

A335375 Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.

Original entry on oeis.org

45, 54, 77, 89, 91, 93, 102, 108, 109, 110, 118, 141, 153, 155, 157, 166, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189, 198, 204, 205, 206, 214, 216, 217, 218, 219, 220, 221, 222, 230, 236, 237, 238, 246, 269, 281, 283, 285, 297, 301, 305, 306, 307, 309

Offset: 1

Gus Wiseman, Jun 04 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. It is co-unimodal if its negation is unimodal.

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 sequence together with the corresponding compositions begins:
   45: (2,1,2,1)
   54: (1,2,1,2)
   77: (3,1,2,1)
   89: (2,1,3,1)
   91: (2,1,2,1,1)
   93: (2,1,1,2,1)
  102: (1,3,1,2)
  108: (1,2,1,3)
  109: (1,2,1,2,1)
  110: (1,2,1,1,2)
  118: (1,1,2,1,2)
  141: (4,1,2,1)
  153: (3,1,3,1)
  155: (3,1,2,1,1)
  157: (3,1,1,2,1)
  166: (2,3,1,2)
  173: (2,2,1,2,1)
  177: (2,1,4,1)
  178: (2,1,3,2)
  179: (2,1,3,1,1)

Non-unimodal compositions are ranked by A335373.
Non-co-unimodal compositions are ranked by A335374.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Co-unimodal compositions are A332578.
Numbers with non-co-unimodal unsorted prime signature are A332642.
Non-co-unimodal compositions are A332669.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!unimodQ[stc[#]]&&!unimodQ[-stc[#]]&]

cp's OEIS Frontend

A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).

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A226541 Number of unimodal compositions of n where the maximal part appears three times.

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A332725 Heinz numbers of integer partitions whose negated first differences are not unimodal.

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A333145 Number of unimodal negated permutations of the multiset of prime indices of n.

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A059619 As upper right triangle, number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing) where initial part is k.

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A229706 Triangular array read by rows: T(n,k) is the number of unimodal compositions of n with greatest part equal to k; n>=1, 1<=k<=n.

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A238871 Number of weakly unimodal compositions of n with absolute difference of successive parts <= 1.

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A288578 q-Expansion of wedge character chi^(2)(q).

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A332871 Number of compositions of n whose run-lengths are not weakly increasing.

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