A332282 -id:A332282 - cp's OEIS frontend

A332283 Number of integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

1, 1, 2, 3, 5, 7, 10, 13, 18, 24, 30, 38, 49, 59, 73, 90, 108, 129, 159, 184, 216, 258, 298, 347, 410, 466, 538, 626, 707, 807, 931, 1043, 1181, 1351, 1506, 1691, 1924, 2132, 2382, 2688, 2971, 3300, 3704, 4073, 4500, 5021, 5510, 6065, 6740, 7362, 8078

Offset: 0

Gus Wiseman, Feb 19 2020

nonn

First differs from A000041 at a(6) = 10, A000041(6) = 11.

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

			The a(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (321)     (421)
                            (11111)  (411)     (511)
                                     (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (31111)
                                               (211111)
                                               (1111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Partitions with unimodal run-lengths are A332280.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
The complement is counted by A332284.
The strict case is A332285.
Heinz numbers of partitions not in this class are A332287.
Cf. A025065, A072706, A115981, A227038, A332288, A332577, A332638, A332642.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],unimodQ[Differences[Append[#,0]]]&]],{n,0,30}]

A332284 Number of integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 18, 28, 42, 62, 86, 123, 168, 226, 306, 411, 534, 704, 908, 1165, 1492, 1898, 2384, 3011, 3758, 4673, 5799, 7168, 8792, 10804, 13192, 16053, 19505, 23633, 28497, 34367, 41283, 49470, 59188, 70675, 84113, 100048, 118689, 140533

Offset: 0

Gus Wiseman, Feb 20 2020

nonn

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

			The a(6) = 1 through a(11) = 18 partitions:
  (2211)  (331)    (431)     (441)      (541)       (551)
          (22111)  (3311)    (4311)     (3322)      (641)
                   (22211)   (32211)    (3331)      (4331)
                   (221111)  (33111)    (4411)      (4421)
                             (222111)   (33211)     (5411)
                             (2211111)  (42211)     (33221)
                                        (43111)     (33311)
                                        (222211)    (44111)
                                        (322111)    (52211)
                                        (331111)    (322211)
                                        (2221111)   (332111)
                                        (22111111)  (422111)
                                                    (431111)
                                                    (2222111)
                                                    (3221111)
                                                    (3311111)
                                                    (22211111)
                                                    (221111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Unimodal Sequence.

The complement is counted by A332283.
The strict version is A332286.
The Heinz numbers of these partitions are A332287.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences appear to be A328509.
Partitions with non-unimodal run-lengths are A332281.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Cf. A001523, A007052, A332280, A332285, A332288, A332579, A332638, A332639, A332640, A332642.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],!unimodQ[Differences[Append[#,0]]]&]],{n,30}]

A332638 Number of integer partitions of n whose negated run-lengths are unimodal.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 52, 70, 91, 118, 151, 195, 246, 310, 388, 484, 600, 743, 909, 1113, 1359, 1650, 1996, 2409, 2895, 3471, 4156, 4947, 5885, 6985, 8260, 9751, 11503, 13511, 15857, 18559, 21705, 25304, 29499, 34259, 39785, 46101, 53360, 61594

Offset: 0

Gus Wiseman, Feb 25 2020

nonn

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

			The a(8) = 21 partitions:
  (8)     (44)     (2222)
  (53)    (332)    (22211)
  (62)    (422)    (32111)
  (71)    (431)    (221111)
  (521)   (3311)   (311111)
  (611)   (4211)   (2111111)
  (5111)  (41111)  (11111111)
Missing from this list is only (3221).

MathWorld, Unimodal Sequence

The non-negated version is A332280.
The complement is counted by A332639.
The Heinz numbers of partitions not in this class are A332642.
The case of 0-appended differences (instead of run-lengths) is A332728.
Unimodal compositions are A001523.
Partitions whose run lengths are not unimodal are A332281.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Cf. A007052, A100883, A115981, A181819, A332283, A332577, A332640, A332669, A332670, A332727.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],unimodQ[-Length/@Split[#]]&]],{n,0,30}]

A072704 Triangle of number of weakly unimodal partitions/compositions of n into exactly k terms.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 8, 7, 5, 1, 1, 6, 12, 12, 9, 6, 1, 1, 7, 16, 20, 16, 11, 7, 1, 1, 8, 21, 30, 28, 20, 13, 8, 1, 1, 9, 27, 42, 45, 36, 24, 15, 9, 1, 1, 10, 33, 58, 68, 60, 44, 28, 17, 10, 1, 1, 11, 40, 77, 98, 95, 75, 52, 32, 19, 11, 1

Offset: 1

Henry Bottomley, Jul 04 2002

			Rows start:
01:  [1]
02:  [1, 1]
03:  [1, 2, 1]
04:  [1, 3, 3, 1]
05:  [1, 4, 5, 4, 1]
06:  [1, 5, 8, 7, 5, 1]
07:  [1, 6, 12, 12, 9, 6, 1]
08:  [1, 7, 16, 20, 16, 11, 7, 1]
09:  [1, 8, 21, 30, 28, 20, 13, 8, 1]
10:  [1, 9, 27, 42, 45, 36, 24, 15, 9, 1]
...
T(6,3)=8 since 6 can be written as 1+1+4, 1+2+3, 1+3+2, 1+4+1, 2+2+2, 2+3+1, 3+2+1, or 4+1+1 but not 2+1+3 or 3+1+2.

Alois P. Heinz, Rows n = 1..141, flattened
Henry Bottomley, Illustration of initial terms
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics (2019) Vol. 342, Issue 11, 3079-3097. See page 3094, Table 4.
Eric Weisstein's World of Mathematics, Unimodal Sequence

Cf. A059623, A072705. Row sums are A001523. First column is A057427, second is A000027 offset, third appears to be A000212 offset, right hand columns include A000012, A000027, A005408 and A008574.
The case of partitions is A072233.
Dominates A332670 (the version for negated compositions).
The strict case is A072705.
The case of constant compositions is A113704.
Unimodal sequences covering an initial interval are A007052.
Partitions whose run-lengths are unimodal are A332280.
Cf. A107429, A115981, A227038, A328509, A332282, A332283, A332578, A332638, A332669, A332728.

b:= proc(n, i) option remember; local q; `if`(i>n, 0,
      `if`(irem(n, i, 'q')=0, x^q, 0) +expand(
      add(b(n-i*j, i+1)*(j+1)*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i ] == 0, x^Quotient[n, i], 0] + Expand[ Sum[b[n-i*j, i+1]*(j+1)*x^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]]; Table[T[n], {n, 1, 12}] // 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}],unimodQ]],{n,0,10},{k,0,n}] (* Gus Wiseman, Mar 06 2020 *)

\\ starting for n=0, with initial column 1, 0, 0, ...:
N=25;  x='x+O('x^N);
T=Vec(1 + sum(n=1, N, t*x^n / ( prod(k=1,n-1, (1 - t*x^k)^2 ) * (1 - t*x^n) ) ) )
for(r=1,#T, print(Vecrev(T[r])) ); \\ Joerg Arndt, Oct 01 2017

G.f. with initial column 1, 0, 0, ...: 1 + Sum_{n>=1} (t*x^n / ( ( Product_{k=1..n-1} (1 - t*x^k)^2 ) * (1 - t*x^n) ) ). - Joerg Arndt, Oct 01 2017

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

A332287 Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

36, 50, 70, 72, 98, 100, 108, 140, 144, 154, 180, 182, 196, 200, 216, 225, 242, 250, 252, 280, 286, 288, 294, 300, 308, 324, 338, 350, 360, 363, 364, 374, 392, 396, 400, 418, 429, 432, 441, 442, 450, 462, 468, 484, 490, 494, 500, 504, 507, 540, 550, 560, 561

Offset: 1

Gus Wiseman, Feb 21 2020

nonn

A sequence of 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), which gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   36: {1,1,2,2}
   50: {1,3,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   98: {1,4,4}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}
  154: {1,4,5}
  180: {1,1,2,2,3}
  182: {1,4,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  216: {1,1,1,2,2,2}
  225: {2,2,3,3}
  242: {1,5,5}
  250: {1,3,3,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
For example, the prime indices of 70 with 0 appended are (4,3,1,0), with differences (-1,-2,-1), which is not unimodal, so 70 belongs to the sequence.

MathWorld, Unimodal Sequence

The enumeration of these partitions by sum is A332284.
Not assuming the last part is zero gives A332725.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Cf. A001523, A007052, A332280, A332282, 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[Append[Reverse[primeMS[#]],0]]]&]

A332288 Number of unimodal 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, 3, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 2, 2, 1, 6, 1, 2, 3, 1, 2, 4, 1, 3, 2, 4, 1, 4, 1, 2, 2, 3, 2, 4, 1, 5, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 22 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 (2,1,2), (2,1,3), and (3,1,2).

			The a(n) permutations for n = 2, 6, 12, 24, 48, 60, 120, 180:
  (1)  (12)  (112)  (1112)  (11112)  (1123)  (11123)  (11223)
       (21)  (121)  (1121)  (11121)  (1132)  (11132)  (11232)
             (211)  (1211)  (11211)  (1231)  (11231)  (11322)
                    (2111)  (12111)  (1321)  (11321)  (12231)
                            (21111)  (2311)  (12311)  (12321)
                                     (3211)  (13211)  (13221)
                                             (23111)  (22311)
                                             (32111)  (23211)
                                                      (32211)

Wikipedia, Permutation pattern
MathWorld, Unimodal Sequence

Dominated by A008480.
A more interesting version is A332294.
The complement is counted by A332671.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Numbers with non-unimodal unsorted prime signature are A332282.
Partitions with unimodal 0-appended first differences are A332283.
Cf. A056239, A112798, A115981, A124010, A227038, A304660, A328509, A332280, A332284, A332294, A332578, A332672.

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}]

A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 5, 5, 7, 9, 12, 15, 22, 23, 31, 40, 47, 58, 72, 81, 100, 122, 144, 171, 206, 236, 280, 333, 381, 445, 522, 593, 694, 802, 914, 1054, 1214, 1376, 1577, 1803, 2040, 2324, 2646, 2973, 3373, 3817, 4287, 4838, 5453, 6096, 6857

Offset: 0

Gus Wiseman, Feb 21 2020

nonn

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

Also the number integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.

			The a(8) = 1 through a(18) = 7 partitions:
  (431)  .  (541)  (641)  (651)   (652)   (752)   (762)   (862)
                          (5421)  (751)   (761)   (861)   (871)
                                  (5431)  (851)   (6531)  (961)
                                          (6431)  (7431)  (6532)
                                          (6521)  (7521)  (6541)
                                                          (7621)
                                                          (8431)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Strict partitions are A000009.
Partitions covering an initial interval are (also) A000009.
The non-strict version is A332284.
The complement is counted by A332285.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Normal partitions whose run-lengths are not unimodal are A332579.
Cf. A007052, A011782, A025065, A072706, A227038, A332282, A332283, A332286, A332287, A332288, A332577, A332638, A332642, A332743.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,!unimodQ[Differences[Append[#,0]]]]&]],{n,0,30}]

A332640 Number of integer partitions of n such that neither the run-lengths nor the negated run-lengths are unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 17, 29, 44, 66, 92, 138, 187, 266, 359, 492, 649, 877, 1140, 1503, 1938, 2517, 3202, 4111, 5175, 6563, 8209, 10297, 12763, 15898, 19568, 24152, 29575, 36249, 44090, 53737, 65022, 78752, 94873, 114294

Offset: 0

Gus Wiseman, Feb 25 2020

nonn

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

			The a(14) = 1 through a(18) = 12 partitions:
  (433211)  (533211)   (443221)    (544211)     (544311)
            (4332111)  (633211)    (733211)     (553221)
                       (5332111)   (4333211)    (644211)
                       (43321111)  (6332111)    (833211)
                                   (53321111)   (4432221)
                                   (433211111)  (5333211)
                                                (5442111)
                                                (7332111)
                                                (43332111)
                                                (63321111)
                                                (533211111)
                                                (4332111111)
For example, the partition (4,3,3,2,1,1) has run-lengths (1,2,1,2), so is counted under a(14).

MathWorld, Unimodal Sequence

Looking only at the original run-lengths gives A332281.
Looking only at the negated run-lengths gives A332639.
The Heinz numbers of these partitions are A332643.
The complement is counted by A332746.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Run-lengths and negated run-lengths are not both unimodal: A332641.
Compositions whose negation is not unimodal are A332669.
Run-lengths and negated run-lengths are both unimodal: A332745.
Cf. A007052, A025065, A100883, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332642, A332726, A332727.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],!unimodQ[Length/@Split[#]]&&!unimodQ[-Length/@Split[#]]&]],{n,0,30}]

A332670 Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 5, 2, 1, 0, 1, 5, 7, 5, 2, 1, 0, 1, 6, 11, 10, 5, 2, 1, 0, 1, 7, 15, 16, 10, 5, 2, 1, 0, 1, 8, 20, 24, 20, 10, 5, 2, 1, 0, 1, 9, 25, 36, 31, 20, 10, 5, 2, 1, 0, 1, 10, 32, 50, 50, 36, 20, 10, 5, 2, 1

Offset: 0

Gus Wiseman, Feb 29 2020

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

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

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  2  1
  0  1  3  2  1
  0  1  4  5  2  1
  0  1  5  7  5  2  1
  0  1  6 11 10  5  2  1
  0  1  7 15 16 10  5  2  1
  0  1  8 20 24 20 10  5  2  1
  0  1  9 25 36 31 20 10  5  2  1
  0  1 10 32 50 50 36 20 10  5  2  1
  0  1 11 38 67 73 59 36 20 10  5  2  1
Column n = 7 counts the following compositions:
  (7)  (16)  (115)  (1114)  (11113)  (111112)  (1111111)
       (25)  (124)  (1123)  (11122)  (211111)
       (34)  (133)  (1222)  (21112)
       (43)  (214)  (2113)  (22111)
       (52)  (223)  (2122)  (31111)
       (61)  (313)  (2212)
             (322)  (2221)
             (331)  (3112)
             (412)  (3211)
             (421)  (4111)
             (511)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Eric Weisstein's World of Mathematics, Unimodal Sequence.

The case of partitions is A072233.
Dominated by A072704 (the non-negated version).
The strict case is A072705.
The case of constant compositions is A113704.
Row sums are A332578.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose negated unsorted prime signature is not unimodal are A332282.
Partitions whose negated run-lengths are unimodal are A332638.
Compositions whose negation is not unimodal are A332669.
Partitions whose negated 0-appended first differences are unimodal: A332728.
Cf. A011782, A107429, A227038, A332280, A332283, A332639, A332642, A332741, A332742, A332744, A332832, 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,{k}],unimodQ[-#]&]],{n,0,10},{k,0,n}]

T(n)={[Vecrev(p) | p<-Vec(1 + sum(j=1, n, y*x^j/((1-y*x^j) * prod(k=j+1, n-j, 1 - y*x^k + O(x*x^(n-j)))^2)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

G.f.: A(x,y) = 1 + Sum_{j>0} y*x^j/((1 - y*x^j)*Product_{k>j} (1 - y*x^k)^2). - Andrew Howroyd, Jan 11 2024

cp's OEIS Frontend

A332283 Number of integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

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A332284 Number of integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

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A332638 Number of integer partitions of n whose negated run-lengths are unimodal.

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A072704 Triangle of number of weakly unimodal partitions/compositions of n into exactly k terms.

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

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A332287 Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.

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A332288 Number of unimodal permutations of the multiset of prime indices of n.

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A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

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