A059204 -id:A059204 - cp's OEIS frontend

A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

2100, 3300, 3900, 4200, 4410, 5100, 5700, 6468, 6600, 6900, 7644, 7800, 8400, 8700, 9300, 9996, 10200, 10500, 10780, 10890, 11100, 11172, 11400, 12300, 12740, 12900, 12936, 13200, 13230, 13524, 13800, 14100, 15210, 15246, 15288, 15600, 15900, 16500, 16660

Offset: 1

Gus Wiseman, Feb 28 2020

nonn

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

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The sequence of terms together with their prime indices begins:
   2100: {1,1,2,3,3,4}
   3300: {1,1,2,3,3,5}
   3900: {1,1,2,3,3,6}
   4200: {1,1,1,2,3,3,4}
   4410: {1,2,2,3,4,4}
   5100: {1,1,2,3,3,7}
   5700: {1,1,2,3,3,8}
   6468: {1,1,2,4,4,5}
   6600: {1,1,1,2,3,3,5}
   6900: {1,1,2,3,3,9}
   7644: {1,1,2,4,4,6}
   7800: {1,1,1,2,3,3,6}
   8400: {1,1,1,1,2,3,3,4}
   8700: {1,1,2,3,3,10}
   9300: {1,1,2,3,3,11}
   9996: {1,1,2,4,4,7}
  10200: {1,1,1,2,3,3,7}
  10500: {1,1,2,3,3,3,4}
  10780: {1,1,3,4,4,5}
  10890: {1,2,2,3,5,5}

MathWorld, Unimodal Sequence

Not requiring non-unimodal negation gives A332282.
These are the Heinz numbers of the partitions counted by A332640.
Not requiring non-unimodality gives A332642.
The case of compositions is A332870.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unsorted prime signature is A124010.
Non-unimodal normal sequences are A328509.
Partitions whose 0-appended first differences are unimodal are A332283, with Heinz numbers the complement of A332287.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Partitions whose 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
Numbers whose signature is neither increasing nor decreasing are A332831.
Cf. A007052, A056239, A072704, A112798, A242031, A242414, A332280, A332281, A332288, A332294, A332639, A332728, A332742.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Select[Range[10000],!unimodQ[Last/@FactorInteger[#]]&&!unimodQ[-Last/@FactorInteger[#]]&]

Intersection of A332282 and A332642.

A332671 Number of non-unimodal permutations of the multiset of prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 6, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0

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.

			The a(n) permutations for n = 18, 30, 36, 42, 50, 54, 60, 66, 70, 72:
  212  213  1212  214  313  2122  1213  215  314  11212
       312  2112  412       2212  1312  512  413  12112
            2121                  2113            12121
                                  2131            21112
                                  3112            21121
                                  3121            21211

MathWorld, Unimodal Sequence

Dominated by A008480.
The complement is counted by A332288.
A more interesting version is A332672.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Compositions whose negation is not unimodal are A332669.
Cf. A007052, A056239, A112798, A124010, A332281, A332284, A332287, A332294, 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]]]];
Table[Length[Select[Permutations[primeMS[n]],!unimodQ[#]&]],{n,100}]

a(n) + A332288(n) = A008480(n).

a(A181821(n)) = A332672(n).

A332743 Number of non-unimodal compositions of n covering an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 14, 35, 83, 193, 417, 890, 1847, 3809, 7805, 15833, 32028, 64513, 129671, 260155, 521775, 1044982, 2092692, 4188168, 8381434, 16767650, 33544423, 67098683, 134213022, 268443023, 536912014, 1073846768, 2147720476, 4295440133, 8590833907

Offset: 0

Gus Wiseman, Mar 02 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(5) = 1 through a(7) = 14 compositions:
  (212)  (213)   (1213)
         (312)   (1312)
         (1212)  (2113)
         (2112)  (2122)
         (2121)  (2131)
                 (2212)
                 (3112)
                 (3121)
                 (11212)
                 (12112)
                 (12121)
                 (21112)
                 (21121)
                 (21211)

MathWorld, Unimodal Sequence

Not requiring non-unimodality gives A107429.
Not requiring the covering condition gives A115981.
The complement is counted by A227038.
A version for partitions is A332579, with complement A332577.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal normal sequences are A328509.
Numbers whose unsorted prime signature is not unimodal are A332282.
Cf. A007052, A072704, A072706, A332281, A332284, A332287, A332578, A332639, A332642, A332669, A332834, A332870.

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}]

For n > 0, a(n) = A107429(n) - A227038(n).

A332672 Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 6, 16, 0, 21, 0, 12, 10, 0, 0, 48, 16, 0, 81, 20, 0, 48, 0, 104, 15, 0, 30, 162, 0, 0, 21, 104, 0, 90, 0, 30, 198, 0, 0, 336, 65, 124, 28, 42, 0, 603, 50, 190, 36, 0, 0, 396, 0, 0, 405, 688, 77, 150, 0, 56, 45, 260, 0

Offset: 1

Gus Wiseman, Feb 23 2020

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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

			The a(n) permutations for n = 8, 9, 12, 15, 16:
  213   1212   1213   11212   1324
  312   2112   1312   12112   1423
        2121   2113   12121   2134
               2131   21112   2143
               3112   21121   2314
               3121   21211   2413
                              3124
                              3142
                              3214
                              3241
                              3412
                              4123
                              4132
                              4213
                              4231
                              4312

MathWorld, Unimodal Sequence

Positions of zeros are one and A001751.
Support is A264828 without one.
Dominated by A318762.
The complement is counted by A332294.
A less interesting version is A332671.
The opposite version is A332742.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Compositions whose negation is not unimodal are A332669.
Cf. A007052, A008480, A056239, A112798, A124010, A181819, A181821, A332281, A332287, A332294, A332642, A332741.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{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[nrmptn[n]],!unimodQ[#]&]],{n,30}]

a(n) = A332671(A181821(n)).

a(n) + A332294(n) = A318762(n).

A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 28, 39, 55, 77, 107, 142, 194, 254, 332, 434, 563, 716, 919, 1162, 1464, 1841, 2305, 2857, 3555, 4383, 5394, 6617, 8099, 9859, 12006, 14551, 17600, 21236, 25574, 30688, 36809, 44007, 52527, 62574, 74430, 88304, 104675, 123799

Offset: 0

Gus Wiseman, Feb 27 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(4) = 1 through a(9) = 17 partitions:
  (211)  (311)   (411)    (322)     (422)      (522)
         (2111)  (2211)   (511)     (611)      (711)
                 (3111)   (3211)    (3221)     (3222)
                 (21111)  (4111)    (3311)     (4221)
                          (22111)   (4211)     (4311)
                          (31111)   (5111)     (5211)
                          (211111)  (22211)    (6111)
                                    (32111)    (32211)
                                    (41111)    (33111)
                                    (221111)   (42111)
                                    (311111)   (51111)
                                    (2111111)  (222111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
For example, the partition y = (4,2,1,1,1) has negated 0-appended first differences (2,1,0,0,1), which is not unimodal, so y is counted under a(9).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Unimodal Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The complement is counted by A332728.
The non-negated version is A332284.
The strict case is A332579.
The case of run-lengths (instead of differences) is A332639.
The Heinz numbers of these partitions are A332832.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Numbers whose negated prime signature is not unimodal are A332642.
Compositions whose negation is not unimodal are A332669.
Cf. A059204, A227038, A332280, A332285, A332286, A332287, A332638, A332670, A332725, A332726.

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}]

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).

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].

cp's OEIS Frontend

A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

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A332671 Number of non-unimodal permutations of the multiset of prime indices of n.

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A332743 Number of non-unimodal compositions of n covering an initial interval of positive integers.

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A332672 Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.

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A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

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

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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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