A332742 -id:A332742 - cp's OEIS frontend

A332578 Number of compositions of n whose negation is unimodal.

1, 1, 2, 4, 7, 13, 21, 36, 57, 91, 140, 217, 323, 485, 711, 1039, 1494, 2144, 3032, 4279, 5970, 8299, 11438, 15708, 21403, 29065, 39218, 52725, 70497, 93941, 124562, 164639, 216664, 284240, 371456, 484004, 628419, 813669, 1050144, 1351757, 1734873, 2221018, 2835613

Offset: 0

Gus Wiseman, Feb 28 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(1) = 1 through a(5) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (112)   (41)
                    (211)   (113)
                    (1111)  (122)
                            (212)
                            (221)
                            (311)
                            (1112)
                            (2111)
                            (11111)

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000 (terms 0..1000 from Andrew Howroyd)

Dominated by A001523 (unimodal compositions).
The strict case is A072706.
The case that is unimodal also is A329398.
The complement is counted by A332669.
Row sums of A332670.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions whose run-lengths are unimodal are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Numbers whose unsorted prime signature is not unimodal are A332642.
Partitions whose negated 0-appended differences are unimodal are A332728.
Cf. A011782, A072704, A107429, A227038, A332282, A332283, A332639, 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],unimodQ[-#]&]],{n,0,10}]
nmax = 50; CoefficientList[Series[1 + Sum[x^j*(1 - x^j)/Product[1 - x^k, {k, j, nmax - j}]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 01 2020 *)

seq(n)={Vec(1 + sum(j=1, n, x^j/((1-x^j)*prod(k=j+1, n-j, 1 - x^k + O(x*x^(n-j)))^2)))} \\ Andrew Howroyd, Mar 01 2020

a(n) + A332669(n) = 2^(n - 1).
G.f.: 1 + Sum_{j>0} x^j/((1 - x^j)*(Product_{k>j} 1 - x^k)^2). - Andrew Howroyd, Mar 01 2020
a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Mar 01 2020

Terms a(26) and beyond from Andrew Howroyd, Mar 01 2020

A332669 Number of compositions of n whose negation is not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 11, 28, 71, 165, 372, 807, 1725, 3611, 7481, 15345, 31274, 63392, 128040, 257865, 518318, 1040277, 2085714, 4178596, 8367205, 16748151, 33515214, 67056139, 134147231, 268341515, 536746350, 1073577185, 2147266984, 4294683056, 8589563136, 17179385180

Offset: 0

Gus Wiseman, Feb 28 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(4) = 1 through a(6) = 11 compositions:
  (121)  (131)   (132)
         (1121)  (141)
         (1211)  (231)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2121)
                 (11121)
                 (11211)
                 (12111)

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

The strict case is A072707.
The complement is counted by A332578.
The version for run-lengths of partitions is A332639.
The version for unsorted prime signature is A332642.
The version for 0-appended first-differences of partitions is A332744.
The case that is not unimodal either is A332870.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose unsorted prime signature is not unimodal are A332282.
A triangle for compositions with unimodal negation is A332670.
Cf. A007052, A072706, A227038, A329398, A332281, A332284, A332638, A332728, A332742, A332832.

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[-#]&]],{n,0,10}]

a(n) + A332578(n) = 2^(n - 1) for n > 0.

Terms a(21) and beyond from Andrew Howroyd, Mar 01 2020

A332639 Number of integer partitions of n whose negated run-lengths are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 10, 17, 25, 36, 51, 75, 102, 143, 192, 259, 346, 462, 599, 786, 1014, 1309, 1670, 2133, 2686, 3402, 4258, 5325, 6623, 8226, 10134, 12504, 15328, 18779, 22878, 27870, 33762, 40916, 49349, 59457, 71394, 85679, 102394

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) = 1 through a(13) = 10 partitions:
  (3221)  (4221)  (5221)   (4331)    (4332)    (5332)
                  (32221)  (6221)    (5331)    (6331)
                           (42221)   (7221)    (8221)
                           (322211)  (43221)   (43321)
                                     (52221)   (53221)
                                     (322221)  (62221)
                                     (422211)  (332221)
                                               (422221)
                                               (522211)
                                               (3222211)

MathWorld, Unimodal Sequence

The version for normal sequences is A328509.
The non-negated complement is A332280.
The non-negated version is A332281.
The complement is counted by A332638.
The case that is not unimodal either is A332640.
The Heinz numbers of these partitions are A332642.
The generalization to run-lengths of compositions is A332727.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Compositions whose negation is not unimodal are A332669.
Cf. A007052, A025065, A100883, A181819, A332282, A332578, A332579, A332641, A332670, A332671, A332726, A332742, A332744.

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}]

A332642 Numbers whose negated unsorted prime signature is not unimodal.

Original entry on oeis.org

90, 126, 198, 234, 270, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 594, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1188, 1206, 1242, 1274, 1275, 1278, 1314, 1350, 1386

Offset: 1

Gus Wiseman, Feb 28 2020

nonn

A sequence of 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:
    90: {1,2,2,3}
   126: {1,2,2,4}
   198: {1,2,2,5}
   234: {1,2,2,6}
   270: {1,2,2,2,3}
   306: {1,2,2,7}
   342: {1,2,2,8}
   350: {1,3,3,4}
   378: {1,2,2,2,4}
   414: {1,2,2,9}
   522: {1,2,2,10}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
   550: {1,3,3,5}
   558: {1,2,2,11}
   594: {1,2,2,2,5}
   630: {1,2,2,3,4}
   650: {1,3,3,6}
   666: {1,2,2,12}
   702: {1,2,2,2,6}
For example, 630 has negated unsorted prime signature (-1,-2,-1,-1), which is not unimodal, so 630 is in the sequence.

Eric Weisstein's World of Mathematics, Unimodal Sequence

These are the Heinz numbers of the partitions counted by A332639.
The case that is not unimodal either is A332643.
The version for compositions is A332669.
The complement is A332282.
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.
The number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n is A332742(n).
Partitions whose negated 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
Cf. A007052, A056239, A112798, A181821, A242031, A329747, A332280, A332281, A332578, A332671, A332831.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Select[Range[2000],!unimodQ[-Last/@FactorInteger[#]]&]

A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 39, 51, 68, 87, 113, 143, 183, 228, 289, 354, 443, 544, 672, 812, 1001, 1202, 1466, 1758, 2123, 2525, 3046, 3606, 4308, 5089, 6054, 7102, 8430, 9855, 11621, 13571, 15915, 18500, 21673, 25103, 29245, 33835, 39296, 45277, 52470

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

Also partitions whose run-lengths and negated run-lengths are both unimodal.

			The a(8) = 21 partitions are:
  (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).

The complement is counted by A332641.
The Heinz numbers of partitions not in this class are A332831.
The case of run-lengths of compositions is A332835.
Only weakly decreasing is A100882.
Only weakly increasing is A100883.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Compositions with unimodal run-lengths are A332726.
Compositions that are neither weakly increasing nor decreasing are A332834.
Cf. A025065, A059204, A181819, A328509, A332281, A332283, A332577, A332578, A332640, A332727, A332742, A332833.

Table[Length[Select[IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@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

A332641 Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 14, 22, 33, 48, 69, 96, 136, 184, 248, 330, 443, 574, 756, 970, 1252, 1595, 2040, 2558, 3236, 4041, 5054, 6256, 7781, 9547, 11782, 14394, 17614, 21423, 26083, 31501, 38158, 45930, 55299, 66262, 79477, 94803, 113214

Offset: 0

Gus Wiseman, Feb 26 2020

nonn

Also partitions whose run-lengths and negated run-lengths are not both unimodal. A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(8) = 1 through a(13) = 14 partitions:
  (3221)  (4221)  (5221)   (4331)    (4332)     (5332)
                  (32221)  (6221)    (5331)     (6331)
                  (33211)  (42221)   (7221)     (8221)
                           (322211)  (43221)    (43321)
                           (332111)  (44211)    (44311)
                                     (52221)    (53221)
                                     (322221)   (62221)
                                     (422211)   (332221)
                                     (3321111)  (333211)
                                                (422221)
                                                (442111)
                                                (522211)
                                                (3222211)
                                                (33211111)

MathWorld, Unimodal Sequence

The complement is counted by A332745.
The Heinz numbers of these partitions are A332831.
The case of run-lengths of compositions is A332833.
Partitions whose run-lengths are weakly increasing are A100883.
Partitions whose run-lengths are weakly decreasing are A100882.
Partitions whose run-lengths are not unimodal are A332281.
Partitions whose negated run-lengths are not unimodal are A332639.
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.
Compositions whose negation is not unimodal are A332669.
The case of run-lengths of compositions is A332833.
Compositions that are neither increasing nor decreasing are A332834.
Cf. A025065, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332640, A332642, A332726, A332727, A332742, A332835.

Table[Length[Select[IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]

A332294 Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 4, 1, 6, 1, 5, 4, 8, 1, 9, 1, 8, 5, 6, 1, 12, 4, 7, 9, 10, 1, 12, 1, 16, 6, 8, 5, 18, 1, 9, 7, 16, 1, 15, 1, 12, 12, 10, 1, 24, 5, 16, 8, 14, 1, 27, 6, 20, 9, 11, 1, 24, 1, 12, 15, 32, 7, 18, 1, 16, 10, 20, 1, 36, 1, 13, 16, 18, 6

Offset: 1

Gus Wiseman, Feb 21 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 positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(12) = 6 permutations:
  {1,1,2,3}
  {1,1,3,2}
  {1,2,3,1}
  {1,3,2,1}
  {2,3,1,1}
  {3,2,1,1}

MathWorld, Unimodal Sequence

Dominated by A318762.
A less interesting version is A332288.
The complement is counted by A332672.
The opposite/negative version is A332741.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Partitions whose run-lengths are unimodal are A332280.
Cf. A007052, A056239, A112798, A115981, A124010, A304660, A328509, A332283, A332578, A332638, A332671, A332742.

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,0,30}]

a(n) + A332672(n) = A318762(n).

a(n) = A332288(A181821(n)).

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

Original entry on oeis.org

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.

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).

cp's OEIS Frontend

A332578 Number of compositions of n whose negation is unimodal.

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A332669 Number of compositions of n whose negation is not unimodal.

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A332639 Number of integer partitions of n whose negated run-lengths are not unimodal.

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A332642 Numbers whose negated unsorted prime signature is not unimodal.

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A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.

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

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A332641 Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

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A332294 Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.

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