A328509 -id:A328509 - cp's OEIS frontend

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

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).

A332872 Number of ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

Original entry on oeis.org

1, 1, 3, 10, 34, 116, 396, 1352, 4616, 15760

Offset: 0

Gus Wiseman, Mar 06 2020

After initial terms, first differs from A291292 at a(7) = 1352, A291292(8) = 1353.

Conjectured to be the same as A007052, shifted right once.

			The a(1) = 1 through a(3) = 10 ordered set partitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{2,3}}
         {{2},{1}}  {{1,2},{3}}
                    {{1,3},{2}}
                    {{2},{1,3}}
                    {{2,3},{1}}
                    {{3},{1,2}}
                    {{1},{2},{3}}
                    {{1},{3},{2}}
                    {{2},{1},{3}}

Row sums of A332673.
Set partitions are A000110.
Ordered set-partitions are A000670.
Unimodal sequences covering an initial interval are A007052.
Cf. A001523, A056242, A097805, A328509, A332280, A332283, A332294, A332724.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]],!MatchQ[#,{_,{_,a_,_},,{_,b_,_},_}/;a>b]&]],{n,0,5}]

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).

A332874 Number of strict compositions of n that are neither unimodal nor is their negation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 20, 30, 50, 150, 180, 290, 420, 630, 860, 1828, 2168, 3326, 4514, 6530, 8576, 12188, 20096, 25314, 35576, 48062, 65592, 86752, 117222, 152060, 237590, 292346, 402798, 524596, 711270, 910606, 1221204, 1554382, 2044460, 2927124

Offset: 0

Gus Wiseman, Mar 04 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. It is strict if there are not repeated parts.

			The a(10) = 10 through a(12) = 20 compositions:
  (1,3,2,4)  (1,3,2,5)  (1,3,2,6)
  (1,4,2,3)  (1,5,2,3)  (1,4,2,5)
  (2,1,4,3)  (2,1,5,3)  (1,5,2,4)
  (2,3,1,4)  (2,3,1,5)  (1,6,2,3)
  (2,4,1,3)  (2,5,1,3)  (2,1,5,4)
  (3,1,4,2)  (3,1,5,2)  (2,1,6,3)
  (3,2,4,1)  (3,2,5,1)  (2,3,1,6)
  (3,4,1,2)  (3,5,1,2)  (2,4,1,5)
  (4,1,3,2)  (5,1,3,2)  (2,5,1,4)
  (4,2,3,1)  (5,2,3,1)  (2,6,1,3)
                        (3,1,6,2)
                        (3,2,6,1)
                        (3,6,1,2)
                        (4,1,5,2)
                        (4,2,5,1)
                        (4,5,1,2)
                        (5,1,4,2)
                        (5,2,4,1)
                        (6,1,3,2)
                        (6,2,3,1)

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

The non-strict version for unsorted prime signature is A332643.
The non-strict version is A332870.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Compositions with neither weakly increasing nor weakly decreasing run-lengths are A332833.
Compositions with weakly increasing or weakly decreasing run-lengths are A332835.
Cf. A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332640, A332641, A332745, A332746, A332831, A332834.

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[#]&&!unimodQ[-#]&]],{n,0,20}]

seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=4, n, (k! - 2^k + 2)*polcoef(p,k,y)), -(n+1))} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=4} (k! - 2^k + 2) * [y^k](Product_{j>=1} 1 + y*x^j). - Andrew Howroyd, Apr 16 2021

Terms a(21) and beyond from Andrew Howroyd, Apr 16 2021

A332673 Triangle read by rows where T(n,k) is the number of length-k ordered set partitions of {1..n} whose non-adjacent blocks are pairwise increasing.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 14, 14, 5, 0, 1, 30, 45, 32, 8, 0, 1, 62, 124, 131, 65, 13, 0, 1, 126, 315, 438, 323, 128, 21, 0, 1, 254, 762, 1305, 1270, 747, 243, 34, 0, 1, 510, 1785, 3612, 4346, 3370, 1629, 452, 55

Offset: 0

Gus Wiseman, Mar 02 2020

In other words, parts of subsequent, non-successive blocks are increasing.

			Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    3
    0    1   14   14    5
    0    1   30   45   32    8
    0    1   62  124  131   65   13
    0    1  126  315  438  323  128   21
    0    1  254  762 1305 1270  747  243   34
    ...
Row n = 4 counts the following ordered set partitions:
  {1234}  {1}{234}  {1}{2}{34}  {1}{2}{3}{4}
          {12}{34}  {1}{23}{4}  {1}{2}{4}{3}
          {123}{4}  {12}{3}{4}  {1}{3}{2}{4}
          {124}{3}  {1}{24}{3}  {2}{1}{3}{4}
          {13}{24}  {12}{4}{3}  {2}{1}{4}{3}
          {134}{2}  {1}{3}{24}
          {14}{23}  {13}{2}{4}
          {2}{134}  {1}{34}{2}
          {23}{14}  {1}{4}{23}
          {234}{1}  {2}{1}{34}
          {24}{13}  {2}{13}{4}
          {3}{124}  {2}{14}{3}
          {34}{12}  {23}{1}{4}
          {4}{123}  {3}{12}{4}

An apparently related triangle is A056242.
Column k = n - 1 is A332724.
Row sums are A332872, which appears to be A007052 shifted right once.
Ordered set-partitions are A000670.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Cf. A072704, A097805, A107429, A227038, A332280, A332283, A332288, A332577.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==k&&!MatchQ[#,{_,{_,a_,_},,{_,b_,_},_}/;a>b]&]],{n,0,5},{k,0,n}]

A332724 Number of length n - 1 ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

Original entry on oeis.org

0, 0, 1, 6, 14, 32, 65, 128, 243, 452, 826, 1490, 2659, 4704, 8261, 14418, 25030, 43252, 74437, 127648, 218199, 371920, 632306, 1072486, 1815239, 3066432, 5170825, 8705118, 14632958, 24562952, 41177801, 68947520, 115313979, 192656924, 321554986, 536191418

Offset: 0

Gus Wiseman, Mar 03 2020

nonn

In other words, parts of not-immediately-subsequent blocks are increasing.

			The a(2) = 1 through a(4) = 14 ordered set partitions:
  {{1,2}}  {{1},{2,3}}  {{1},{2},{3,4}}
           {{1,2},{3}}  {{1},{2,3},{4}}
           {{1,3},{2}}  {{1,2},{3},{4}}
           {{2},{1,3}}  {{1},{2,4},{3}}
           {{2,3},{1}}  {{1,2},{4},{3}}
           {{3},{1,2}}  {{1},{3},{2,4}}
                        {{1,3},{2},{4}}
                        {{1},{3,4},{2}}
                        {{1},{4},{2,3}}
                        {{2},{1},{3,4}}
                        {{2},{1,3},{4}}
                        {{2},{1,4},{3}}
                        {{2,3},{1},{4}}
                        {{3},{1,2},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Column k = n - 1 of A332673, which has row-sums A332872.
Ordered set-partitions are A000670.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal normal sequences are A328509.
Cf. A000045, A001286, A001629, A056242, A227038, A332577.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==n-1&&!MatchQ[#,{_,{_,a_,_},,{_,b_,_},_}/;a>b]&]],{n,0,8}]

\\ here b(n) is A001629(n).
b(n) = {((n+1)*fibonacci(n-1) + (n-1)*fibonacci(n+1))/5}
a(n) = {if(n==0, 0, b(n) + 4*b(n-1) + b(n-2))} \\ Andrew Howroyd, Apr 17 2021

From Andrew Howroyd, Apr 17 2021: (Start)
a(n) = A001629(n) + 4*A001629(n+1) + A001629(n+2) for n > 0.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n > 4.
G.f.: x*(1 + 4*x + x^2)/(1 - x - x^2)^2.
(End)

Terms a(9) and beyond from Andrew Howroyd, Apr 17 2021

A332873 Number of non-unimodal, non-co-unimodal sequences of length n covering an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 22, 340, 3954, 44716, 536858, 7056252, 102140970, 1622267196, 28090317226, 526854073564, 10641328363722, 230283141084220, 5315654511587498, 130370766447282204, 3385534661270087178, 92801587312544823804, 2677687796221222845802, 81124824998424994578652

Offset: 0

Gus Wiseman, Mar 03 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 negative is unimodal.

			The a(4) = 22 sequences:
  (1,2,1,2)  (2,3,1,3)
  (1,2,1,3)  (2,3,1,4)
  (1,3,1,2)  (2,4,1,3)
  (1,3,2,3)  (3,1,2,1)
  (1,3,2,4)  (3,1,3,2)
  (1,4,2,3)  (3,1,4,2)
  (2,1,2,1)  (3,2,3,1)
  (2,1,3,1)  (3,2,4,1)
  (2,1,3,2)  (3,4,1,2)
  (2,1,4,3)  (4,1,3,2)
  (2,3,1,2)  (4,2,3,1)

Andrew Howroyd, Table of n, a(n) for n = 0..200

Not requiring non-co-unimodality gives A328509.
Not requiring non-unimodality also gives A328509.
The version for run-lengths of partitions is A332640.
The version for unsorted prime signature is A332643.
The version for compositions is A332870.
Unimodal compositions are A001523.
Unimodal sequences covering an initial interval are A007052.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unimodal compositions covering an initial interval are A227038.
Numbers whose unsorted prime signature is not unimodal are A332282.
Numbers whose negated prime signature is not unimodal are A332642.
Compositions whose run-lengths are not unimodal are A332727.
Non-unimodal compositions covering an initial interval are A332743.
Cf. A000225, A000670, A060223, A072704, A329398, A332281, A332284, A332577, A332578, A332639, A332672, A332834.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Union@@Permutations/@allnorm[n],!unimodQ[#]&&!unimodQ[-#]&]],{n,0,5}]

seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - (1 - 6*x + 12*x^2 - 6*x^3)/((1 - x)*(1 - 2*x)*(1 - 4*x + 2*x^2)), -(n+1)) \\ Andrew Howroyd, Jan 28 2024

a(n) = A000670(n) + A000225(n) - 2*A007052(n-1) for n > 0. - Andrew Howroyd, Jan 28 2024

a(9) onwards from Andrew Howroyd, Jan 28 2024

A333146 Number of non-unimodal negated 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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 7, 0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 0, 8, 0, 0, 0

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.

			The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 96:
  (121)  (1121)  (1212)  (1132)  (11212)  (1232)  (111121)
         (1211)  (1221)  (1213)  (11221)  (1322)  (111211)
                 (2121)  (1231)  (12112)  (2132)  (112111)
                         (1312)  (12121)  (2231)  (121111)
                         (1321)  (12211)  (2312)
                         (2131)  (21121)  (2321)
                         (2311)  (21211)
                         (3121)

Eric Weisstein's World of Mathematics, Unimodal Sequence

Dominated by A008480.
The non-negated version is A332671.
A more interesting version is A332742.
The complement is counted by A333145.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Compositions whose negation is unimodal are A332578.
Partitions with unimodal negated run-lengths are A332638.
Numbers with non-unimodal negated unsorted prime signature are A332642.
Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332288, A332294, A332639, A332669, A332670, A332741.

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) + A333145(n) = A008480(n).

cp's OEIS Frontend

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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A332872 Number of ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

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

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A332874 Number of strict compositions of n that are neither unimodal nor is their negation.

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A332673 Triangle read by rows where T(n,k) is the number of length-k ordered set partitions of {1..n} whose non-adjacent blocks are pairwise increasing.

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A332724 Number of length n - 1 ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

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