A332578 -id:A332578 - cp's OEIS frontend

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

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

A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 29, 56, 101, 181, 327, 583, 1023, 1820, 3207, 5631, 9905, 17394, 30489, 53481, 93725, 164169, 287606, 503672, 881834, 1544018, 2703161, 4731860, 8283291, 14499392, 25379278, 44422866, 77754798, 136093756, 238204369, 416923752, 729728031

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

			The a(6) = 29 compositions:
  (6)    (141)  (213)   (1113)  (21111)
  (51)   (114)  (132)   (222)   (12111)
  (15)   (33)   (123)   (2211)  (11121)
  (42)   (321)  (3111)  (2121)  (11112)
  (24)   (312)  (1311)  (1212)  (111111)
  (411)  (231)  (1131)  (1122)
Missing are: (2112), (1221), (11211).

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

The version for the compositions themselves (not run-lengths) is A329398.
Compositions with equal run-lengths are A329738.
The case of partitions is A332745.
The version for unsorted prime signature is the complement of A332831.
The complement is counted by A332833.
Unimodal compositions are A001523.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Neither weakly increasing nor weakly decreasing compositions are A332834.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A181819, A329744, A329766, A332273, A332640, A332641, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,20}]

a(n) = 2 * A332836(n) - A329738(n).

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

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}]

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

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}]

A332870 Number of compositions of n that are neither unimodal nor is their negation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 9, 32, 92, 243, 587, 1361, 3027, 6564, 13928, 29127, 60180, 123300, 250945, 508326, 1025977, 2065437, 4150056, 8327344, 16692844, 33438984, 66951671, 134004892, 268148573, 536486146, 1073227893, 2146800237, 4294061970, 8588740071, 17178298617

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(6) = 2 and a(7) = 9 compositions:
  (1212)  (1213)
  (2121)  (1312)
          (2131)
          (3121)
          (11212)
          (12112)
          (12121)
          (21121)
          (21211)

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

The case of run-lengths of partitions is A332640.
The version for unsorted prime signature is A332643.
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.
Partitions with weakly increasing or decreasing run-lengths are A332745.
Compositions that are neither weakly increasing nor decreasing are A332834.
Compositions with weakly increasing or decreasing run-lengths are A332835.
Cf. A000005, A000041, A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332641, A332746, A332831, A332833.

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

a(n) = 2^(n-1) - A001523(n) - A332578(n) + 2*A000041(n) - A000005(n) for n > 0. - Andrew Howroyd, Dec 30 2020

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

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

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)).

cp's OEIS Frontend

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

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A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

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

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

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A332870 Number of compositions of n that are neither unimodal nor is their negation.

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

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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.