A332639 -id:A332639 - cp's OEIS frontend

A328509 Number of non-unimodal sequences of length n covering an initial interval of positive integers.

0, 0, 0, 3, 41, 425, 4287, 45941, 541219, 7071501, 102193755, 1622448861, 28090940363, 526856206877, 10641335658891, 230283166014653, 5315654596751659, 130370766738143517, 3385534662263335179, 92801587315936355325, 2677687796232803000171, 81124824998464533181661

Offset: 0

Gus Wiseman, Feb 19 2020

nonn

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

			The a(3) = 3 sequences are (2,1,2), (2,1,3), (3,1,2).
The a(4) = 41 sequences:
  (1212)  (2113)  (2134)  (2413)  (3142)  (3412)
  (1213)  (2121)  (2143)  (3112)  (3212)  (4123)
  (1312)  (2122)  (2212)  (3121)  (3213)  (4132)
  (1323)  (2123)  (2213)  (3122)  (3214)  (4213)
  (1324)  (2131)  (2312)  (3123)  (3231)  (4231)
  (1423)  (2132)  (2313)  (3124)  (3241)  (4312)
  (2112)  (2133)  (2314)  (3132)  (3312)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Not requiring non-unimodality gives A000670.
The complement is counted by A007052.
The case where the negation is not unimodal either is A332873.
Unimodal compositions are A001523.
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.
Covering partitions with unimodal run-lengths are A332577.
Non-unimodal compositions covering an initial interval are A332743.
Cf. A060223, A255906, A332281, A332284, A332639, A332672, A332834, A332870.

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[#]&]],{n,0,5}]

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

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

a(9) from Robert Price, Jun 19 2021

a(10) onwards from Andrew Howroyd, Jan 28 2024

A332578 Number of compositions of n whose negation is unimodal.

Original entry on oeis.org

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

A332282 Numbers whose unsorted prime signature is not unimodal.

Original entry on oeis.org

300, 588, 600, 980, 1176, 1200, 1452, 1500, 1960, 2028, 2100, 2205, 2352, 2400, 2420, 2904, 2940, 3000, 3300, 3380, 3388, 3468, 3900, 3920, 4056, 4116, 4200, 4332, 4410, 4704, 4732, 4800, 4840, 5100, 5445, 5700, 5780, 5808, 5880, 6000, 6348, 6468, 6600, 6615

Offset: 1

Gus Wiseman, Feb 19 2020

nonn

The unsorted prime signature of a positive integer (row n of A124010) is the sequence of exponents it is prime factorization.
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also Heinz numbers of integer partitions with non-unimodal run-lengths. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   300: {1,1,2,3,3}
   588: {1,1,2,4,4}
   600: {1,1,1,2,3,3}
   980: {1,1,3,4,4}
  1176: {1,1,1,2,4,4}
  1200: {1,1,1,1,2,3,3}
  1452: {1,1,2,5,5}
  1500: {1,1,2,3,3,3}
  1960: {1,1,1,3,4,4}
  2028: {1,1,2,6,6}
  2100: {1,1,2,3,3,4}
  2205: {2,2,3,4,4}
  2352: {1,1,1,1,2,4,4}
  2400: {1,1,1,1,1,2,3,3}
  2420: {1,1,3,5,5}
  2904: {1,1,1,2,5,5}
  2940: {1,1,2,3,4,4}
  3000: {1,1,1,2,3,3,3}
  3300: {1,1,2,3,3,5}
  3380: {1,1,3,6,6}

MathWorld, Unimodal Sequence

The opposite version is A332642.
These are the Heinz numbers of the partitions counted by A332281.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Cf. A001523, A007052, A056239, A112798, A124010, A227038, A332280, A332284, A332286, A332639, A332643, A332671.

unimodQ[q_]:=Or[Length[q]==1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Select[Range[1000],!unimodQ[Last/@FactorInteger[#]]&]

A332281 Number of integer partitions of n whose run-lengths are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 6, 10, 16, 24, 33, 51, 70, 100, 137, 189, 250, 344, 450, 597, 778, 1019, 1302, 1690, 2142, 2734, 3448, 4360, 5432, 6823, 8453, 10495, 12941, 15968, 19529, 23964, 29166, 35525, 43054, 52173, 62861, 75842, 91013, 109208

Offset: 0

Gus Wiseman, Feb 19 2020

nonn

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

			The a(10) = 1 through a(15) = 10 partitions:
  (33211)  (332111)  (44211)    (44311)     (55211)      (44322)
                     (3321111)  (333211)    (433211)     (55311)
                                (442111)    (443111)     (443211)
                                (33211111)  (3332111)    (533211)
                                            (4421111)    (552111)
                                            (332111111)  (4332111)
                                                         (4431111)
                                                         (33321111)
                                                         (44211111)
                                                         (3321111111)

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

The complement is counted by A332280.
The Heinz numbers of these partitions are A332282.
The opposite version is A332639.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Cf. A007052, A025065, A072706, A100883, A332283, A332284, A332286, A332287, A332579, A332638, A332640, A332641, A332642.

b:= proc(n, i, m, t) option remember; `if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1, j, t and j>=m),
      j=1..min(`if`(t, [][], m), n/i))+b(n, i-1, m, t)))
    end:
a:= n-> combinat[numbpart](n)-b(n$2, 0, true):
seq(a(n), n=0..65);  # Alois P. Heinz, Feb 20 2020

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}]
(* Second program: *)
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, j, t && j >= m], {j, 1, Min[If[t, Infinity, m], n/i]}] + b[n, i - 1, m, t]]];
a[n_] := PartitionsP[n] - b[n, n, 0, True];
a /@ Range[0, 65] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A335373 Numbers k such that the k-th composition in standard order (A066099) is not unimodal.

Original entry on oeis.org

22, 38, 44, 45, 46, 54, 70, 76, 77, 78, 86, 88, 89, 90, 91, 92, 93, 94, 102, 108, 109, 110, 118, 134, 140, 141, 142, 148, 150, 152, 153, 154, 155, 156, 157, 158, 166, 172, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 198

Offset: 1

Gus Wiseman, Jun 03 2020

nonn

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

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
  22: (2,1,2)
  38: (3,1,2)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  54: (1,2,1,2)
  70: (4,1,2)
  76: (3,1,3)
  77: (3,1,2,1)
  78: (3,1,1,2)
  86: (2,2,1,2)
  88: (2,1,4)
  89: (2,1,3,1)
  90: (2,1,2,2)
  91: (2,1,2,1,1)
  92: (2,1,1,3)
  93: (2,1,1,2,1)
  94: (2,1,1,1,2)

The dual version (non-co-unimodal compositions) is A335374.
The case that is not co-unimodal either is A335375.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Partitions with non-unimodal 0-appended first differences are A332284.
Non-unimodal permutations of the multiset of prime indices of n are A332671.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A072704, A332281, A332286, A332287, A332639, A332642, A332669, A332672.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,200],!unimodQ[stc[#]]&]

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[#]]&]

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 20 2020

nonn

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 25 2020

nonn

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

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

MathWorld, Unimodal Sequence

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

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

A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 25, 30, 36, 40, 45, 54, 59, 68, 79, 86, 96, 112, 121, 135, 155, 168, 188, 214, 230, 253, 284, 308, 337, 380, 407, 445, 497, 533, 580, 645, 689, 748, 828, 885, 956, 1053, 1124, 1212, 1330, 1415, 1519, 1665, 1771

Offset: 0

Gus Wiseman, Feb 24 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(1) = 1 through a(9) = 8 partitions:
  1  11  21   211   221    321     2221     3221      3321
         111  1111  2111   2211    3211     22211     22221
                    11111  21111   22111    32111     32211
                           111111  211111   221111    222111
                                   1111111  2111111   321111
                                            11111111  2211111
                                                      21111111
                                                      111111111

MathWorld, Unimodal Sequence

Not requiring unimodality gives A000009.
A version for compositions is A227038.
Not requiring the partition to cover an initial interval gives A332280.
The complement is counted by A332579.
Unimodal compositions are A001523.
Cf. A007052, A011782, A025065, A100883, A107429, A115981, A332281, A332283, A332638, A332639, A332728.

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[IntegerPartitions[n],normQ[#]&&unimodQ[Length/@Split[#]]&]],{n,0,30}]

cp's OEIS Frontend

A328509 Number of non-unimodal sequences of length n covering an initial interval of positive integers.

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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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A332282 Numbers whose unsorted prime signature is not unimodal.

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A332281 Number of integer partitions of n whose run-lengths are not unimodal.

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A335373 Numbers k such that the k-th composition in standard order (A066099) is not unimodal.

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A332642 Numbers whose negated unsorted prime signature is not unimodal.

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