A332578 -id:A332578 - cp's OEIS frontend

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

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

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].

A332832 Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

12, 20, 24, 28, 36, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 165, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 195, 196, 198

Offset: 1

Gus Wiseman, Mar 02 2020

nonn

First differs from A065201 in having 165.
First differs from A316597 in having 36.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   90: {1,2,2,3}
For example, 60 is the Heinz number of (3,2,1,1), with negated 0-appended first-differences (1,1,0,1), which are not unimodal, so 60 is in the sequence.

MathWorld, Unimodal Sequence
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The non-negated version is A332287.
The version for of run-lengths (instead of differences) is A332642.
The enumeration of these partitions by sum is A332744.
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.
Compositions whose negation is not unimodal are A332669.
Cf. A059204, A227038, A332284, A332285, A332286, A332578, A332638, A332639, A332670, A332725, A332728.

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]]]];
Select[Range[100],!unimodQ[Differences[Prepend[primeMS[#],0]]]&]

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

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

A333148 Number of compositions of n whose non-adjacent parts are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 19, 30, 46, 69, 102, 149, 214, 304, 428, 596, 823, 1127, 1532, 2068, 2774, 3697, 4900, 6460, 8474, 11061, 14375, 18600, 23970, 30770, 39354, 50153, 63702, 80646, 101783, 128076, 160701, 201076, 250933, 312346, 387832, 480409, 593716, 732105, 900810, 1106063, 1355336, 1657517, 2023207, 2464987, 2997834, 3639464

Offset: 0

Gus Wiseman, May 16 2020

nonn

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (211)   (131)    (51)
                    (1111)  (212)    (141)
                            (221)    (222)
                            (311)    (231)
                            (1211)   (312)
                            (2111)   (321)
                            (11111)  (411)
                                     (1311)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (12111)
                                     (21111)
                                     (111111)
For example, (2,3,1,2) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), all of which are weakly decreasing.

Alois P. Heinz, Table of n, a(n) for n = 0..700

Unimodal compositions are A001523.
The case of normal sequences appears to be A028859.
A version for ordered set partitions is A332872.
The case of strict compositions is A333150.
The version for strictly decreasing parts is A333193.
Standard composition numbers (A066099) of these compositions are A334966.
Cf. A056242, A059204, A072706, A107429, A115981, A329398, A332578, A332669, A332673, A332724, A332834.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,,y_,_}/;y>x]&]],{n,0,15}]

def a333148(n): return number_of_partitions(n) + sum( Partitions(m, max_part=l, length=k).cardinality() * Partitions(n-m-l^2, min_length=k+2*l).cardinality() for l in range(1, (n+1).isqrt()) for m in range((n-l^2-2*l)*l//(l+1)+1) for k in range(ceil(m/l), min(m,n-m-l^2-2*l)+1) ) # Max Alekseyev, Oct 31 2024

See Sage code for the formula. - Max Alekseyev, Oct 31 2024

Edited and terms a(21)-a(51) added by Max Alekseyev, Oct 30 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).

A335374 Numbers k such that the k-th composition in standard order (A066099) is not co-unimodal.

Original entry on oeis.org

13, 25, 27, 29, 41, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 81, 82, 83, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 145, 153, 155, 157, 161, 162, 163, 165, 166, 167, 169, 173, 177

Offset: 1

Gus Wiseman, Jun 03 2020

nonn

A sequence of integers is co-unimodal if it is the concatenation of a weakly decreasing and a weakly increasing sequence, implying that its negation is unimodal.

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:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  41: (2,3,1)
  45: (2,1,2,1)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  81: (2,4,1)
  82: (2,3,2)
  83: (2,3,1,1)
  89: (2,1,3,1)

This is the dual version of A335373.
The case that is not 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.
Co-unimodal compositions are A332578.
Numbers with non-co-unimodal unsorted prime signature are A332642.
Non-co-unimodal compositions are A332669.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.

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[0,100],!unimodQ[-stc[#]]&]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A333147 Number of compositions of n that are either strictly increasing or strictly decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A333150 Number of strict compositions of n whose non-adjacent parts are strictly decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A072705 Triangle of number of unimodal compositions of n into exactly k distinct terms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A332832 Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica