A001523 -id:A001523 - 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

A072706 Number of unimodal partitions/compositions of n into distinct terms.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 33, 39, 55, 69, 93, 127, 159, 201, 261, 327, 411, 537, 653, 819, 1011, 1257, 1529, 1899, 2331, 2829, 3441, 4179, 5031, 6093, 7305, 8767, 10575, 12573, 14997, 17847, 21223, 25089, 29757, 35055, 41379, 48801, 57285, 67131

Offset: 0

Henry Bottomley, Jul 04 2002

nonn

Also the number of ways to partition a strict integer partition of n into two unordered blocks. - Gus Wiseman, Dec 31 2019

			a(6)=9 since 6 can be written as 1+2+3, 1+3+2, 1+5, 2+3+1, 2+4, 3+2+1, 4+2, 5+1, or 6, but not for example 1+4+1 (which does not have distinct terms) nor 2+1+3 (which is not unimodal).
From _Joerg Arndt_, Mar 25 2014: (Start)
The a(10) = 33 such compositions of 10 are:
01:  [ 1 2 3 4 ]
02:  [ 1 2 4 3 ]
03:  [ 1 2 7 ]
04:  [ 1 3 4 2 ]
05:  [ 1 3 6 ]
06:  [ 1 4 3 2 ]
07:  [ 1 4 5 ]
08:  [ 1 5 4 ]
09:  [ 1 6 3 ]
10:  [ 1 7 2 ]
11:  [ 1 9 ]
12:  [ 2 3 4 1 ]
13:  [ 2 3 5 ]
14:  [ 2 4 3 1 ]
15:  [ 2 5 3 ]
16:  [ 2 7 1 ]
17:  [ 2 8 ]
18:  [ 3 4 2 1 ]
19:  [ 3 5 2 ]
20:  [ 3 6 1 ]
21:  [ 3 7 ]
22:  [ 4 3 2 1 ]
23:  [ 4 5 1 ]
24:  [ 4 6 ]
25:  [ 5 3 2 ]
26:  [ 5 4 1 ]
27:  [ 6 3 1 ]
28:  [ 6 4 ]
29:  [ 7 2 1 ]
30:  [ 7 3 ]
31:  [ 8 2 ]
32:  [ 9 1 ]
33:  [ 10 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The non-strict version is A001523.

Cf. A000009, A000041, A001970, A032020, A059618, A072705, A072707, A270995, A294617.

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:
a:= n->(p->add(coeff(p, x, i)*ceil(2^(i-1)), i=0..degree(p)))(b(n$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 25 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]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*Ceiling[2^(i - 1)], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
Table[If[n==0,1,Sum[2^(Length[ptn]-1),{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,15}] (* Gus Wiseman, Dec 31 2019 *)

N=66; q='q+O('q^N); Vec( 1 + sum(n=1, N, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ) ) \\ Joerg Arndt, Mar 25 2014

a(n) = sum_k A072705(n, k) = A032020(n)-A072707(k) = A032302(n)/2 (n>0).
G.f.: 1/2*(1+Product_{k>0} (1+2*x^k)). - Vladeta Jovovic, Jun 24 2003
G.f.: 1 + sum(n>=1, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Jan 20 2014]
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi)*n^(3/4)), where c = -polylog(2, -2) = A266576 = 1.436746366883680946362902023893583354... - Vaclav Kotesovec, Sep 22 2019

A028859 a(n+2) = 2a(n+1) + 2a(n); a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968

Offset: 0

N. J. A. Sloane

Number of words of length n without adjacent 0's from the alphabet {0,1,2}. For example, a(2) counts 01,02,10,11,12,20,21,22. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 12 2001
Individually, both this sequence and A002605 are convergents to 1+sqrt(3). Mutually, both sequences are convergents to 2+sqrt(3) and 1+sqrt(3)/2. - Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001 [Can someone clarify what is meant by the obscure second phrase, "Mutually..."? - M. F. Hasler, Aug 06 2018]
Add a loop at two vertices of the graph C_3=K_3. a(n) counts walks of length n+1 between these vertices. - Paul Barry, Oct 15 2004
Prefaced with a 1 as (1 + x + 3x^2 + 8x^3 + 22x^4 + ...) = 1 / (1 - x - 2x^2 - 3x^3 - 5x^4 - 8x^5 - 13x^6 - 21x^7 - ...). - Gary W. Adamson, Jul 28 2009
Equals row 2 of the array in A180165, and the INVERTi transform of A125145. - Gary W. Adamson, Aug 14 2010
Pisano period lengths: 1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1, 144, 9, 180, 24, .... - R. J. Mathar, Aug 10 2012
Also the number of independent vertex sets and vertex covers in the n-centipede graph. - Eric W. Weisstein, Sep 21 2017
From Gus Wiseman, May 19 2020: (Start)
Conjecture: Also the number of length n + 1 sequences that cover an initial interval of positive integers and whose non-adjacent parts are weakly decreasing. For example, (3,2,3,1,2) has non-adjacent pairs (3,3), (3,1), (3,2), (2,1), (2,2), (3,2), all of which are weakly decreasing, so is counted under a(11). The a(1) = 1 through a(3) = 8 sequences are:
  (1)  (11)  (111)
       (12)  (121)
       (21)  (211)
             (212)
             (221)
             (231)
             (312)
             (321)
The case of compositions is A333148, or A333150 for strict compositions, or A333193 for strictly decreasing parts. A version for ordered set partitions is A332872. Standard composition numbers of these compositions are A334966. Unimodal normal sequences are A227038. See also: A001045, A001523, A032020, A100471, A100881, A115981, A329398, A332836, A332872.
(End)
Number of 2-compositions of n+1 restricted to parts 1 and 2 (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
The number of ternary strings of length n not containing 00. Complement of A186244. - R. J. Mathar, Feb 13 2022

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 73).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
Joerg Arndt, Matters Computational (The Fxtbook), section 14.9 "Strings with no two consecutive zeros", pp.318-320.
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #12.7.8.
Moussa Benoumhani, On the Modes of the Independence Polynomial of the Centipede, Journal of Integer Sequences, Vol. 15 (2012), #12.5.1.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 Example 7.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. Z. Chinn, R. Grimaldi, and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 1-13.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
J. Shallit, Proof of Irvine's conjecture via mechanized guessing, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.
Eric Weisstein's World of Mathematics, Centipede Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (2,2).

Cf. A180165, A125145, A026150, A030195, A080040, A083337, A106435, A108898.
Cf. A155020 (same sequence with term 1 prepended).
Cf. A002605.

a028859 n = a028859_list !! n
a028859_list =
   1 : 3 : map (* 2) (zipWith (+) a028859_list (tail a028859_list))
-- Reinhard Zumkeller, Oct 15 2011

a[0]:=1:a[1]:=3:for n from 2 to 24 do a[n]:=2*a[n-1]+2*a[n-2] od: seq(a[n],n=0..24); # Emeric Deutsch

a[n_]:=(MatrixPower[{{1,3},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Table[2^(n - 1) Hypergeometric2F1[(1 - n)/2, -n/2, -n, -2], {n, 20}] (* Eric W. Weisstein, Jun 14 2017 *)
LinearRecurrence[{2, 2}, {1, 3}, 20] (* Eric W. Weisstein, Jun 14 2017 *)

a(n)=([1,3;1,1]^n*[2;1])[2,1] \\ Charles R Greathouse IV, Mar 27 2012

A028859(n)=([1,1]*[2,2;1,0]^n)[1] \\ M. F. Hasler, Aug 06 2018

a(n) = a(n-1) + A052945(n) = A002605(n) + A002605(n-1).
G.f.: -(x+1)/(2*x^2+2*x-1).
a(n) = [(1+sqrt(3))^(n+2)-(1-sqrt(3))^(n+2)]/(4*sqrt(3)). - Emeric Deutsch, Feb 01 2005
If p[i]=fibonacci(i+1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n) = 3^n - A186244(n). - Toby Gottfried, Mar 07 2013
E.g.f.: exp(x)*(cosh(sqrt(3)*x) + 2*sinh(sqrt(3)*x)/sqrt(3)). - Stefano Spezia, Mar 02 2024

Definition completed by M. F. Hasler, Aug 06 2018

A227038 Number of (weakly) unimodal compositions of n where all parts 1, 2, ..., m appear where m is the largest part.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 13, 19, 30, 44, 71, 98, 147, 205, 294, 412, 575, 783, 1077, 1456, 1957, 2634, 3492, 4627, 6082, 7980, 10374, 13498, 17430, 22451, 28767, 36806, 46803, 59467, 75172, 94839, 119285, 149599, 187031, 233355, 290340, 360327, 446222, 551251, 679524, 835964, 1026210

Offset: 0

Joerg Arndt, Jun 28 2013

nonn

			There are a(8) = 30 such compositions of 8:
01:  [ 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 2 2 ]
06:  [ 1 1 1 2 1 1 1 ]
07:  [ 1 1 1 2 2 1 ]
08:  [ 1 1 1 2 3 ]
09:  [ 1 1 1 3 2 ]
10:  [ 1 1 2 1 1 1 1 ]
11:  [ 1 1 2 2 1 1 ]
12:  [ 1 1 2 2 2 ]
13:  [ 1 1 2 3 1 ]
14:  [ 1 1 3 2 1 ]
15:  [ 1 2 1 1 1 1 1 ]
16:  [ 1 2 2 1 1 1 ]
17:  [ 1 2 2 2 1 ]
18:  [ 1 2 2 3 ]
19:  [ 1 2 3 1 1 ]
20:  [ 1 2 3 2 ]
21:  [ 1 3 2 1 1 ]
22:  [ 1 3 2 2 ]
23:  [ 2 1 1 1 1 1 1 ]
24:  [ 2 2 1 1 1 1 ]
25:  [ 2 2 2 1 1 ]
26:  [ 2 2 3 1 ]
27:  [ 2 3 1 1 1 ]
28:  [ 2 3 2 1 ]
29:  [ 3 2 1 1 1 ]
30:  [ 3 2 2 1 ]
From _Gus Wiseman_, Mar 05 2020: (Start)
The a(1) = 1 through a(6) = 13 compositions:
  (1)  (11)  (12)   (112)   (122)    (123)
             (21)   (121)   (221)    (132)
             (111)  (211)   (1112)   (231)
                    (1111)  (1121)   (321)
                            (1211)   (1122)
                            (2111)   (1221)
                            (11111)  (2211)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)
(End)

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Composition (combinatorics)
Eric Weisstein's World of Mathematics, Unimodal Sequence
Wikipedia, Unimodality

Cf. A001523 (unimodal compositions), A001522 (smooth unimodal compositions with first and last part 1), A001524 (unimodal compositions such that each up-step is by at most 1 and first part is 1).
Organizing by length rather than sum gives A007052.
The complement is counted by A332743.
The case of run-lengths of partitions is A332577, with complement A332579.
Compositions covering an initial interval are A107429.
Non-unimodal compositions are A115981.
Cf. A000009, A055932, A072704, A317086, A329766, A332578, A332669, A332670.

b:= proc(n,i) option remember;
      `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+
      add(b(n-i*j, i+1)*(j+1), j=1..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i] == 0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 1, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)
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}] (* Gus Wiseman, Mar 05 2020 *)

a(n) ~ c * exp(Pi*sqrt(r*n)) / n, where r = 0.9409240878664458093345791978063..., c = 0.05518035191234679423222212249... - Vaclav Kotesovec, Mar 04 2020

a(n) + A332743(n) = 2^(n - 1). - Gus Wiseman, Mar 05 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[#]]&]

A332834 Number of compositions of n that are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 14, 36, 88, 199, 432, 914, 1900, 3896, 7926, 16036, 32311, 64944, 130308, 261166, 523040, 1046996, 2095152, 4191796, 8385466, 16773303, 33549564, 67102848, 134210298, 268426328, 536859712, 1073728142, 2147466956, 4294947014, 8589909976

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

			The a(4) = 1 through a(6) = 14 compositions:
  (121)  (131)   (132)
         (212)   (141)
         (1121)  (213)
         (1211)  (231)
                 (312)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2112)
                 (2121)
                 (11121)
                 (11211)
                 (12111)

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

The version for unsorted prime signature is A332831.
The version for run-lengths of compositions is A332833.
The complement appears to be counted by A329398.
Unimodal compositions are A001523.
Compositions that are not unimodal are A115981.
Partitions with weakly increasing or decreasing run-lengths are A332745.
Compositions with weakly increasing or decreasing run-lengths are A332835.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A007052, A072704, A107429, A328509, A329744, A332281, A332284, A332578, A332640, A332641, A332643, A332669, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Or[LessEqual@@#,GreaterEqual@@#]&]],{n,0,10}]

a(n)={if(n==0, 0, 2^(n-1) - 2*numbpart(n) + numdiv(n))} \\ Andrew Howroyd, Dec 30 2020

a(n) = 2^(n - 1) - 2 * A000041(n) + A000005(n).

A329398 Number of compositions of n with uniform Lyndon factorization and uniform co-Lyndon factorization.

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 28, 40, 57, 80, 110, 148, 200, 266, 348, 457, 592, 764, 978, 1248, 1580, 2000, 2508, 3142, 3913

Offset: 1

Gus Wiseman, Nov 13 2019

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).
A sequence of words is uniform if they all have the same length.
Conjecture: Also the number of compositions of n that are either weakly increasing or weakly decreasing. Hence a(n) = 2 * A000041(n) - A000005(n). - Gus Wiseman, Mar 05 2020

			The a(1) = 1 through a(6) = 18 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (112)   (41)     (42)
                    (211)   (113)    (51)
                    (1111)  (122)    (114)
                            (221)    (123)
                            (311)    (222)
                            (1112)   (321)
                            (2111)   (411)
                            (11111)  (1113)
                                     (1122)
                                     (2211)
                                     (3111)
                                     (11112)
                                     (21111)
                                     (111111)

Lyndon and co-Lyndon compositions are (both) counted by A059966.
Lyndon compositions that are not weakly increasing are A329141.
Lyndon compositions whose reverse is not co-Lyndon are A329324.
Cf. A000740, A001037, A001523, A008965, A059204, A060223, A211100, A328596, A329312, A329318, A329396, A329397, A329399, A332578, A332669, A332834.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Length/@lynfac[#]&&SameQ@@Length/@colynfac[#]&]],{n,10}]

a(19)-a(25) from Robert Price, Jun 20 2021

A332280 Number of integer partitions of n with unimodal run-lengths.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 97, 129, 166, 215, 273, 352, 439, 557, 692, 865, 1066, 1325, 1614, 1986, 2413, 2940, 3546, 4302, 5152, 6207, 7409, 8862, 10523, 12545, 14814, 17562, 20690, 24397, 28615, 33645, 39297, 46009, 53609, 62504, 72581, 84412

Offset: 0

Gus Wiseman, Feb 18 2020

nonn

First differs from A000041 at a(10) = 41, A000041(10) = 42.

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) = 41 partitions (A = 10) are:
  (A)     (61111)   (4321)     (3211111)
  (91)    (55)      (43111)    (31111111)
  (82)    (541)     (4222)     (22222)
  (811)   (532)     (42211)    (222211)
  (73)    (5311)    (421111)   (2221111)
  (721)   (5221)    (4111111)  (22111111)
  (7111)  (52111)   (3331)     (211111111)
  (64)    (511111)  (3322)     (1111111111)
  (631)   (442)     (331111)
  (622)   (4411)    (32221)
  (6211)  (433)     (322111)
Missing from this list is only (33211).

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 A332281.
Heinz numbers of these partitions are the complement of A332282.
Taking 0-appended first-differences instead of run-lengths gives A332283.
The normal case is A332577.
The opposite version is A332638.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Numbers whose unsorted prime signature is unimodal are A332288.
Cf. A007052, A025065, A072706, A100883, A115981, A227038, A317086, A328509, A329398, A332284, A332285, A332294, A332578, A332579.

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-> 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_] := b[n, n, 0, True];
a /@ Range[0, 65] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

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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A072706 Number of unimodal partitions/compositions of n into distinct terms.

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A028859 a(n+2) = 2*a(n+1) + 2*a(n); a(0) = 1, a(1) = 3.

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A227038 Number of (weakly) unimodal compositions of n where all parts 1, 2, ..., m appear where m is the largest part.

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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.

A028859 a(n+2) = 2a(n+1) + 2a(n); a(0) = 1, a(1) = 3.