A329398 -id:A329398 - cp's OEIS frontend

A001523 Number of stacks, or planar partitions of n; also weakly unimodal compositions of n.

1, 1, 2, 4, 8, 15, 27, 47, 79, 130, 209, 330, 512, 784, 1183, 1765, 2604, 3804, 5504, 7898, 11240, 15880, 22277, 31048, 43003, 59220, 81098, 110484, 149769, 202070, 271404, 362974, 483439, 641368, 847681, 1116325, 1464999, 1916184, 2498258, 3247088, 4207764

Offset: 0

N. J. A. Sloane

a(n) counts stacks of integer-length boards of total length n where no board overhangs the board underneath.
Number of graphical partitions on 2n nodes that contain a 1. E.g. a(3)=4 and so there are 4 graphical partitions of 6 that contain a 1, namely (111111), (21111), (2211) and (3111). Only (222) fails. - Jon Perry, Jul 25 2003
It would seem from Stanley that he regards a(0)=0 for this sequence and A001522. - Michael Somos, Feb 22 2015
In the article by Auluck is a typo in the formula (24), 2*Pi is missing in an exponent on the left side of the equation for Q(n). The correct term is exp(2*Pi*sqrt(n/3)), not just exp(sqrt(n/3)). - Vaclav Kotesovec, Jun 22 2015

			For a(4)=8 we have the following stacks:
x
x x. .x
x x. .x x.. .x. ..x xx
x xx xx xxx xxx xxx xx xxxx
G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 27*x^6 + 47*x^7 + 79*x^8 + ...
From _Gus Wiseman_, Mar 04 2020: (Start)
The a(1) = 1 through a(5) = 15 unimodal compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (112)   (41)
                    (121)   (113)
                    (211)   (122)
                    (1111)  (131)
                            (221)
                            (311)
                            (1112)
                            (1121)
                            (1211)
                            (2111)
                            (11111)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686, g(x).
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
H. Bottomley, Illustration of initial terms
Shouvik Datta, Matthias R. Gaberdiel, Wei Li, and Cheng Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, U(1,z).
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 46.
Rigoberto Flórez, José L. Ramírez, and Diego Villamizar, Restricted bargraphs and unimodal compositions, J. Comb. Theory, Series A, (2024) Vol. 208, Art. No. 105934.
R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms
Alan D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Unimodal Sequence
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A054250, A059618, A059623, A001522, A001524.
Cf. A000569. Bisections give A100505, A100506.
Row sums of A247255.
Row sums of A072704.
The strict case is A072706.
The complement is counted by A115981.
The case covering an initial interval is A227038.
The version whose negation is unimodal as well appears to be A329398.
Unimodal sequences covering an initial interval are A007052.
Non-unimodal permutations are A059204.
Non-unimodal sequences covering an initial interval are A328509.
Partitions with unimodal run-lengths are A332280.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
The number of unimodal permutations of the prime indices of n is A332288.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Cf. A107429, A156253, A332285, A332294, A332577, A332642, A332669.

m:=100;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1 + (&+[ x^n*(1-x^n)/(&*[(1-x^j)^2: j in [1..n]]): n in [1..m+2]]) )); // G. C. Greubel, Apr 03 2023

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=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

max = 40; s = 1 + Sum[(-1)^(k + 1)*q^(k*(k + 1)/2), {k, 1, max}] / QPochhammer[q]^2 + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Jan 25 2012, updated Nov 29 2015 *)
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, 0, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 24 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],unimodQ[#]&]],{n,0,10}] (* Gus Wiseman, Mar 04 2020 *)

{a(n) = if( n<1, n==0, polcoeff( sum(k=1, (sqrt(1 + 8*n) - 1)\2, -(-1)^k * x^((k + k^2)/2)) / eta(x + x * O(x^n))^2 ,n))}; /* Michael Somos, Jul 22 2003 */

def b(n, i):
    if i>n: return 0
    if n%i==0: x=1
    else: x=0
    return x + sum([b(n - i*j, i + 1)*(j + 1) for j in range(n//i + 1)])
def a(n): return 1 if n==0 else b(n, 1) # Indranil Ghosh, Jun 09 2017, after Maple code by Alois P. Heinz

a(n) = Sum_{k=1..n} f(k, n-k), where f(n, k) (= A054250) = 1 if k = 0; Sum_{j=1..min(n, k)} (n-j+1)*f(j, k-j) if k > 0. - David W. Wilson, May 05 2000
a(n) = Sum_{k} A059623(n, k) for n > 0. - Henry Bottomley, Feb 01 2001
A006330(n) + a(n) = A000712(n). - Michael Somos, Jul 22 2003
G.f.: 1 + (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k))^2. - Michael Somos, Jul 22 2003
G.f.: 1 + Sum_{n>=1} (x^n / ( ( Product_{k=1..n-1} (1 - x^k)^2 ) * (1-x^n) ) ). - Joerg Arndt, Oct 01 2012
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015
a(n) + A115981(n) = 2^(n - 1). - Gus Wiseman, Mar 04 2020

More terms from David W. Wilson, May 05 2000

Definition corrected by Wolfdieter Lang, Dec 05 2018

A115981 The number of compositions of n which cannot be viewed as stacks.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 17, 49, 126, 303, 694, 1536, 3312, 7009, 14619, 30164, 61732, 125568, 254246, 513048, 1032696, 2074875, 4163256, 8345605, 16717996, 33473334, 66998380, 134067959, 268233386, 536599508, 1073378850, 2147000209

Offset: 0

Alford Arnold, Feb 12 2006

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. - Gus Wiseman, Mar 05 2020

			a(5) = 1 counting {212}.
a(6) = 5 counting {1212, 2112,2121,213,312}.
a(7) = 17 counting {11212, 12112,12121, 21211, 21121, 21112, 2122, 2212, 2113, 3112, 2131, 3121, 1213, 1312, 412, 214, 313}.
a(8) = 49 = 128 - 79.
a(9) = 126 = 256 - 130.

Eric Weisstein's World of Mathematics, Unimodal Sequence

Cf. A011782, A115982.
The complement is counted by A001523.
The strict case is A072707.
The case covering an initial interval is A332743.
The version whose negation is not unimodal either is A332870.
Non-unimodal permutations are A059204.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are not unimodal are A332284.
Non-unimodal permutations of the prime indices of n are A332671.
Cf. A007052, A072704, A227038, A329398, A332280, A332283, A332672, A332578, A332669, 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],!unimodQ[#]&]],{n,0,10}] (* Gus Wiseman, Mar 05 2020 *)

a(n) = A011782(n) - A001523(n).

More terms from Brian Kuehn (brk158(AT)psu.edu), Apr 20 2006

a(25) corrected by Georg Fischer, Jun 29 2021

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

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

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

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).

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 *)

A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 8, 27, 75, 185, 441, 1025, 2276, 4985, 10753, 22863, 48142, 100583, 208663, 430563, 884407, 1809546, 3690632, 7506774, 15233198, 30851271, 62377004, 125934437, 253936064, 511491634, 1029318958, 2069728850, 4158873540, 8351730223, 16762945432

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) = 3 and a(7) = 8 compositions:
  (1221)   (2113)
  (2112)   (3112)
  (11211)  (11311)
           (12112)
           (21112)
           (21121)
           (111211)
           (112111)

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

The case of partitions is A332641.
The version for unsorted prime signature is A332831.
The version for the compositions themselves (not run-lengths) is A332834.
The complement is counted by A332835.
Unimodal compositions are A001523.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Compositions whose run-lengths are not unimodal are A332727.
Partitions with weakly increasing or weakly decreasing run-lengths: A332745.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A072704, A072706, A107429, A181819, A329398, A329744, A329746, A329766, A332273, A332640, A332746.

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

a(n) = 2^(n - 1) - 2 * A332836(n) + A329738(n).

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

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

A329395 Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 13, 15, 16, 22, 25, 31, 32, 36, 42, 46, 49, 53, 59, 63, 64, 76, 82, 94, 97, 109, 115, 127, 128, 136, 148, 156, 162, 166, 169, 170, 172, 181, 182, 190, 193, 201, 202, 211, 213, 214, 217, 221, 227, 235, 247, 255, 256, 280, 292, 306, 308

Offset: 1

Gus Wiseman, Nov 13 2019

nonn

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).
Conjecture: also numbers k such that the k-th composition in standard order (A066099) is a palindrome, cf. A025065, A242414, A317085, A317086, A317087, A335373. - Gus Wiseman, Jun 06 2020

			The sequence of terms together with their trimmed binary expansions and their co-Lyndon and Lyndon factorizations begins:
   1:      () =               0 = 0
   2:     (0) =             (0) = (0)
   3:     (1) =             (1) = (1)
   4:    (00) =          (0)(0) = (0)(0)
   7:    (11) =          (1)(1) = (1)(1)
   8:   (000) =       (0)(0)(0) = (0)(0)(0)
  10:   (010) =         (0)(10) = (01)(0)
  13:   (101) =         (10)(1) = (1)(01)
  15:   (111) =       (1)(1)(1) = (1)(1)(1)
  16:  (0000) =    (0)(0)(0)(0) = (0)(0)(0)(0)
  22:  (0110) =        (0)(110) = (011)(0)
  25:  (1001) =        (100)(1) = (1)(001)
  31:  (1111) =    (1)(1)(1)(1) = (1)(1)(1)(1)
  32: (00000) = (0)(0)(0)(0)(0) = (0)(0)(0)(0)(0)
  36: (00100) =     (0)(0)(100) = (001)(0)(0)
  42: (01010) =     (0)(10)(10) = (01)(01)(0)
  46: (01110) =       (0)(1110) = (0111)(0)
  49: (10001) =       (1000)(1) = (1)(0001)
  53: (10101) =     (10)(10)(1) = (1)(01)(01)
  59: (11011) =     (110)(1)(1) = (1)(1)(011)
  63: (11111) = (1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)

Lyndon and co-Lyndon compositions are (both) counted by A059966.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Cf. A001037, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329394, A329398.
Cf. A329358, A329399, A329400, A329401.

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, #]]&]]];
Select[Range[100],Length[lynfac[Rest[IntegerDigits[#,2]]]]==Length[colynfac[Rest[IntegerDigits[#,2]]]]&]

cp's OEIS Frontend

A001523 Number of stacks, or planar partitions of n; also weakly unimodal compositions of n.

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A115981 The number of compositions of n which cannot be viewed as stacks.

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A332578 Number of compositions of n whose negation is unimodal.

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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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A332669 Number of compositions of n whose negation is not unimodal.

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

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