A001523 -id:A001523 - cp's OEIS frontend

A332294 Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 1, 3, 1, 4, 3, 4, 1, 6, 1, 5, 4, 8, 1, 9, 1, 8, 5, 6, 1, 12, 4, 7, 9, 10, 1, 12, 1, 16, 6, 8, 5, 18, 1, 9, 7, 16, 1, 15, 1, 12, 12, 10, 1, 24, 5, 16, 8, 14, 1, 27, 6, 20, 9, 11, 1, 24, 1, 12, 15, 32, 7, 18, 1, 16, 10, 20, 1, 36, 1, 13, 16, 18, 6

Offset: 1

Gus Wiseman, Feb 21 2020

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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

			The a(12) = 6 permutations:
  {1,1,2,3}
  {1,1,3,2}
  {1,2,3,1}
  {1,3,2,1}
  {2,3,1,1}
  {3,2,1,1}

MathWorld, Unimodal Sequence

Dominated by A318762.
A less interesting version is A332288.
The complement is counted by A332672.
The opposite/negative version is A332741.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Partitions whose run-lengths are unimodal are A332280.
Cf. A007052, A056239, A112798, A115981, A124010, A304660, A328509, A332283, A332578, A332638, A332671, A332742.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]],unimodQ]],{n,0,30}]

a(n) + A332672(n) = A318762(n).

a(n) = A332288(A181821(n)).

A332579 Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 7, 8, 10, 14, 19, 22, 30, 36, 43, 56, 69, 80, 101, 121, 141, 172, 202, 234, 282, 332, 384, 452, 527, 602, 706, 815, 929, 1077, 1236, 1403, 1615, 1842, 2082, 2379, 2702, 3044, 3458, 3908, 4388, 4963, 5589, 6252

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.

Also the number of strict integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

			The a(10) = 1 through a(16) = 7 partitions:
  33211  332111  3321111  333211    433211     443211      443221
                          33211111  3332111    4332111     3333211
                                    332111111  33321111    4432111
                                               3321111111  33322111
                                                           43321111
                                                           333211111
                                                           33211111111

MathWorld, Unimodal Sequence

The complement is counted by A332577.
Not requiring the partition to cover an initial interval gives A332281.
The opposite version is A332286.
A version for compositions is A332743.
Partitions covering an initial interval of positive integers are A000009.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negated run-lengths are not unimodal are A332727.
Cf. A007052, A100883, A107429, A227038, A332280, A332284, A332638, A332639, A332640, A332671, A332672, 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}]

A056242 Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 16, 8, 1, 14, 41, 44, 16, 1, 20, 85, 146, 112, 32, 1, 27, 155, 377, 456, 272, 64, 1, 35, 259, 833, 1408, 1312, 640, 128, 1, 44, 406, 1652, 3649, 4712, 3568, 1472, 256, 1, 54, 606, 3024, 8361, 14002, 14608, 9312, 3328, 512, 1, 65, 870, 5202

Offset: 1

Colin Mallows, Aug 23 2000

Generalized Riordan array (1/(1-x), x/(1-x) + x*dif(x/1-x),x)). - Paul Barry, Dec 26 2007
Reversal of A117317. - Philippe Deléham, Feb 11 2012
Essentially given by (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 11 2012
This sequence is given in the Strehl presentation with the o.g.f. (1-z)/[1-2(1+t)z+(1+t)z^2], with offset 0, along with a recursion relation, a combinatorial interpretation, and relations to Hermite and Laguerre polynomials. Note that the o.g.f. is related to that of A049310. - Tom Copeland, Jan 08 2017
From Gus Wiseman, Mar 06 2020: (Start)
T(n,k) is also the number of unimodal length-n sequences covering an initial interval of positive integers with maximum part k, where a sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. For example, the sequences counted by row n = 4 are:
  (1111)  (1112)  (1123)  (1234)
          (1121)  (1132)  (1243)
          (1122)  (1223)  (1342)
          (1211)  (1231)  (1432)
          (1221)  (1232)  (2341)
          (1222)  (1233)  (2431)
          (2111)  (1321)  (3421)
          (2211)  (1322)  (4321)
          (2221)  (1332)
                  (2231)
                  (2311)
                  (2321)
                  (2331)
                  (3211)
                  (3221)
                  (3321)
(End)
T(n,k) is the number of hexagonal directed-column convex polyominoes of area n with k columns (see Baril et al. at page 9). - Stefano Spezia, Oct 14 2023

			Triangle begins:
  1;
  1,    2;
  1,    5,    4;
  1,    9,   16,    8;
  1,   14,   41,   44,   16;
  1,   20,   85,  146,  112,   32;
  1,   27,  155,  377,  456,  272,   64;
  1,   35,  259,  833, 1408, 1312,  640,  128;
  1,   44,  406, 1652, 3649, 4712, 3568, 1472,  256;
T(3,2)=5 because we have {1}{23}, {23}{1}, {12}{3}, {3}{12} and {2}{13}.
Triangle (1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   5,   4,   0;
  1,   9,  16,   8,   0;
  1,  14,  41,  44,  16,   0;
  1,  20,  85, 146, 112,  32,   0;
  1,  27, 155, 377, 456, 272,  64,   0;

Reinhard Zumkeller, Rows n = 1..125 of table, flattened
Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, Bijections between Directed-Column Convex Polyominoes and Restricted Compositions, September 29, 2023.
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
Finn Bjarne Jost, Tautological Intersection Numbers and Order-Consecutive Partition Sequences, arXiv:2307.15825 [math.CO], 2023. See p. 9.
V. Strehl, Combinatoire rétrospective et créative, an on-line presentation, slide 36, SLC 71, Bertinoro,, September 18, 2013.
Volker Strehl, Lacunary Laguerre Series from a Combinatorial Perspective, Séminaire Lotharingien de Combinatoire, B76c (2017).

Row sums are A007052.
Column k = n - 1 is A053220.
Ordered set-partitions are A000670.
Cf. A001523, A049310, A072704, A084938, A097805, A117317, A227038, A328509, A332294, A332673, A332724, A332872.

a056242 n k = a056242_tabl !! (n-1)!! (k-1)
a056242_row n = a056242_tabl !! (n-1)
a056242_tabl = [1] : [1,2] : f [1] [1,2] where
   f us vs = ws : f vs ws where
     ws = zipWith (-) (map (* 2) $ zipWith (+) ([0] ++ vs) (vs ++ [0]))
                      (zipWith (+) ([0] ++ us ++ [0]) (us ++ [0,0]))
-- Reinhard Zumkeller, May 08 2014

T:=proc(n,k) if k=1 then 1 elif k<=n then sum((-1)^(k-1-j)*binomial(k-1,j)*binomial(n+2*j-1,2*j),j=0..k-1) else 0 fi end: seq(seq(T(n,k),k=1..n),n=1..12);

rows = 11; t[n_, k_] := (-1)^(k+1)*HypergeometricPFQ[{1-k, (n+1)/2, n/2}, {1/2, 1}, 1]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* Jean-François Alcover, Nov 17 2011 *)

The Hwang and Mallows reference gives explicit formulas.
T(n,k) = Sum_{j=0..k-1} (-1)^(k-1-j)*binomial(k-1, j)*binomial(n+2j-1, 2j) (1<=k<=n); this is formula (11) in the Huang and Mallows reference.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(1,1) = 1, T(2,1) = 1, T(2,2) = 2. - Philippe Deléham, Feb 11 2012
G.f.: -(-1+x)*x*y/(1-2*x-2*x*y+x^2*y+x^2). - R. J. Mathar, Aug 11 2015

A332836 Number of compositions of n whose run-lengths are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 24, 40, 73, 128, 230, 399, 712, 1241, 2192, 3833, 6746, 11792, 20711, 36230, 63532, 111163, 194782, 340859, 596961, 1044748, 1829241, 3201427, 5604504, 9808976, 17170112, 30051470, 52601074, 92063629, 161140256, 282033124, 493637137, 863982135, 1512197655

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

Also compositions whose run-lengths are weakly decreasing.

			The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (121)   (41)
                        (211)   (122)
                        (1111)  (131)
                                (212)
                                (311)
                                (1211)
                                (2111)
                                (11111)
For example, the composition (2,3,2,2,1,1,2,2,2) has run-lengths (1,1,2,2,3) so is counted under a(17).

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

The version for the compositions themselves (not run-lengths) is A000041.
The case of partitions is A100883.
The case of unsorted prime signature is A304678, with dual A242031.
Permitting the run-lengths to be weakly decreasing also gives A332835.
The complement is counted by A332871.
Unimodal compositions are A001523.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Cf. A001462, A072704, A072706, A100882, A181819, A329744, A329766, A332641, A332727, A332745, A332833, A332834.

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

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
seq(n)={my(M=matrix(n+1, n, i, j, i==1)); for(m=1, n, M=step(M, m)); M[1,n]=0; vector(n+1, i, vecsum(M[i,]))/(n-1)} \\ Andrew Howroyd, Dec 31 2020

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

A027349 Number of partitions of n into distinct odd parts, the least being 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 4, 4, 5, 6, 6, 6, 8, 8, 9, 9, 11, 12, 13, 13, 16, 17, 18, 19, 22, 24, 25, 27, 30, 33, 35, 37, 41, 46, 47, 51, 56, 61, 64, 69, 75, 82, 86, 92, 100, 109, 114, 122, 133, 143, 151, 161, 174, 187, 198

Offset: 1

Clark Kimberling

Column 1 of A116860. - Emeric Deutsch, Feb 27 2006
Also number of partitions of n such that the largest part occurs exactly once and each number smaller than the largest part occurs an even nonzero number of times. Example: a(17)=3 because we have [3,2,2,2,2,2,2,1,1],[3,2,2,2,2,1,1,1,1,1,1] and [3,2,2,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 13 2006
a(n) is the number of symmetric stack polyominoes of area n with square core. The core of a stack is the set of all maximal columns. The core is a square when the number of columns is equal to their height. Equivalently, a(n) is the number of symmetric unimodal compositions of n, where the number of the parts of maximum value equal the maximum value itself. For instance, for n = 20, we have the following stacks: (2,4,4,4,4,2), (1,1,4,4,4,4,1,1), (1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1). - Emanuele Munarini, Apr 08 2011

			a(17) = 3 because we have [13,3,1], [11,5,1] and [9,7,1].
G.f. = x + x^4 + x^6 + x^8 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + x^14 + 2*x^15 + ...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Abdulaziz Alanazi, Augustine O. Munagi, and Andrew V. Sills, "Overpartitionized" Rogers-Ramanujan type identities, arXiv:2501.17826 [math.CO], 2025. See pp. 5, 13.
Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=1. - _N. J. A. Sloane_, Aug 31 2014
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. A000700, A116860, A001523, A096441, A188674.

N := 100; t1 := series(mul(1+x^(2*k+1),k=1..N),x,N); A027349 := proc(n) coeff(t1,x,n); end;

a[n_]:=CoefficientList[Series[1+Sum[x^((k+1)^2)/Product[(1-x^(2i)),{i,1,k}],{k,0,n}],{x,0,n}],x] (* Emanuele Munarini, Apr 08 2011 *)
a[ n_] := SeriesCoefficient[ x QHypergeometricPFQ[ {}, {}, x^2, -x^3], {x, 0, n}]; (* Michael Somos, Feb 02 2015 *)
nmax = 100; Rest[CoefficientList[Series[x/(1+x) * Product[1+x^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)
(2/((1 + x) QPochhammer[-1, -x]) + O[x]^70)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)

G.f.: x*Product_{i>=2} 1+x^(2*i-1). - Emeric Deutsch, Feb 27 2006
G.f.: (Sum_{k>=1} x^(k^2))/Product_{j=1..k-1} 1-x^(2*j). - Emeric Deutsch, Mar 13 2006
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(11/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015
G.f.: 2/((1 + x)*(-1; -x)inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 22 2016
If n > 1, a(n) = A000700(n - 1) - a(n - 1). - Álvar Ibeas, Aug 03 2020
G.f.: x*Sum_{n >= 0} x^(n*(n+2))/Product_{k = 1..n} (1 - x^(2*k)) = x*(1 + x^3) * Sum_{n >= 0} x^(n*(n+4))/Product_{k = 1..n} (1 - x^(2*k)) = x*(1 + x^3)*(1 + x^5) * Sum_{n >= 0} x^(n*(n+6))/ Product_{k = 1..n} (1 - x^(2*k)) = .... - Peter Bala, Jan 15 2021

A059618 Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing).

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 10, 15, 21, 30, 43, 59, 82, 111, 148, 199, 263, 344, 451, 584, 751, 965, 1230, 1560, 1973, 2483, 3110, 3885, 4834, 5990, 7405, 9123, 11202, 13724, 16762, 20417, 24815, 30081, 36377, 43900, 52860, 63511, 76166, 91157, 108886, 129842

Offset: 0

Henry Bottomley, Jan 31 2001

			a(6) = 10 since 6 can be written as 6, 5+1, 4+2, 3+2+1, 2+4, 2+3+1, 1+5, 1+4+1, 1+3+2 or 1+2+3 (but for example neither 2+2+1+1 nor 1+2+2+1 which are only weakly unimodal).
From _Joerg Arndt_, Dec 09 2012: (Start)
The a(7) = 15 strongly unimodal compositions of 7 are
[ #]   composition
[ 1]   [ 1 2 3 1 ]
[ 2]   [ 1 2 4 ]
[ 3]   [ 1 3 2 1 ]
[ 4]   [ 1 4 2 ]
[ 5]   [ 1 5 1 ]
[ 6]   [ 1 6 ]
[ 7]   [ 2 3 2 ]
[ 8]   [ 2 4 1 ]
[ 9]   [ 2 5 ]
[10]   [ 3 4 ]
[11]   [ 4 2 1 ]
[12]   [ 4 3 ]
[13]   [ 5 2 ]
[14]   [ 6 1 ]
[15]   [ 7 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms

Cf. A000009, A000041, A001523, A059607, A059619.

b:= proc(n, i, t) option remember; `if`(t=0 and n>i*(i-1)/2, 0,
      `if`(n=0, 1, add(b(n-j, j, 0), j=1..min(n, i-1))+
      `if`(t=1, add(b(n-j, j, 1), j=i+1..n), 0)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 21 2014

s[n_?Positive, k_] := s[n, k] = Sum[s[n - k, j], {j, 0, k - 1}]; s[0, 0] = 1; s[0, ] = 0; s[?Negative, ] = 0; t[n, k_] := t[n, k] = s[n, k] + Sum[t[n - k, j], {j, k + 1, n}]; a[n_] := t[n, 0]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Dec 06 2012, after Vladeta Jovovic *)

N=66; x='x+O('x^N); Vec(sum(n=0,N,x^(n) * prod(k=1,n-1,1+x^k)^2)) \\ Joerg Arndt, Mar 26 2014

a(n) = A059619(n,0) = Sum_k A059619(n,k) for k>0 when n>0.

G.f.: sum(k>=0, x^k * prod(i=1..k-1, 1 + x^i)^2 ). - Vladeta Jovovic, Dec 05 2003

A186085 Number of 1-dimensional sandpiles with n grains.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 13, 22, 36, 60, 100, 166, 277, 461, 769, 1282, 2137, 3565, 5945, 9916, 16540, 27589, 46022, 76769, 128062, 213628, 356366, 594483, 991706, 1654352, 2759777, 4603843, 7680116, 12811951, 21372882, 35654237, 59478406, 99221923, 165522118, 276124217, 460630839

Offset: 0

Joerg Arndt, Feb 12 2011

Number of compositions of n where the first and the last parts are 1 and the absolute difference between consecutive parts is <=1 (smooth compositions).
Such a composition [c1,c2,c3,...] corresponds to a sandpile with c1(=1) grains in the first positions, c2 in the second, and so on.  Assuming the critical slope is 1 (for the pile to be stable) we obtain the conditions on the compositions.
With the additional requirement of unimodality one gets A001522. [Joerg Arndt, Dec 09 2012]
Dropping the requirement that the first and last parts are 1 gives A034297. Restriction to weakly increasing (or decreasing) sums gives A034296. [Joerg Arndt, Jun 02 2013]
Also the number of compositions of n with first part 1, up-steps of at most 1, and no two consecutive up-steps.  The sandpiles are recovered by shifting the rows above the bottom row to the left by one position relative to the next lower row. [Joerg Arndt, Mar 30 2014]
Also fountains of coins (cf. A005169) with no consecutive up-steps. Shift the top rows in the previous comment by half a position. [Joerg Arndt, Mar 30 2014]

			The a(7)=8 smooth compositions of 7 are:
:   1:      [ 1 1 1 1 1 1 1 ]  (composition)
:
: ooooooo  (rendering of sandpile)
:
:   2:      [ 1 1 1 1 2 1 ]
:
:     o
: oooooo
:
:   3:      [ 1 1 1 2 1 1 ]
:
:    o
: oooooo
:
:   4:      [ 1 1 2 1 1 1 ]
:
:   o
: oooooo
:
:   5:      [ 1 1 2 2 1 ]
:
:   oo
: ooooo
:
:   6:      [ 1 2 1 1 1 1 ]
:
:  o
: oooooo
:
:   7:      [ 1 2 1 2 1 ]
:
:  o o
: ooooo
:
:   8:      [ 1 2 2 1 1 ]
:
:  oo
: ooooo

Seiichi Manyama, Table of n, a(n) for n = 0..4502 (terms 0..1000 from Alois P. Heinz)

Cf. A186084 (sandpiles by base length).
Cf. A005169 (compositions of n with c(1)=1 and c(i+1)<=c(i)+1).
Cf. A186505 (antidiagonal sums of triangle A186084).
Cf. A001522, A001523, A001524.
Cf. A129181.

b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),
      `if`(n<0 or i<1, 0, add(b(n-i, i+j), j=-1..1)))
    end:
a:= n-> `if`(n=0, 1, b(n-1, 1)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 11 2013

b[n_, i_] := b[n, i] = If[n == 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, Sum[b[n-i, i+j], {j, -1, 1}]]]; a[n_] := If[n == 0, 1, b[n-1, 1]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)

{a(n)=local(Txy=1+x*y); for(i=1, n, Txy=1/(1-x*y-x^3*y^2*subst(Txy, y, x*y+x*O(x^n)))); polcoeff(subst(1+x*Txy, y, 1), n, x)} /* Paul D. Hanna */

/* continued fraction for terms up to 460630839: */
Vec(1/ (1-x/ (1-x^3/ (1-x^2/ (1-x^3/ (1-x^7/ (1-x^4/ (1-x^5/ (1-x^11/ (1-x^6/(1-x*O(x^0) ))))))))))) /* Paul D. Hanna */

N = 66; x = 'x + O('x^N);
Q(k) = if(k>N, 1, 1/x^(k+1) - 1 - 1/Q(k+1) );
gf = 1 + 1/Q(0);
Vec(gf) /* Joerg Arndt, May 07 2013 */

G.f.: 1 + x/(1-x - x^3*B(x)) where B(x) equals the g.f. of the antidiagonal sums of triangle A186084 [Paul D. Hanna].
G.f.: 1 + x/(1-x - x^3/(1-x^2 - x^5/(1-x^3 - x^7/(1-x^4 - x^9/(1 -...))))) (continued fraction).  [Paul D. Hanna].
G.f.: 1/(1 - x/(1-x^3/(1-x^2/(1 - x^3/(1-x^7/(1-x^4/(1 - x^5/(1-x^11/(1-x^6/(1 -...)))))))))) (continued fraction).  [Paul D. Hanna].
The g.f. T(x,y) of triangle A186084 satisfies: T(x,y) = 1/(1 - x*y - x^3*y^2*T(x,x*y)); therefore, the g.f. of this sequence is A(x) = 1 + x*T(x,1). [Paul D. Hanna]
a(n) ~ c/r^n where r = 0.5994477646147968266874606710272382... and c = 0.213259838728143595595398989847345... [Paul D. Hanna]
G.f.: 1 + 1/Q(0), where Q(k)= 1/x^(k+1) - 1 - 1/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 07 2013
G.f.: G(1), where G(k) = 1 + x^k/( 1 - x^k * G(k+1) ) (continued fraction). [Joerg Arndt, Jun 29 2013]
a(n) = Sum_{j=1..n} A129181(n-j,j-1) for n>=1. - Alois P. Heinz, Jun 25 2023

A247255 Triangular array read by rows: T(n,k) is the number of weakly unimodal partitions of n in which the greatest part occurs exactly k times, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 6, 1, 0, 1, 12, 2, 0, 0, 1, 21, 4, 1, 0, 0, 1, 38, 6, 2, 0, 0, 0, 1, 63, 11, 3, 1, 0, 0, 0, 1, 106, 16, 5, 2, 0, 0, 0, 0, 1, 170, 27, 7, 3, 1, 0, 0, 0, 0, 1, 272, 40, 11, 4, 2, 0, 0, 0, 0, 0, 1, 422, 63, 16, 6, 3, 1, 0, 0, 0, 0, 0, 1, 653, 92, 24, 8, 4, 2, 0, 0, 0, 0, 0, 0, 1, 986, 141, 34, 12, 5, 3, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Geoffrey Critzer, Nov 29 2014

These are called stack polyominoes in the Flajolet and Sedgewick reference.

			    1;
    1,  1;
    3,  0, 1;
    6,  1, 0, 1;
   12,  2, 0, 0, 1;
   21,  4, 1, 0, 0, 1;
   38,  6, 2, 0, 0, 0, 1;
   63, 11, 3, 1, 0, 0, 0, 1;
  106, 16, 5, 2, 0, 0, 0, 0, 1;
  170, 27, 7, 3, 1, 0, 0, 0, 0, 1;

P. Flajolet and R Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 46.

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=1-10 give: A006330, A114921, A226541, A320315, A320316, A320317, A320318, A320319, A320320, A320321.
Row sums give A001523.
Main diagonal gives A000012.

b:= proc(n, i) option remember; local r; expand(
      `if`(i>n, 0, `if`(irem(n, i, 'r')=0, x^r, 0)+
      add(b(n-i*j, i+1)*(j+1), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..14);  # Alois P. Heinz, Nov 29 2014

nn = 14; Table[
  Take[Drop[
     CoefficientList[
      Series[ Sum[
        u z^k/(1 - u z^k) Product[1/(1 - z^i), {i, 1, k - 1}]^2, {k,
         1, nn}], {z, 0, nn}], {z, u}], 1], n, {2, n + 1}][[n]], {n,
   1, nn}] // Grid

G.f.: Sum_{k>=1} y*x^k/(1 - y*x^k)/(Product_{i=1..k-1} (1 - x^i))^2.

For fixed k>=1, T(n,k) ~ Pi^(k-1) * (k-1)! * exp(2*Pi*sqrt(n/3)) / (2^(k+2) * 3^(k/2 + 1/4) * n^(k/2 + 3/4)). - Vaclav Kotesovec, Oct 24 2018

A332727 Number of compositions of n whose run-lengths are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 8, 28, 74, 188, 468, 1120, 2596, 5944, 13324, 29437, 64288, 138929, 297442, 632074, 1333897, 2798352, 5840164, 12132638, 25102232, 51750419, 106346704, 217921161, 445424102, 908376235, 1848753273, 3755839591, 7617835520, 15428584567, 31207263000

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

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

			The a(6) = 1 through a(8) = 8 compositions:
  (11211)  (11311)   (11411)
           (111211)  (111311)
           (112111)  (112112)
                     (113111)
                     (211211)
                     (1111211)
                     (1112111)
                     (1121111)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not its run-lengths) gives A115981.
The case of partitions is A332281, with complement counted by A332280.
The complement is counted by A332726.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negation is not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A007052, A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332578, A332638, A332639, A332670, A332741, A332833.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[Length/@Split[#]]&]],{n,0,10}]

a(n) + A332726(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A332742 Number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 3, 2, 0, 8, 0, 3, 7, 16, 0, 24, 0, 16, 12, 4, 0, 52, 16, 5, 81, 26, 0, 54, 0, 104, 18, 6, 31, 168, 0, 7, 25, 112, 0, 99, 0, 38, 201, 8, 0, 344, 65, 132, 33, 52, 0, 612, 52, 202, 42, 9, 0, 408, 0, 10, 411, 688, 80, 162, 0, 68, 52, 272

Offset: 1

Gus Wiseman, Mar 09 2020

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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

			The a(n) permutations for n = 6, 8, 9, 10, 12, 14, 15, 16:
  121  132  1212  1121  1132  11121  11212  1243
       231  1221  1211  1213  11211  11221  1324
            2121        1231  12111  12112  1342
                        1312         12121  1423
                        1321         12211  1432
                        2131         21121  2143
                        2311         21211  2314
                        3121                2341
                                            2413
                                            2431
                                            3142
                                            3241
                                            3412
                                            3421
                                            4132
                                            4231

Eric Weisstein's World of Mathematics, Unimodal Sequence

Dominated by A318762.
The complement of the non-negated version is counted by A332294.
The non-negated version is A332672.
The complement is counted by A332741.
A less interesting version is A333146.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal 0-appended first differences are A332284.
Compositions whose negation is unimodal are A332578.
Partitions with non-unimodal negated run-lengths are A332639.
Numbers whose negated prime signature is not unimodal are A332642.
Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332283, A332288, A332638, A332669, A333145.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]],!unimodQ[#]&]],{n,30}]

a(n) + A332741(n) = A318762(n).

cp's OEIS Frontend

A332294 Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.

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A332579 Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.

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A056242 Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).

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A332836 Number of compositions of n whose run-lengths are weakly increasing.

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A027349 Number of partitions of n into distinct odd parts, the least being 1.

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A059618 Number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing).

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A186085 Number of 1-dimensional sandpiles with n grains.

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