A332294 -id:A332294 - 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

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 *)

A332287 Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

36, 50, 70, 72, 98, 100, 108, 140, 144, 154, 180, 182, 196, 200, 216, 225, 242, 250, 252, 280, 286, 288, 294, 300, 308, 324, 338, 350, 360, 363, 364, 374, 392, 396, 400, 418, 429, 432, 441, 442, 450, 462, 468, 484, 490, 494, 500, 504, 507, 540, 550, 560, 561

Offset: 1

Gus Wiseman, Feb 21 2020

nonn

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), which gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   36: {1,1,2,2}
   50: {1,3,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   98: {1,4,4}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}
  154: {1,4,5}
  180: {1,1,2,2,3}
  182: {1,4,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  216: {1,1,1,2,2,2}
  225: {2,2,3,3}
  242: {1,5,5}
  250: {1,3,3,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
For example, the prime indices of 70 with 0 appended are (4,3,1,0), with differences (-1,-2,-1), which is not unimodal, so 70 belongs to the sequence.

MathWorld, Unimodal Sequence

The enumeration of these partitions by sum is A332284.
Not assuming the last part is zero gives A332725.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Cf. A001523, A007052, A332280, A332282, A332283, A332285, A332286, A332288, A332294, A332579, A332639, A332642.

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[1000],!unimodQ[Differences[Append[Reverse[primeMS[#]],0]]]&]

A332288 Number of unimodal permutations of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 2, 2, 1, 6, 1, 2, 3, 1, 2, 4, 1, 3, 2, 4, 1, 4, 1, 2, 2, 3, 2, 4, 1, 5, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 22 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.
Also permutations of the multiset of prime indices of n avoiding the patterns (2,1,2), (2,1,3), and (3,1,2).

			The a(n) permutations for n = 2, 6, 12, 24, 48, 60, 120, 180:
  (1)  (12)  (112)  (1112)  (11112)  (1123)  (11123)  (11223)
       (21)  (121)  (1121)  (11121)  (1132)  (11132)  (11232)
             (211)  (1211)  (11211)  (1231)  (11231)  (11322)
                    (2111)  (12111)  (1321)  (11321)  (12231)
                            (21111)  (2311)  (12311)  (12321)
                                     (3211)  (13211)  (13221)
                                             (23111)  (22311)
                                             (32111)  (23211)
                                                      (32211)

Wikipedia, Permutation pattern
MathWorld, Unimodal Sequence

Dominated by A008480.
A more interesting version is A332294.
The complement is counted by A332671.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Numbers with non-unimodal unsorted prime signature are A332282.
Partitions with unimodal 0-appended first differences are A332283.
Cf. A056239, A112798, A115981, A124010, A227038, A304660, A328509, A332280, A332284, A332294, A332578, A332672.

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}]

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

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).

A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

Original entry on oeis.org

2100, 3300, 3900, 4200, 4410, 5100, 5700, 6468, 6600, 6900, 7644, 7800, 8400, 8700, 9300, 9996, 10200, 10500, 10780, 10890, 11100, 11172, 11400, 12300, 12740, 12900, 12936, 13200, 13230, 13524, 13800, 14100, 15210, 15246, 15288, 15600, 15900, 16500, 16660

Offset: 1

Gus Wiseman, Feb 28 2020

nonn

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

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The sequence of terms together with their prime indices begins:
   2100: {1,1,2,3,3,4}
   3300: {1,1,2,3,3,5}
   3900: {1,1,2,3,3,6}
   4200: {1,1,1,2,3,3,4}
   4410: {1,2,2,3,4,4}
   5100: {1,1,2,3,3,7}
   5700: {1,1,2,3,3,8}
   6468: {1,1,2,4,4,5}
   6600: {1,1,1,2,3,3,5}
   6900: {1,1,2,3,3,9}
   7644: {1,1,2,4,4,6}
   7800: {1,1,1,2,3,3,6}
   8400: {1,1,1,1,2,3,3,4}
   8700: {1,1,2,3,3,10}
   9300: {1,1,2,3,3,11}
   9996: {1,1,2,4,4,7}
  10200: {1,1,1,2,3,3,7}
  10500: {1,1,2,3,3,3,4}
  10780: {1,1,3,4,4,5}
  10890: {1,2,2,3,5,5}

MathWorld, Unimodal Sequence

Not requiring non-unimodal negation gives A332282.
These are the Heinz numbers of the partitions counted by A332640.
Not requiring non-unimodality gives A332642.
The case of compositions is A332870.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unsorted prime signature is A124010.
Non-unimodal normal sequences are A328509.
Partitions whose 0-appended first differences are unimodal are A332283, with Heinz numbers the complement of A332287.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Partitions whose 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
Numbers whose signature is neither increasing nor decreasing are A332831.
Cf. A007052, A056239, A072704, A112798, A242031, A242414, A332280, A332281, A332288, A332294, A332639, A332728, A332742.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Select[Range[10000],!unimodQ[Last/@FactorInteger[#]]&&!unimodQ[-Last/@FactorInteger[#]]&]

Intersection of A332282 and A332642.

A332671 Number of non-unimodal 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, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 6, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 22 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 = 18, 30, 36, 42, 50, 54, 60, 66, 70, 72:
  212  213  1212  214  313  2122  1213  215  314  11212
       312  2112  412       2212  1312  512  413  12112
            2121                  2113            12121
                                  2131            21112
                                  3112            21121
                                  3121            21211

MathWorld, Unimodal Sequence

Dominated by A008480.
The complement is counted by A332288.
A more interesting version is A332672.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Compositions whose negation is not unimodal are A332669.
Cf. A007052, A056239, A112798, A124010, A332281, A332284, A332287, A332294, A332639, A332642.

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,100}]

a(n) + A332288(n) = A008480(n).

a(A181821(n)) = A332672(n).

A332672 Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 6, 16, 0, 21, 0, 12, 10, 0, 0, 48, 16, 0, 81, 20, 0, 48, 0, 104, 15, 0, 30, 162, 0, 0, 21, 104, 0, 90, 0, 30, 198, 0, 0, 336, 65, 124, 28, 42, 0, 603, 50, 190, 36, 0, 0, 396, 0, 0, 405, 688, 77, 150, 0, 56, 45, 260, 0

Offset: 1

Gus Wiseman, Feb 23 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 = 8, 9, 12, 15, 16:
  213   1212   1213   11212   1324
  312   2112   1312   12112   1423
        2121   2113   12121   2134
               2131   21112   2143
               3112   21121   2314
               3121   21211   2413
                              3124
                              3142
                              3214
                              3241
                              3412
                              4123
                              4132
                              4213
                              4231
                              4312

MathWorld, Unimodal Sequence

Positions of zeros are one and A001751.
Support is A264828 without one.
Dominated by A318762.
The complement is counted by A332294.
A less interesting version is A332671.
The opposite version is A332742.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Compositions whose negation is not unimodal are A332669.
Cf. A007052, A008480, A056239, A112798, A124010, A181819, A181821, A332281, A332287, A332294, A332642, A332741.

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) = A332671(A181821(n)).

a(n) + A332294(n) = A318762(n).

A332741 Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 8, 4, 2, 9, 4, 1, 6, 1, 16, 3, 2, 4, 12, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 16, 5, 8, 3, 4, 1, 18, 4, 8, 3, 2, 1, 12, 1, 2, 9, 32, 4, 6, 1, 4, 3, 8, 1, 24, 1, 2, 12, 4, 5, 6, 1, 16, 27, 2, 1

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

			The a(12) = 4 permutations:
  {1,1,2,3}
  {2,1,1,3}
  {3,1,1,2}
  {3,2,1,1}

Eric Weisstein's World of Mathematics, Unimodal Sequence

Dominated by A318762.
The non-negated version is A332294.
The complement is counted by A332742.
A less interesting version is A333145.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Numbers with non-unimodal negated prime signature are A332642.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Partitions with unimodal negated run-lengths are A332638.
Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332288, A332639, A332669, A332672.

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) + A332742(n) = A318762(n).

cp's OEIS Frontend

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

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

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A332287 Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.

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A332288 Number of unimodal permutations of the multiset of prime indices of n.

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

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A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

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