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

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

Author

Gus Wiseman, Feb 21 2020

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• MathWorld, Unimodal Sequence

Crossrefs

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.

Programs

Mathematica

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]]]&]