A332832 Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.

12, 20, 24, 28, 36, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 165, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 195, 196, 198

Offset: 1

Author

Gus Wiseman, Mar 02 2020

Keywords

• nonn

Comments

First differs from A065201 in having 165.

First differs from A316597 in having 36.

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

Examples

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   90: {1,2,2,3}
For example, 60 is the Heinz number of (3,2,1,1), with negated 0-appended first-differences (1,1,0,1), which are not unimodal, so 60 is in the sequence.
		

Links

• MathWorld, Unimodal Sequence
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

The non-negated version is A332287.

The version for of run-lengths (instead of differences) is A332642.

The enumeration of these partitions by sum is A332744.

Unimodal compositions are A001523.

Non-unimodal compositions are A115981.

Heinz numbers of partitions with non-unimodal run-lengths are A332282.

Partitions whose 0-appended first differences are unimodal are A332283.

Compositions whose negation is unimodal are A332578.

Compositions whose negation is not unimodal are A332669.

Cf. A059204, A227038, A332284, A332285, A332286, A332578, A332638, A332639, A332670, A332725, A332728.

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[100],!unimodQ[Differences[Prepend[primeMS[#],0]]]&]