A332725 Heinz numbers of integer partitions whose negated first differences are not unimodal.

90, 126, 180, 198, 234, 252, 270, 306, 342, 350, 360, 378, 396, 414, 450, 468, 504, 522, 525, 540, 550, 558, 594, 612, 630, 650, 666, 684, 700, 702, 720, 738, 756, 774, 792, 810, 825, 828, 846, 850, 882, 900, 910, 918, 936, 950, 954, 975, 990, 1008, 1026, 1044

Offset: 1

Author

Gus Wiseman, Feb 26 2020

Keywords

• nonn

Comments

A sequence of positive 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:
    90: {1,2,2,3}
   126: {1,2,2,4}
   180: {1,1,2,2,3}
   198: {1,2,2,5}
   234: {1,2,2,6}
   252: {1,1,2,2,4}
   270: {1,2,2,2,3}
   306: {1,2,2,7}
   342: {1,2,2,8}
   350: {1,3,3,4}
   360: {1,1,1,2,2,3}
   378: {1,2,2,2,4}
   396: {1,1,2,2,5}
   414: {1,2,2,9}
   450: {1,2,2,3,3}
   468: {1,1,2,2,6}
   504: {1,1,1,2,2,4}
   522: {1,2,2,10}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
For example, 350 is the Heinz number of (4,3,3,1), with negated first differences (1,0,2), which is not unimodal, so 350 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 complement is too full.

The enumeration of these partitions by sum is A332284.

The version where the last part is taken to be 0 is A332832.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences are A328509.

Partitions with non-unimodal run-lengths are A332281.

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

Heinz numbers of partitions with weakly increasing differences are A325360.

Cf. A001523, A007052, A240026, A332280, 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[primeMS[#]]]&]