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

%I A332287 #9 Feb 16 2025 08:33:59
%S A332287 36,50,70,72,98,100,108,140,144,154,180,182,196,200,216,225,242,250,
%T A332287 252,280,286,288,294,300,308,324,338,350,360,363,364,374,392,396,400,
%U A332287 418,429,432,441,442,450,462,468,484,490,494,500,504,507,540,550,560,561
%N A332287 Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.
%C A332287 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332287 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.
%H A332287 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332287 The sequence of terms together with their prime indices begins:
%e A332287    36: {1,1,2,2}
%e A332287    50: {1,3,3}
%e A332287    70: {1,3,4}
%e A332287    72: {1,1,1,2,2}
%e A332287    98: {1,4,4}
%e A332287   100: {1,1,3,3}
%e A332287   108: {1,1,2,2,2}
%e A332287   140: {1,1,3,4}
%e A332287   144: {1,1,1,1,2,2}
%e A332287   154: {1,4,5}
%e A332287   180: {1,1,2,2,3}
%e A332287   182: {1,4,6}
%e A332287   196: {1,1,4,4}
%e A332287   200: {1,1,1,3,3}
%e A332287   216: {1,1,1,2,2,2}
%e A332287   225: {2,2,3,3}
%e A332287   242: {1,5,5}
%e A332287   250: {1,3,3,3}
%e A332287   252: {1,1,2,2,4}
%e A332287   280: {1,1,1,3,4}
%e A332287 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.
%t A332287 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A332287 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332287 Select[Range[1000],!unimodQ[Differences[Append[Reverse[primeMS[#]],0]]]&]
%Y A332287 The enumeration of these partitions by sum is A332284.
%Y A332287 Not assuming the last part is zero gives A332725.
%Y A332287 Non-unimodal permutations are A059204.
%Y A332287 Non-unimodal compositions are A115981.
%Y A332287 Non-unimodal normal sequences are A328509.
%Y A332287 Partitions with non-unimodal run-lengths are A332281.
%Y A332287 Cf. A001523, A007052, A332280, A332282, A332283, A332285, A332286, A332288, A332294, A332579, A332639, A332642.
%K A332287 nonn
%O A332287 1,1
%A A332287 _Gus Wiseman_, Feb 21 2020