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

%I A332725 #5 Feb 16 2025 08:33:59
%S A332725 90,126,180,198,234,252,270,306,342,350,360,378,396,414,450,468,504,
%T A332725 522,525,540,550,558,594,612,630,650,666,684,700,702,720,738,756,774,
%U A332725 792,810,825,828,846,850,882,900,910,918,936,950,954,975,990,1008,1026,1044
%N A332725 Heinz numbers of integer partitions whose negated first differences are not unimodal.
%C A332725 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332725 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.
%H A332725 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%H A332725 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A332725 The sequence of terms together with their prime indices begins:
%e A332725     90: {1,2,2,3}
%e A332725    126: {1,2,2,4}
%e A332725    180: {1,1,2,2,3}
%e A332725    198: {1,2,2,5}
%e A332725    234: {1,2,2,6}
%e A332725    252: {1,1,2,2,4}
%e A332725    270: {1,2,2,2,3}
%e A332725    306: {1,2,2,7}
%e A332725    342: {1,2,2,8}
%e A332725    350: {1,3,3,4}
%e A332725    360: {1,1,1,2,2,3}
%e A332725    378: {1,2,2,2,4}
%e A332725    396: {1,1,2,2,5}
%e A332725    414: {1,2,2,9}
%e A332725    450: {1,2,2,3,3}
%e A332725    468: {1,1,2,2,6}
%e A332725    504: {1,1,1,2,2,4}
%e A332725    522: {1,2,2,10}
%e A332725    525: {2,3,3,4}
%e A332725    540: {1,1,2,2,2,3}
%e A332725 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.
%t A332725 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A332725 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332725 Select[Range[1000],!unimodQ[Differences[primeMS[#]]]&]
%Y A332725 The complement is too full.
%Y A332725 The enumeration of these partitions by sum is A332284.
%Y A332725 The version where the last part is taken to be 0 is A332832.
%Y A332725 Non-unimodal permutations are A059204.
%Y A332725 Non-unimodal compositions are A115981.
%Y A332725 Non-unimodal normal sequences are A328509.
%Y A332725 Partitions with non-unimodal run-lengths are A332281.
%Y A332725 Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y A332725 Heinz numbers of partitions with weakly increasing differences are A325360.
%Y A332725 Cf. A001523, A007052, A240026, A332280, A332283, A332285, A332286, A332288, A332294, A332579, A332639, A332642.
%K A332725 nonn
%O A332725 1,1
%A A332725 _Gus Wiseman_, Feb 26 2020