A332725 - OEIS

%I #4 Feb 28 2020 22:55:14

%S 90,126,180,198,234,252,270,306,342,350,360,378,396,414,450,468,504,

%T 522,525,540,550,558,594,612,630,650,666,684,700,702,720,738,756,774,

%U 792,810,825,828,846,850,882,900,910,918,936,950,954,975,990,1008,1026,1044

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

%C A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%C 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 MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The sequence of terms together with their prime indices begins:

%e     90: {1,2,2,3}

%e    126: {1,2,2,4}

%e    180: {1,1,2,2,3}

%e    198: {1,2,2,5}

%e    234: {1,2,2,6}

%e    252: {1,1,2,2,4}

%e    270: {1,2,2,2,3}

%e    306: {1,2,2,7}

%e    342: {1,2,2,8}

%e    350: {1,3,3,4}

%e    360: {1,1,1,2,2,3}

%e    378: {1,2,2,2,4}

%e    396: {1,1,2,2,5}

%e    414: {1,2,2,9}

%e    450: {1,2,2,3,3}

%e    468: {1,1,2,2,6}

%e    504: {1,1,1,2,2,4}

%e    522: {1,2,2,10}

%e    525: {2,3,3,4}

%e    540: {1,1,2,2,2,3}

%e 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 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t Select[Range[1000],!unimodQ[Differences[primeMS[#]]]&]

%Y The complement is too full.

%Y The enumeration of these partitions by sum is A332284.

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

%Y Non-unimodal permutations are A059204.

%Y Non-unimodal compositions are A115981.

%Y Non-unimodal normal sequences are A328509.

%Y Partitions with non-unimodal run-lengths are A332281.

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

%Y Heinz numbers of partitions with weakly increasing differences are A325360.

%Y Cf. A001523, A007052, A240026, A332280, A332283, A332285, A332286, A332288, A332294, A332579, A332639, A332642.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 26 2020