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A332287
Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.
17
36, 50, 70, 72, 98, 100, 108, 140, 144, 154, 180, 182, 196, 200, 216, 225, 242, 250, 252, 280, 286, 288, 294, 300, 308, 324, 338, 350, 360, 363, 364, 374, 392, 396, 400, 418, 429, 432, 441, 442, 450, 462, 468, 484, 490, 494, 500, 504, 507, 540, 550, 560, 561
OFFSET
1,1
COMMENTS
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), which gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The sequence of terms together with their prime indices begins:
36: {1,1,2,2}
50: {1,3,3}
70: {1,3,4}
72: {1,1,1,2,2}
98: {1,4,4}
100: {1,1,3,3}
108: {1,1,2,2,2}
140: {1,1,3,4}
144: {1,1,1,1,2,2}
154: {1,4,5}
180: {1,1,2,2,3}
182: {1,4,6}
196: {1,1,4,4}
200: {1,1,1,3,3}
216: {1,1,1,2,2,2}
225: {2,2,3,3}
242: {1,5,5}
250: {1,3,3,3}
252: {1,1,2,2,4}
280: {1,1,1,3,4}
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.
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[Append[Reverse[primeMS[#]], 0]]]&]
CROSSREFS
The enumeration of these partitions by sum is A332284.
Not assuming the last part is zero gives A332725.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Sequence in context: A260008 A261292 A335249 * A050691 A211720 A261285
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 21 2020
STATUS
approved