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A317143
In the ranked poset of integer partitions ordered by refinement, row n lists the Heinz numbers of integer partitions finer (less) than or equal to the integer partition with Heinz number n.
3
1, 2, 3, 4, 4, 5, 6, 8, 6, 8, 7, 9, 10, 12, 16, 8, 9, 12, 16, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 12, 16, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 14, 18, 20, 24, 32, 15, 18, 20, 24, 32, 16, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128
OFFSET
1,2
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
If x and y are partitions of the same integer and it is possible to produce x by further partitioning the parts of y, flattening, and sorting, then x <= y.
EXAMPLE
The partitions finer than or equal to (2,2) are (2,2), (2,1,1), (1,1,1,1), with Heinz numbers 9, 12, 16, so the 9th row is {9, 12, 16}.
Triangle begins:
1
2
3 4
4
5 6 8
6 8
7 9 10 12 16
8
9 12 16
10 12 16
11 14 15 18 20 24 32
12 16
13 21 22 25 27 28 30 36 40 48 64
14 18 20 24 32
15 18 20 24 32
16
17 26 33 35 42 44 45 50 54 56 60 72 80 96 128
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Union[Times@@@Map[Prime, Join@@@Tuples[IntegerPartitions/@primeMS[n]], {2}]], {n, 12}]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jul 22 2018
STATUS
approved