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A384180 Irregular triangle read by rows where row n lists the Heinz numbers of all uniform (equal multiplicities) and normal (covering an initial interval) multisets of length n.

%I A384180 #5 May 28 2025 09:19:54
%S A384180 2,4,6,8,30,16,36,210,32,2310,64,216,900,30030,128,510510,256,1296,
%T A384180 44100,9699690,512,27000,223092870,1024,7776,5336100,6469693230,2048,
%U A384180 200560490130,4096,46656,810000,9261000,901800900,7420738134810,8192,304250263527210
%N A384180 Irregular triangle read by rows where row n lists the Heinz numbers of all uniform (equal multiplicities) and normal (covering an initial interval) multisets of length n.
%C A384180 A permutation of A100778 (powers of primorials).
%C A384180 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A384180 An integer partition is uniform iff all parts appear with the same multiplicity, and normal iff it covers an initial interval of positive integers.
%e A384180 The uniform normal multisets of length 6 are: {1,1,1,1,1,1}, {1,1,1,2,2,2}, {1,1,2,2,3,3}, {1,2,3,4,5,6}, so row 6 is: 64, 216, 900, 30030.
%e A384180 Triangle begins:
%e A384180     2
%e A384180     4       6
%e A384180     8      30
%e A384180    16      36    210
%e A384180    32    2310
%e A384180    64     216    900    30030
%e A384180   128  510510
%e A384180   256    1296  44100  9699690
%t A384180 Table[Table[Times@@Prime/@Range[d]^(n/d),{d,Divisors[n]}],{n,10}]
%Y A384180 Row lengths are A000005.
%Y A384180 Final term in each row is A002110.
%Y A384180 The union is A100778.
%Y A384180 Reversing rows gives A322792.
%Y A384180 For just normal multisets we have A324939.
%Y A384180 A047966 counts uniform partitions.
%Y A384180 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384180 A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
%Y A384180 A381431 is the section-sum transform.
%Y A384180 Cf. A000009, A001694, A056239, A106529, A112798, A116540, A122111, A258280, A325326, A325337.
%K A384180 nonn,tabf
%O A384180 1,1
%A A384180 _Gus Wiseman_, May 25 2025