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.

2, 4, 6, 8, 30, 16, 36, 210, 32, 2310, 64, 216, 900, 30030, 128, 510510, 256, 1296, 44100, 9699690, 512, 27000, 223092870, 1024, 7776, 5336100, 6469693230, 2048, 200560490130, 4096, 46656, 810000, 9261000, 901800900, 7420738134810, 8192, 304250263527210

Offset

1,1

Comments

A permutation of A100778 (powers of primorials).

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.

An integer partition is uniform iff all parts appear with the same multiplicity, and normal iff it covers an initial interval of positive integers.

Links

Table of n, a(n) for n=1..37.

Example

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.
Triangle begins:
    2
    4       6
    8      30
   16      36    210
   32    2310
   64     216    900    30030
  128  510510
  256    1296  44100  9699690

Mathematica

Table[Table[Times@@Prime/@Range[d]^(n/d), {d, Divisors[n]}], {n, 10}]

Crossrefs

Row lengths are A000005.

Final term in each row is A002110.

The union is A100778.

Reversing rows gives A322792.

For just normal multisets we have A324939.

A047966 counts uniform partitions.

A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.

A098859 counts Wilf partitions, ranks A130091, conjugate A383512.

A381431 is the section-sum transform.

Cf. A000009, A001694, A056239, A106529, A112798, A116540, A122111, A258280, A325326, A325337.

Sequence in context: A117912 A092047 A057809 * A135632 A068541 A329887

Adjacent sequences: A384177 A384178 A384179 * A384181 A384182 A384183

Keyword

nonn,tabf

Author

Gus Wiseman, May 25 2025

Status

approved