RFE Jan 2026: Solid partitions

[Sean A. Irvine]

Hi,

Thanks to a great effort by seqfan and particularly Christian Sievers
and Thomas Scheurle, December's request has been resolved.

This month I'm giving you a break from large groups of sequences, to
look at just three old sequences relating to solid partitions. These
sequences appeared in Neil's original Handbook.

Provide an explanation, more terms, formulas, programs, and/or more
specific names for A002043, A002044, and A002045. In 2018, I attempted
to reproduce these sequences using the recurrences in the Chandra and
Nanda papers, but failed. Perhaps someone on this list will have more
luck!

https://oeis.org/A002043

I track these requests for enhancement here:

https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements

Sean.

[Geoffrey Caveney]

Observation:

As the annotated scanned copy of the Chandra paper indicates, there is also the sequence for m=1 (in Chandra's notation), which is OEIS sequence A000294.

However, a comment on OEIS sequence A000293, "a(n) = number of solid (i.e., three-dimensional) partitions of n" points out the following:

"Finding a g.f. for this sequence is an unsolved problem. At first it was thought that it was given by A000294."

It would be useful to know in which time frame the belief existed in the mathematical community that the generation function for A000293 was given by A000294. If Chandra, Nanda, et al., were operating under that premise, now known to be false, in the 1950s, then perhaps this may provide part of the explanation for why we are now having difficulty reproducing Chandra's and Nanda's methods to generate sequences A002043, et al., and relating them to solid partitions.

Geoffrey

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[Martin Fuller]

This generating function reproduces the sequences using m=2,3,4 respectively:

(1 / (1 - x^m)^binomial(m+1,2) - 1) * Product_{r>m} (1 / (1 - x^r)^binomial(r+1,2))

(PARI) a(n,m=2)=Vec(prod(r=m,n,1/(1-'x^r+O('x^(n+1)))^binomial(r+1,2)-(r==m)));

It is based on a generating function from the papers with a change to the first multiplicand.

Martin Fuller

[Wouter Meeussen]

Well done Martin,

checks out for all values in the table by R. Chandra (https://oeis.org/A000294/a000294_1.pdf)

Wouter.

Op 3/01/2026 om 15:40 schreef 'Martin Fuller' via SeqFan:

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[Ruud H.G. van Tol]

On 2026-01-03 15:40, 'Martin Fuller' via SeqFan wrote:
> This generating function reproduces the sequences using m=2,3,4
> respectively:
>
> (1 / (1 - x^m)^binomial(m+1,2) - 1) * Product_{r>m} (1 / (1 -
> x^r)^binomial(r+1,2))
>
> (PARI)
> a(n,m=2)=Vec(prod(r=m,n,1/(1-'x^r+O('x^(n+1)))^binomial(r+1,2)-(r==m)));
>
> It is based on a generating function from the papers with a change to
> the first multiplicand.

To wrap it:

lista(n, m=2)= my(s=a(n,m)); concat(vector(n-#s), s);


Examples:

? lista(20,1)
% [1, 1, 4, 10, 26, 59, 141, 310, ...]

? lista(20,2)
% [0, 3, 0,  6, 18, 40,  81, 201, ...]

? lista(20,3)
% [0, 0, 6,  0,  0, 21,  60,  90, ...]

? lista(20,4)
% [0, 0, 0, 10,  0,  0,   0,  55, ...]

? lista(20,5)
% [0, 0, 0,  0, 15,  0,   0,   0, ...]

? lista(20,6)
% [0, 0, 0,  0,  0, 21,   0,   0, ...]

? lista(20,7)
% [0, 0, 0,  0,  0,  0,  28,   0, ...]

1 = 1 + 0
4 = 1 + 3 + 0
10 = 4 + 0 + 6 + 0
...
310 = 141 + 81 + 60 + 28 + 0
(etc.)

-- Ruud

[Jod Frut]

Maybe a better title for these could be "Number of partitions of n with least part m and r*(r+1)/2 kinds of parts r." 

John R. 

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[Martin Fuller]

On Saturday, 3 January 2026 at 19:05:07 UTC John Rascoe wrote:

Maybe a better title for these could be "Number of partitions of n with least part m and r*(r+1)/2 kinds of parts r." 

John R.

 

Could you give more detail? How do you interpret the case where m=n and the result is n(n+1)/2?

There is a footnote on the first page of Nanda (1953) which seems relevant, but I don't understand it.

I had a go at calculating T(n,m) = Number of solid partitions of n where the smallest part is m.

Example: T(7,2)=9 from {5, 2} in 3 orientations, {3, 2, 2} in 3 orientations, and {{3, 2}, {2}} plane partition in 3 orientations.

If I've got it right, the triangle starts as follows. Row sums should equal A000293.

1
3, 1
9, 0, 1
22, 3, 0, 1
55, 3, 0, 0, 1
127, 9, 3, 0, 0, 1
294, 9, 3, 0, 0, 0, 1
646, 31, 3, 3, 0, 0, 0, 1
1420, 31, 9, 3, 0, 0, 0, 0, 1
3018, 88, 9, 3, 3, 0, 0, 0, 0, 1
6366, 109, 18, 3, 3, 0, 0, 0, 0, 0, 1
13138, 241, 31, 9, 3, 3, 0, 0, 0, 0, 0, 1
26861, 331, 40, 9, 3, 3, 0, 0, 0, 0, 0, 0, 1
54031, 681, 64, 18, 3, 3, 3, 0, 0, 0, 0, 0, 0, 1
107708, 942, 118, 18, 9, 3, 3, 0, 0, 0, 0, 0, 0, 0, 1
212033, 1837, 142, 40, 9, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 1
413866, 2680, 235, 40, 18, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1
799811, 4818, 385, 73, 18, 9, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1
1533723, 7248, 526, 94, 27, 9, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
2916690, 12633, 784, 151, 40, 18, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

[Jod Frut]

The case when m=n is a partition with a single part m and there are m(m+1)/2 kinds of m's. 

For A002043 this would be the "Number of partitions of n with least part 2 and r*(r+1)/2 kinds of parts r." maybe adding for r >= 2?

A002043(2) = 3 counts the three kinds of a single part 2: (2_a), (2_b),  and (2_c).

A002043(4) = 6 counts: (2_a,2_a), (2_a,2_b), (2_a,2_c), (2_b,2_b), (2_b,2_c), (2_c,2_c).

Nice, column m=1 of the correct triangle for solid partitions should also be added as a new sequence. 

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[Jod Frut]

So the three existing sequences A002043, A002044, and A002045 count what the paper calls MacMahon's solid partitions with the smallest summand m =2,3,4. Macmahon's solid partitions being A000294 not A000293. Most sequences using the name "Solid Partition" are referencing A000293, I just saw the changes made to A002043 and I think the name should make this clear. 

John Rascoe