login
A320451
Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.
6
1, 1, 3, 5, 8, 7, 19, 11, 24, 26, 38, 28, 85, 46, 89, 99, 146, 110, 246, 163, 326, 305, 416, 376, 816, 591, 903, 971, 1450, 1295, 2517, 1916, 3045, 3141, 4042, 4117, 7073, 5736, 8131, 9026, 12658, 11514, 19459, 16230, 24638, 27129, 33747, 32279, 55778, 45761, 71946
OFFSET
0,3
COMMENTS
An integer partitions is uniform if all parts appear with the same multiplicity.
Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - Andrew Howroyd, Feb 04 2022
LINKS
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.
EXAMPLE
The a(9) = 26 multiset partitions:
{{9}}
{{1,8}}
{{2,7}}
{{3,6}}
{{4,5}}
{{1,2,6}}
{{1,3,5}}
{{1},{8}}
{{2,3,4}}
{{2},{7}}
{{3,3,3}}
{{3},{6}}
{{4},{5}}
{{1},{2},{6}}
{{1},{3},{5}}
{{2},{3},{4}}
{{3},{3},{3}}
{{1,1,1,2,2,2}}
{{1,1,1},{2,2,2}}
{{1,1,2},{1,2,2}}
{{1,1},{1,2},{2,2}}
{{1,2},{1,2},{1,2}}
{{1,1,1,1,1,1,1,1,1}}
{{1,1,1},{1,1,1},{1,1,1}}
{{1},{1},{1},{2},{2},{2}}
{{1},{1},{1},{1},{1},{1},{1},{1},{1}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n], And[SameQ@@Length/@Split[Sort[Join@@#]], SameQ@@Length/@#]&]], {n, 10}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 12 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Feb 04 2022
STATUS
approved