OFFSET
0,4
COMMENTS
A set partition is balanced if it has exactly as many blocks as the greatest size of a block.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
EXAMPLE
The a(1) = 1 through a(5) = 10 balanced set partitions (empty column indicated by dot):
{{1}} . {{1},{2,3}} {{1,2},{3,4}} {{1},{2},{3,4,5}}
{{1,2},{3}} {{1,3},{2,4}} {{1},{2,3,4},{5}}
{{1,3},{2}} {{1,4},{2,3}} {{1,2,3},{4},{5}}
{{1},{2,3,5},{4}}
{{1,2,4},{3},{5}}
{{1},{2,4,5},{3}}
{{1,2,5},{3},{4}}
{{1,3,4},{2},{5}}
{{1,3,5},{2},{4}}
{{1,4,5},{2},{3}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[sps[Range[n]], Length[#]==Max@@Length/@#&]], {n, 0, 8}]
PROG
(PARI) \\ D(n, k) counts balanced set partitions with k blocks.
D(n, k)={my(t=sum(i=1, k, x^i/i!) + O(x*x^n)); n!*polcoef(t^k - (t-x^k/k!)^k, n)/k!}
a(n)={sum(k=sqrtint(n), (n+1)\2, D(n, k))} \\ Andrew Howroyd, Mar 14 2021
CROSSREFS
A000110 counts set partitions.
A000670 counts ordered set partitions.
A113547 counts set partitions by maximin.
Other balance-related sequences:
- A098124 counts balanced integer compositions.
- A340596 counts co-balanced factorizations.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 20 2021
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Mar 14 2021
STATUS
approved