A340598 - OEIS

%I #12 Mar 14 2021 12:36:53

%S 0,1,0,3,3,10,60,210,700,3556,19845,105567,550935,3120832,19432413,

%T 127949250,858963105,5882733142,41636699676,307105857344,

%U 2357523511200,18694832699907,152228641035471,1270386473853510,10872532998387918,95531590347525151

%N Number of balanced set partitions of {1..n}.

%C A set partition is balanced if it has exactly as many blocks as the greatest size of a block.

%H Andrew Howroyd, <a href="/A340598/b340598.txt">Table of n, a(n) for n = 0..200</a>

%e The a(1) = 1 through a(5) = 10 balanced set partitions (empty column indicated by dot):

%e   {{1}}  .  {{1},{2,3}}  {{1,2},{3,4}}  {{1},{2},{3,4,5}}

%e             {{1,2},{3}}  {{1,3},{2,4}}  {{1},{2,3,4},{5}}

%e             {{1,3},{2}}  {{1,4},{2,3}}  {{1,2,3},{4},{5}}

%e                                         {{1},{2,3,5},{4}}

%e                                         {{1,2,4},{3},{5}}

%e                                         {{1},{2,4,5},{3}}

%e                                         {{1,2,5},{3},{4}}

%e                                         {{1,3,4},{2},{5}}

%e                                         {{1,3,5},{2},{4}}

%e                                         {{1,4,5},{2},{3}}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Select[sps[Range[n]],Length[#]==Max@@Length/@#&]],{n,0,8}]

%o (PARI) \\ D(n,k) counts balanced set partitions with k blocks.

%o 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!}

%o a(n)={sum(k=sqrtint(n), (n+1)\2, D(n,k))} \\ _Andrew Howroyd_, Mar 14 2021

%Y The unlabeled version is A047993 (A106529).

%Y A000110 counts set partitions.

%Y A000670 counts ordered set partitions.

%Y A113547 counts set partitions by maximin.

%Y Other balance-related sequences:

%Y - A010054 counts balanced strict integer partitions (A002110).

%Y - A098124 counts balanced integer compositions.

%Y - A340596 counts co-balanced factorizations.

%Y - A340599 counts alt-balanced factorizations.

%Y - A340600 counts unlabeled balanced multiset partitions.

%Y - A340653 counts balanced factorizations.

%Y Cf. A000258, A001055, A006141, A008275, A008277, A008278, A095149.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 20 2021

%E Terms a(12) and beyond from _Andrew Howroyd_, Mar 14 2021