%I
%S 1,1,1,4,9,16,36,144,400,900,3600,11025,28224,78400,254016,705600,
%T 2286144,6350400,25401600,85377600,250905600,768398400,3073593600,
%U 10600761600,32464832400,129859329600,456536705625
%N Maximum number of binary strings with symmetrically partitioned n 1's and n 0's, counted up to isomorphism.
%C The number of binary strings, counted up to isomorphism, that can be constructed by taking an equal number (n) of 0's and 1's and partitioning both the 0's and the 1's into m runs using the same partition, can be written as:
%C dualseq[partition[n]]=m!^2/(Prod_j(m_j!))^2
%C where m_j is the multiplicity of runs of length j.
%C The numbers satisfy the relations Sum_j(m_j)=m, Sum_j(j*m_j)=n.
%C The strings obtained in this manner are a subset of those in A247651.
%C Both the finest and coarsest partitions of n minimize dualseq[partition[n]]. In this sense, dualseq[partition[n]] is another relative measure of the complexity of the partition and the associated binary strings.
%C a[n] is the number of strings, counted up to isomorphism, that can be generated based on the partition that maximizes dualseq[partition[n]].
%F a[n]=Max[m!^2/(Prod_j(m_j!))^2] where Sum_j(m_j)=m, Sum_j(j*m_j)=n, over all partitions of n.
%e n=0 gives the empty string.
%e n=1 and the only possible partition generate 01 (and the isomorphic 10).
%e For n=2, both possible partitions generate, up to isomorphism, 1 string, 0011 (1100), and respectively 0101 (1010).
%e For n=3, the optimal partition is {1,2}, generating, up to isomorphism, 4 strings: 001011 (110100), 001101 (110010), 010011 (101100) and 011001 (100110).
%e For n=4, the optimal partition is {1,1,2}, generating, up to isomorphism, 9 strings: 00101011 (11010100), 00101101 (11010010), 00110101 (11001010), 01001011 (10110100), 01001101 (10110010), 01010011 (10101100), 01011001 (10100110), 01100101 (10011010) and 01101001 (10010110).
%t dualseq[p_]:=Factorial[Length[p]]^2/Apply[Times,Map[Factorial[Count[p,#1]]&,Range[Max[Length[p]]]]]^2
%t a[n_]:=Max[Map[dualseq,IntegerPartitions[n]]]
%t Table[a[n],{n,0,25}] (* after A130670 *)
%Y Cf. A247651, A046952, A092266, A136404, A176471, A065886.
%K nonn
%O 0,4
%A _Andrei Cretu_, Nov 11 2014
