cp's OEIS Frontend

A250029 Maximum number of binary strings with symmetrically partitioned n 1's and n 0's, counted up to isomorphism.

%I A250029 #19 Nov 19 2014 03:22:06
%S A250029 1,1,1,4,9,16,36,144,400,900,3600,11025,28224,78400,254016,705600,
%T A250029 2286144,6350400,25401600,85377600,250905600,768398400,3073593600,
%U A250029 10600761600,32464832400,129859329600,456536705625
%N A250029 Maximum number of binary strings with symmetrically partitioned n 1's and n 0's, counted up to isomorphism.
%C A250029 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 A250029 dualseq[partition[n]]=m!^2/(Prod_j(m_j!))^2
%C A250029 where m_j is the multiplicity of runs of length j.
%C A250029 The numbers satisfy the relations Sum_j(m_j)=m, Sum_j(j*m_j)=n.
%C A250029 The strings obtained in this manner are a subset of those in A247651.
%C A250029 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 A250029 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 A250029 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 A250029 n=0 gives the empty string.
%e A250029 n=1 and the only possible partition generate 01 (and the isomorphic 10).
%e A250029 For n=2, both possible partitions generate, up to isomorphism, 1 string, 0011 (1100), and respectively 0101 (1010).
%e A250029 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 A250029 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 A250029 dualseq[p_]:=Factorial[Length[p]]^2/Apply[Times,Map[Factorial[Count[p,#1]]&,Range[Max[Length[p]]]]]^2
%t A250029 a[n_]:=Max[Map[dualseq,IntegerPartitions[n]]]
%t A250029 Table[a[n],{n,0,25}] (* after A130670 *)
%Y A250029 Cf. A247651, A046952, A092266, A136404, A176471, A065886.
%K A250029 nonn
%O A250029 0,4
%A A250029 _Andrei Cretu_, Nov 11 2014