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Maximal length of a set partition of the binary indices of n into blocks all having the same sum.
4

%I #7 Apr 15 2024 09:47:59

%S 0,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,2,1,1,2,1,2,1,1,2,3,1,1,

%T 1,1,1,1,1,2,1,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,1,1,1,3,2,1,2,3,1,1,1,1,

%U 1,1,1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,2,1

%N Maximal length of a set partition of the binary indices of n into blocks all having the same sum.

%C A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

%C If a(n) = k then the binary indices of n (row n of A048793) are k-quanimous (counted by A371783).

%e The binary indices of 119 are {1,2,3,5,6,7}, and the set partitions into blocks with the same sum are:

%e {{1,7},{2,6},{3,5}}

%e {{1,5,6},{2,3,7}}

%e {{1,2,3,6},{5,7}}

%e {{1,2,3,5,6,7}}

%e So a(119) = 3.

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

%t bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Table[Max[Length/@Select[sps[bix[n]],SameQ@@Total/@#&]],{n,0,100}]

%Y Set partitions of this type are counted by A035470, A336137.

%Y A version for factorizations is A371733.

%Y Positions of 1's are A371738.

%Y Positions of terms > 1 are A371784.

%Y A001055 counts factorizations.

%Y A002219 (aerated) counts biquanimous partitions, ranks A357976.

%Y A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

%Y A070939 gives length of binary expansion.

%Y A321452 counts quanimous partitions, ranks A321454.

%Y A326031 gives weight of the set-system with BII-number n.

%Y A371783 counts k-quanimous partitions.

%Y A371789 counts non-quanimous sets, differences A371790.

%Y A371796 counts quanimous sets, differences A371797.

%Y Cf. A006827, A038041, A096111, A279787, A305551, A321451, A321455, A322794, A326534, A371731, A371734.

%K nonn

%O 0,8

%A _Gus Wiseman_, Apr 14 2024