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Let U be the set $U=\{(1,2,3,\ldots,2^m)\}$. Let $A$ and $B$ partitions of $U$, such that $A \cup B$ is the set $U$, and their intersection is empty, and adding the elements of the first set is the same number of the addition of the elements of the second set $B$.

How many $A$ and $B$ is there?

I mean, if $U=\{1,2,3,4\}$, there exist just a single pair of sets $A$, and $B$. $A=(3,2)$ and $B=(4,1)$, because $3+2=4+1$, i want to know how many $A$ and $B$s exists for bigger sets $U$.

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    @Ross, yes, thanks.2012-06-27

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If instead we take $U=\{{1,2,3,\dots,m\}}$, then the answer is easily seen to be nonzero only if $m$, reduced modulo 4, is 0 or 3. The sequence of values is tabulated at the Online Encyclopedia of Integer Sequences; it starts $\eqalign{&0, 0, 1, 1, 0, 0, 4, 7, 0, 0, 35, 62, 0, 0, 361, 657, 0, 0, 4110, 7636, 0, 0, 49910, 93846, 0, 0,\cr &632602, 1199892, 0, 0, 8273610, 15796439, 0, 0, 110826888, 212681976, 0, 0,\cr &1512776590, 2915017360, 0, 0, 20965992017, 40536016030, 0, 0, 294245741167,\cr &570497115729\cr}$ No formula is given.

If we restrict to powers of two, the sequence starts $0,0,1,7,657=9\times73,15796439=41\times385279,24435006625667338=(2)(727)(182764397)(91951)$ This sequence is not tabulated. I am not optimistic about finding any useful formula for these numbers.