[math-fun] doublet strings

From the formula for A(n, k) given in http://oeis.org/A183135 it can easily be deduced that the number of "doublet strings" of length 2 n on k = 2 symbols equals the number of binary strings with n zeros and n ones. But if you were not told this, what constructive bijection between these sets might establish it directly? For n = 3 the sets are (in lexicographical order!) : AAAAAA OOOIII AAAABB OOIOII AAABBA OOIIOI AABAAB OOIIIO AABBAA OIOOII AABBBB OIOIOI ABAABA OIOIIO ABBAAA OIIOOI ABBABB OIIOIO ABBBBA OIIIOO BAAAAB IOOOII BAABAA IOOIOI BAABBB IOOIIO BABBAB IOIOOI BBAAAA IOIOIO BBAABB IOIIOO BBABBA IIOOOI BBBAAB IIOOIO BBBBAA IIOIOO BBBBBB IIIOOO Fred Lunnon