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%I #16 Nov 23 2021 03:39:07
%S 1,2,2,12,48,164,348,6000,56202,529434,4378410,50607424,465231342,
%T 4414360152,28075191592
%N Number of permutations of the base-2 representations of 1, 2, ..., n that are unambiguous when written without delimiters.
%e For n = 3 the base-2 representations of 1, 2, 3 are the words 1, 10, 11. Of the 6 permutations of these words, the a(3) = 2 permutations 1,10,11 and 11,10,1 are unambiguous when the commas are removed, since 11011 and 11101 each arise from a unique permutation. On the other hand, the permutations 1,11,10 and 11,1,10 are ambiguous when the commas are removed because they become the same word; similarly, 10,1,11 and 10,11,1 become the same word.
%t Table[Count[Counts[Join @@@ Permutations[(IntegerDigits[#, 2] &) /@ Range[1, n]]], 1], {n, 1, 8}]
%o (Python)
%o from collections import Counter
%o from itertools import permutations
%o def A349529(n): return len(list(filter(lambda x: x == 1, Counter(''.join(d) for d in permutations(bin(i)[2:] for i in range(1,n+1))).values()))) # _Chai Wah Wu_, Nov 22 2021
%K nonn,base,more
%O 1,2
%A _Eric Rowland_, Nov 20 2021
%E a(13)-a(15) from _Martin Ehrenstein_, Nov 23 2021
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