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A091444 Concatenate binary vectors ordered first by length, then by the number of 1's and finally lexicographically.

%I A091444 #19 Jul 28 2025 20:18:02
%S A091444 0,1,0,0,0,1,1,0,1,1,0,0,0,0,0,1,0,1,0,1,0,0,0,1,1,1,0,1,1,1,0,1,1,1,
%T A091444 0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,1,0,1,0,1,0,1,1,0,1,0,
%U A091444 0,1,1,0,1,0,1,1,0,0,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1
%N A091444 Concatenate binary vectors ordered first by length, then by the number of 1's and finally lexicographically.
%H A091444 Robert P. P. McKone, <a href="/A091444/b091444.txt">Table of n, a(n) for n = 0..18433</a>
%t A091444 c[n_, k_] := If[k == 0, {0}, If[k == n, {2^n - 1}, Join[c[n - 1, k], c[n - 1, k - 1] + 2^(n - 1)]]];
%t A091444 b[n_] := If[n == 1, {0, 1}, Flatten[Table[c[n, k], {k, 0, n}]]];
%t A091444 a[n_] := Map[PadLeft, IntegerDigits[Array[b, n], 2]];
%t A091444 a[4] // Flatten (* _Robert P. P. McKone_, Aug 13 2021 *)
%t A091444 Flatten[Table[SortBy[IntegerDigits[Range[0, 2^k - 1], 2, k], Total], {k, 4}]] (* _Paolo Xausa_, Jul 28 2025 *)
%o A091444 (Python)
%o A091444 from itertools import product
%o A091444 def sortby(x): return (len(x), x.count(1), x)
%o A091444 def agen(maxvecdigits):
%o A091444     for i in range(1, maxvecdigits+1):
%o A091444         for t in sorted([p for p in product([0, 1], repeat=i)], key=sortby):
%o A091444             yield from t
%o A091444 print([an for an in agen(4)]) # _Michael S. Branicky_, Aug 13 2021
%Y A091444 Cf. A076478, A091445, A091446, A091447.
%Y A091444 Cf. A294648.
%K A091444 easy,nonn
%O A091444 0,1
%A A091444 _Paul Boddington_, Jan 10 2004