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A214676 A(n,k) is n represented in bijective base-k numeration; square array A(n,k), n>=1, k>=1, read by antidiagonals.

%I A214676 #27 Feb 16 2025 08:33:18
%S A214676 1,1,11,1,2,111,1,2,11,1111,1,2,3,12,11111,1,2,3,11,21,111111,1,2,3,4,
%T A214676 12,22,1111111,1,2,3,4,11,13,111,11111111,1,2,3,4,5,12,21,112,
%U A214676 111111111,1,2,3,4,5,11,13,22,121,1111111111
%N A214676 A(n,k) is n represented in bijective base-k numeration; square array A(n,k), n>=1, k>=1, read by antidiagonals.
%C A214676 The digit set for bijective base-k numeration is {1, 2, ..., k}.
%H A214676 Alois P. Heinz, <a href="/A214676/b214676.txt">Antidiagonals n = 1..18, flattened</a>
%H A214676 R. R. Forslund, <a href="http://www.emis.de/journals/SWJPAM/Vol1_1995/rrf01.ps">A logical alternative to the existing positional number system</a>, Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1995, 27-29.
%H A214676 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Zerofree.html">Zerofree</a>
%H A214676 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bijective_numeration">Bijective numeration</a>
%e A214676 Square array A(n,k) begins:
%e A214676 :         1,   1,  1,  1,  1,  1,  1,  1, ...
%e A214676 :        11,   2,  2,  2,  2,  2,  2,  2, ...
%e A214676 :       111,  11,  3,  3,  3,  3,  3,  3, ...
%e A214676 :      1111,  12, 11,  4,  4,  4,  4,  4, ...
%e A214676 :     11111,  21, 12, 11,  5,  5,  5,  5, ...
%e A214676 :    111111,  22, 13, 12, 11,  6,  6,  6, ...
%e A214676 :   1111111, 111, 21, 13, 12, 11,  7,  7, ...
%e A214676 :  11111111, 112, 22, 14, 13, 12, 11,  8, ...
%p A214676 A:= proc(n, b) local d, l, m; m:= n; l:= NULL;
%p A214676       while m>0 do  d:= irem(m, b, 'm');
%p A214676         if d=0 then d:=b; m:=m-1 fi;
%p A214676         l:= d, l
%p A214676       od; parse(cat(l))
%p A214676     end:
%p A214676 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A214676 A[n_, b_] := Module[{d, l, m}, m = n; l = Nothing; While[m > 0, {m, d} = QuotientRemainder[m, b]; If[d == 0, d = b; m--]; l = {d, l}]; FromDigits @ Flatten @ l];
%t A214676 Table[A[n, d-n+1], {d, 1, 12}, {n, 1, d}] // Flatten (* _Jean-François Alcover_, May 28 2019, from Maple *)
%Y A214676 Columns k=1-9 give: A000042, A007931, A007932, A084544, A084545, A057436, A214677, A214678, A052382.
%Y A214676 A(n+1,n) gives A010850.
%K A214676 nonn,tabl
%O A214676 1,3
%A A214676 _Alois P. Heinz_, Jul 25 2012