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A340245 Order array of the Wythoff B-array (A340244): an interspersion, read by antidiagonals.

%I A340245 #6 Jan 05 2021 14:45:13
%S A340245 1,2,4,3,7,5,6,11,9,8,10,19,15,14,12,17,31,26,23,20,13,28,51,43,39,32,
%T A340245 22,16,46,81,71,64,53,36,27,18,75,110,101,94,83,60,44,30,21,105,138,
%U A340245 130,123,112,90,73,49,35,24,133,165,157,151,140,119,103,79
%N A340245 Order array of the Wythoff B-array (A340244): an interspersion, read by antidiagonals.
%C A340245 Suppose that (r(n,k)), for n >= 1 and k >= 1 is an array of distinct numbers.  Replace each r(n,k) with its position when all the numbers r(n,k) are ordered by <.  The resulting array is the order array of (r(n,k)).  The order array A340245 is an interspersion and a dispersion, and, as a sequence, a permutation of the natural numbers; see A333029 and A191450.
%e A340245 Corner:
%e A340245    1    2    3    6   10    17    28    46
%e A340245    4    7   11   19   31    51    81   110
%e A340245    5    9   15   26   43    71   101   130
%e A340245    8   14   23   39   64    94   123   151
%e A340245   12   20   32   53   83   112   140   167
%t A340245 r = GoldenRatio; f[n_] := Fibonacci[n];
%t A340245 a[n_] := Floor[r*n]; b[n_] := Floor[r^2*n];
%t A340245 c[n_] := a[a[b[n]]]; d[n_] := b[a[b[n]]];
%t A340245 w[n_, k_] := f[k - 2] c[n] + f[k - 1] d[n];
%t A340245 Grid[Table[w[n, k], {n, 1, 9}, {k, 1, 15}]] (* A340244 array *)
%t A340245 u = Table[w[n - k + 1, k], {n, 30}, {k, n, 1, -1}]//Flatten  (* A340244 sequence *)
%t A340245 rk[n_] := Position[Sort[u], u[[n]]];
%t A340245 Take[Flatten[Table[rk[n], {n, 1, 350}]], 100]  (* A340245 sequence *)
%Y A340245 Cf. A191450, A333029, A340244.
%K A340245 nonn,tabl
%O A340245 1,2
%A A340245 _Clark Kimberling_, Jan 02 2021