cp's OEIS Frontend

A071223 Triangle T(n,k) (n >= 2, 1 <= k <= n) read by rows: number of linearly inducible orderings of n points in k-dimensional Euclidean space.

%I A071223 #34 Sep 02 2023 19:07:36
%S A071223 2,2,6,2,12,24,2,20,72,120,2,30,172,480,720,2,42,352,1512,3600,5040,2,
%T A071223 56,646,3976,14184,30240,40320,2,72,1094,9144,45992,143712,282240,
%U A071223 362880,2,90,1742,18990,128288,557640,1575648,2903040,3628800,2,110,2642,36410,318188,1840520,7152048,18659520,32659200,39916800
%N A071223 Triangle T(n,k) (n >= 2, 1 <= k <= n) read by rows: number of linearly inducible orderings of n points in k-dimensional Euclidean space.
%C A071223 This can also be regarded as the lower triangular part of an infinite square array - see Example section and A198889.
%C A071223 Second and third columns are A002378 and A087645.
%H A071223 T. M. Cover, <a href="http://www.jstor.org/stable/2946294">The number of linearly inducible orderings of points in d-space</a>, SIAM J. Applied Math., 15 (1967), 434-439.
%F A071223 T(n, 1) = 2 for n >= 2, T(2, k) = 2 for k >= 1, T(n+1, k) = T(n, k) + n*T(n, k-1). Also T(n, k) = n! for k >= n-1.
%e A071223 Triangle begins:
%e A071223   2
%e A071223   2  6
%e A071223   2 12  24
%e A071223   2 20  72 120
%e A071223   2 30 172 480 720
%e A071223   ...
%e A071223 This triangle is the lower triangular part of a square array which begins
%e A071223   2   2   2   2   2 ...
%e A071223   2   6   6   6   6 ...
%e A071223   2  12  24  24  24 ...
%e A071223   2  20  72 120 120 ...
%e A071223   2  30 172 480 720 ...
%e A071223   ...
%p A071223 T:=proc(n,k) if k>=n then 0 elif k=1 and n>=2 then 2 elif n=2 and k>=1 then 2 elif k=n-1 then n! else T(n-1,k)+(n-1)*T(n-1,k-1) fi end:seq(seq(T(n,k),k=1..n-1),n=2..12);
%t A071223 T[n_ /; n >= 2, 1] = 2; T[2, k_ /; k >= 1] = 2;
%t A071223 T[n_, k_] := T[n, k] = T[n-1, k] + (n-1)*T[n-1, k-1];
%t A071223 T[n_, k_] /; k >= n-1 = n!;
%t A071223 Flatten[Table[T[n, k], {n, 2, 11}, {k, 1, n-1}]]  (* _Jean-François Alcover_, Apr 27 2011 *)
%Y A071223 Cf. A087644, A002378, A087645.
%K A071223 nonn,tabl,easy,nice
%O A071223 2,1
%A A071223 _N. J. A. Sloane_, Oct 26 2003
%E A071223 More terms from _Emeric Deutsch_, May 24 2004