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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.
3

%I #34 Sep 02 2023 19:07:36

%S 2,2,6,2,12,24,2,20,72,120,2,30,172,480,720,2,42,352,1512,3600,5040,2,

%T 56,646,3976,14184,30240,40320,2,72,1094,9144,45992,143712,282240,

%U 362880,2,90,1742,18990,128288,557640,1575648,2903040,3628800,2,110,2642,36410,318188,1840520,7152048,18659520,32659200,39916800

%N 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 This can also be regarded as the lower triangular part of an infinite square array - see Example section and A198889.

%C Second and third columns are A002378 and A087645.

%H 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 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 Triangle begins:

%e 2

%e 2 6

%e 2 12 24

%e 2 20 72 120

%e 2 30 172 480 720

%e ...

%e This triangle is the lower triangular part of a square array which begins

%e 2 2 2 2 2 ...

%e 2 6 6 6 6 ...

%e 2 12 24 24 24 ...

%e 2 20 72 120 120 ...

%e 2 30 172 480 720 ...

%e ...

%p 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 T[n_ /; n >= 2, 1] = 2; T[2, k_ /; k >= 1] = 2;

%t T[n_, k_] := T[n, k] = T[n-1, k] + (n-1)*T[n-1, k-1];

%t T[n_, k_] /; k >= n-1 = n!;

%t Flatten[Table[T[n, k], {n, 2, 11}, {k, 1, n-1}]] (* _Jean-François Alcover_, Apr 27 2011 *)

%Y Cf. A087644, A002378, A087645.

%K nonn,tabl,easy,nice

%O 2,1

%A _N. J. A. Sloane_, Oct 26 2003

%E More terms from _Emeric Deutsch_, May 24 2004