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.
2, 2, 6, 2, 12, 24, 2, 20, 72, 120, 2, 30, 172, 480, 720, 2, 42, 352, 1512, 3600, 5040, 2, 56, 646, 3976, 14184, 30240, 40320, 2, 72, 1094, 9144, 45992, 143712, 282240, 362880, 2, 90, 1742, 18990, 128288, 557640, 1575648, 2903040, 3628800, 2, 110, 2642, 36410, 318188, 1840520, 7152048, 18659520, 32659200, 39916800
Offset: 2
Examples
Triangle begins: 2 2 6 2 12 24 2 20 72 120 2 30 172 480 720 ... This triangle is the lower triangular part of a square array which begins 2 2 2 2 2 ... 2 6 6 6 6 ... 2 12 24 24 24 ... 2 20 72 120 120 ... 2 30 172 480 720 ... ...
Links
- T. M. Cover, The number of linearly inducible orderings of points in d-space, SIAM J. Applied Math., 15 (1967), 434-439.
Programs
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Maple
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);
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Mathematica
T[n_ /; n >= 2, 1] = 2; T[2, k_ /; k >= 1] = 2; T[n_, k_] := T[n, k] = T[n-1, k] + (n-1)*T[n-1, k-1]; T[n_, k_] /; k >= n-1 = n!; Flatten[Table[T[n, k], {n, 2, 11}, {k, 1, n-1}]] (* Jean-François Alcover, Apr 27 2011 *)
Formula
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.
Extensions
More terms from Emeric Deutsch, May 24 2004
Comments