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COMMENT
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Triangle in which the n-th row begins with n and the k-th term is obtained by multiplying the (k-1)-th term by (n-k+1) until n-k+1 = 1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 11 2002
Has many diagonals in common with A105725. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 14 2006
Also: the array of rising factorials A(n,k)=n(n+1)(n+2)*..(n+k-1) read by antidiagonals. There are no perfect squares in T(n,k) for k>=2 [O. Rigge, 9th Congr. Math. Scan., Helsingfors, 1938, Mercator, 1939 p 155-160]. T(n,k) is divisible by a prime exceeding k, if n>=2*k [N. Saradha and T. N. Shorey, Composito Mathematica 138 (1) (2003) 113-124]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 02 2007
T(n,k) is the number of injective functions f from {1,...,k} into {1,...,n}, since there are n choices for f(1), then (n-1) choices for f(2),... and (n-k+1) choices for f(k). E.g. T(3,2)=6 because there are exactly 6 injective functions f:{1,2}->{1,2,3}, namely, f1={(1,1),(2,2)}, f2={(1,1),(2,3)}, f3={(1,2),(2,1)}, f4={(1,2},(2,3)}, f5={(1,3),(2,1)} and f6={(1,3),(2,2)}. - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 18 2007
Permuted words defined by the connectivity of regular simplices are related to T by T = A135278 * (1!, 2!, 3!, 4!, ...). E.g. for T(4,k) with k-1 = simplex number, label the vertices of a tetrahedron with a, b, c, d, then the 0-simplex, the points, a,b,c,d gives 4 * 1 = 4 words; the 1-simplex, the edges: (ab or ba), (ac or ca), (ad or da), (bc or cb), (bd or db), (cd or dc) gives 6 * 2 = 12 words; the 2-simplex, the faces: (abc or ...), (acd or ...), (adb or ...), (bcd or ...) gives 4 * 6 = 24 words; the 3-simplex, (abcd or ....) gives 1 * 24 = 24 words. - Tom Copeland (tcjpn(AT)msn.com), Dec 08 2007
Reversal of the triangle by rows = (n+1) * n-th row of triangle A094587. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 03 2009]
The rectangular array R(n,k), read by diagonals is the number of ways n people can queue up in k (possibly empty) distinct queues. R(n,k)=(n+k-1)!/(k-1)!; R(n,k)=(n+k-1)*R(n-1,k) Northwest corner: 1, 2, 3, 4, 5, ... 2, 6, 12, 20, 30, ... 6, 24, 60, 120, 210, ... 24, 120, 360, 840, 1680, ... 120, 720, 2520, 6720, 15120,... . . . . . . Example: R(2,2)=6 because there are six ways that two people can get in line at a fast food restaurant that has two order windows open. Let 1 and 2 represent the two people and a | will seperate the lines. 12|; 21|; |12; |21; 1|2; 2|1. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 06 2009]
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