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A135814 Triangle of numbers of coincidence-free length n-m lists of m-tuples with all numbers 1,...,n-m in tuple position k, for k=1..m.

%I A135814 #13 Aug 29 2019 17:48:04
%S A135814 1,0,1,0,0,1,0,1,0,1,0,2,3,0,1,0,9,26,7,0,1,0,44,453,194,15,0,1,0,265,
%T A135814 11844,13005,1250,31,0,1,0,1854,439975,1660964,326685,7682,63,0,1,0,
%U A135814 14833,22056222,363083155,205713924,7931709,46466,127,0,1,0,133496
%N A135814 Triangle of numbers of coincidence-free length n-m lists of m-tuples with all numbers 1,...,n-m in tuple position k, for k=1..m.
%C A135814 The column sequences (without leading zeros) give A000007, A000166 (subfactorials), A089041, A135809 - A135813, for m=0..7.
%C A135814 a(n,m), n >= m, enumerates (ordered) length n-m lists of m-tuples such that every number from 1 to n-m appears once at each of the n-m tuple positions and the j-th list member is not the tuple (j,j,...,j) (m times j), for every j=1,...,n-m. Called coincidence-free m-tuple lists of length n-m. See the Charalambides reference for this combinatorial interpretation.
%D A135814 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a).
%H A135814 W. Lang, <a href="/A135814/a135814.txt">First 10 rows and more</a>.
%F A135814 a(n,m) = Sum_{j=0..n-m} (-1)^(n-m-j)*binomial(n-m,j)*(j!)^m, n >= m >= 0, otherwise 0.
%e A135814 [1]; [0,1]; [0,0,1]; [0,1,0,1]; [,0,2,3,0,1]; [0,9,26,7,0,1]; ...
%e A135814 The a(5,3)=7 lists of length 5-3=2 with coincidence-free 3-tuples are [(1,1,2),(2,2,1)], [(1,2,1),(2,1,2)], [(2,1,1),(1,2,2)], [(1,2,2),(2,1,1)], [(2,1,2),(1,2,1)], [(2,2,1),(1,1,2)] and [(2,2,2),(1,1,1)]. The list [(1,1,1),(2,2,2)] is not coincidence-free because (1,1,1) appears at position 1 and also because (2,2,2) appears at position 2.
%K A135814 nonn,easy,tabl
%O A135814 0,12
%A A135814 _Wolfdieter Lang_, Jan 21 2008, Feb 22 2008, May 21 2008