0,12

The column sequences (without leading zeros) give A000007, A000166 (subfactorials), A089041, A135809 - A135813, for m=0..7.

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.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a).

W. Lang, First 10 rows and more.

a(n,m)= sum(((-1)^(n-m-j))*binomial(n-m,j)*(j!)^m,j=0..n-m), n >= m >= 0, else 0.

[1];[0,1];[0,0,1];[0,1,0,1];[,0,2,3,0,1];[0,9,26,7,0,1],...

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.

Sequence in context: A064722 A123735 A155839 this_sequence A038570 A103498 A030386

Adjacent sequences: A135811 A135812 A135813 this_sequence A135815 A135816 A135817

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 21 2008, Feb 22 2008, May 21 2008