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.

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, 11844, 13005, 1250, 31, 0, 1, 0, 1854, 439975, 1660964, 326685, 7682, 63, 0, 1, 0, 14833, 22056222, 363083155, 205713924, 7931709, 46466, 127, 0, 1, 0, 133496

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008, Feb 22 2008, May 21 2008

Keywords

• nonn
• easy
• tabl

Comments

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.

Examples

			[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.
		

References

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

Links

• W. Lang, First 10 rows and more.

Formula

a(n,m) = Sum_{j=0..n-m} (-1)^(n-m-j)*binomial(n-m,j)*(j!)^m, n >= m >= 0, otherwise 0.