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A135814
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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.
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3
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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
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OFFSET
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0,12
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COMMENTS
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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.
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REFERENCES
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Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a).
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LINKS
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FORMULA
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a(n,m) = Sum_{j=0..n-m} (-1)^(n-m-j)*binomial(n-m,j)*(j!)^m, n >= m >= 0, otherwise 0.
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EXAMPLE
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[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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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