

A135814


Triangle of numbers of coincidencefree length nm lists of mtuples with all numbers 1,...,nm in tuple position k, for k=1..m.


3



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

0,12


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 nm lists of mtuples such that every number from 1 to nm appears once at each of the nm tuple positions and the jth list member is not the tuple (j,j,...,j) (m times j), for every j=1,...,nm. Called coincidencefree mtuple lists of length nm. See the Charalambides reference for this combinatorial interpretation.


REFERENCES

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


LINKS

Table of n, a(n) for n=0..56.
W. Lang, First 10 rows and more.


FORMULA

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


EXAMPLE

[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 53=2 with coincidencefree 3tuples 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 coincidencefree because (1,1,1) appears at position 1 and also because (2,2,2) appears at position 2.


CROSSREFS

Sequence in context: A123735 A155839 A229615 * A038570 A103498 A030386
Adjacent sequences: A135811 A135812 A135813 * A135815 A135816 A135817


KEYWORD

nonn,easy,tabl


AUTHOR

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


STATUS

approved



