A135809 -id:A135809 - cp's OEIS frontend

A135810 Number of coincidence-free length n lists of 4-tuples with all numbers 1..n in tuple position k, for k=1..4.

1, 0, 15, 1250, 326685, 205713924, 267499350955, 643364455632870, 2637753876195952185, 17316358344270678304520, 173227930768100416550798151, 2536860701329458663625695526890

Offset: 0

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

a(n) enumerates (ordered) lists of n 4-tuples such that every number from 1 to n appears once at each of the four tuple positions and the j-th list member is not the tuple (j,j,j,j), for every j=1..n. Called coincidence-free 4-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.

			4-tuple combinatorics: a(1)=0 because the only list of 4-tuples composed of 1 is [(1,1,1,1)] and this is a coincidence for j=1.
4-tuple combinatorics: from the 2^4 possible 4-tuples of numbers 1 and 2 all except (1,1,1,1) appear as first members of the length 2 lists. The second members are the 4-tuples obtained by interchanging 1 and 2. E.g., one of the a(2)=15 lists is [(1,1,1,2),(2,2,2,1)]. The list [(1,1,1,1),(2,2,2,2)] does not qualify because it has in fact two coincidences, those for j=1 and j=2.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=4.

G. C. Greubel, Table of n, a(n) for n = 0..143

Cf. A135809 (coincidence-free 3-tuples), A135811 (coincidence-free 5-tuples).

Table[Sum[(-1)^k Binomial[n,k](n-k)!^4,{k,0,n}],{n,0,11}] (* Geoffrey Critzer, Jun 17 2013 *)

a(n) = Sum_{j=0..n} ( ((-1)^(n-j))*binomial(n,j)*(j!)^4 ). See the Charalambides reference a(n) = B_{n,4}.

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.

Original entry on oeis.org

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

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

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.

			[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.

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

cp's OEIS Frontend

A135810 Number of coincidence-free length n lists of 4-tuples with all numbers 1..n in tuple position k, for k=1..4.

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