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A057524 Number of 3 x n binary matrices without unit columns up to row and column permutations. 16
1, 3, 7, 14, 25, 41, 64, 95, 136, 189, 256, 339, 441, 564, 711, 885, 1089, 1326, 1600, 1914, 2272, 2678, 3136, 3650, 4225, 4865, 5575, 6360, 7225, 8175, 9216, 10353, 11592, 12939, 14400, 15981, 17689, 19530, 21511, 23639, 25921, 28364, 30976 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 3 points of that set uniquely (if offset is 3).
LINKS
FORMULA
(1/6)*(Z(S_n; 5, 5, ...)+3*Z(S_n; 3, 5, 3, 5, ...)+2*Z(S_n; 2, 2, 5, 2, 2, 5, ...)) where Z(S_n; x_1, x_2, x_3, ...) is cycle index of symmetric group S_n of degree n.
G.f.: 1/(1-x^3)/(1-x^2)/(1-x)^3.
Let P(i,k) be the number of integer partitions of n into k parts, then with k=3 we have a(n) = Sum_{m=1..n} Sum_{i=k..m} P(i,k). - Thomas Wieder, Feb 18 2007
a(n) = Sum_{m=0..n} (n-m+1)*floor(((m+3)^2+3)/12). [Renzo Benedetti, Sep 30 2009]
a(n) = floor( ((n+2)*(n+6)/12)^2 ) = round( ((n+2)*(n+6)/12)^2 ). [Renzo Benedetti, Jul 25 2012]
Partial sums of A000601. - R. J. Mathar, Jul 25 2012
EXAMPLE
There are 7 binary 3x2 matrices without unit columns up to row and column permutations:
[0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
[0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
[0 0] [0 1] [1 1] [0 1] [1 1] [1 1] [1 1].
MATHEMATICA
CoefficientList[ Series[ 1/(1 - x^3)/(1 - x^2)/(1 - x)^3, {x, 0, 42}], x] (* Jean-François Alcover, Mar 26 2013 *)
CROSSREFS
Cf. A038846 for labeled case.
Sequence in context: A365641 A004006 A089240 * A293467 A011795 A051170
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Sep 02 2000
EXTENSIONS
More terms from James A. Sellers, Sep 07 2000
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)