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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

Author?, Table of n, a(n) for n = 0..1374

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,0,1,2,-3,1).

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-Fran├žois Alcover, Mar 26 2013 *)

CROSSREFS

Cf. A038846 for labeled case.

Cf. A000217, A002623, A002620.

Sequence in context: A253895 A004006 A089240 * A293467 A011795 A051170

Adjacent sequences:  A057521 A057522 A057523 * A057525 A057526 A057527

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 25 20:38 EDT 2019. Contains 321477 sequences. (Running on oeis4.)