A057222 Number of 4 X n binary matrices with 1 unit column up to row and column permutations.

1, 6, 27, 102, 333, 969, 2572, 6309, 14472, 31333, 64500, 127011, 240475, 439626, 778848, 1341286, 2251350, 3691629, 5925443, 9326451, 14417175, 21918490, 32812572, 48422262, 70510271, 101402091, 144137322, 202654565, 282015876, 388677651, 530815688, 718713015, 965220510

Offset: 1

Author

Vladeta Jovovic, Sep 18 2000

Keywords

• nonn

Comments

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 4-covers of an unlabeled n-set that cover 5 points of that set uniquely (if offset is 5).

Generally, the number b(n, k) of 4 X n binary matrices with k=0, 1, ..., n unit columns, up to row and column permutations, is coefficient of x^k in 1/24*(Z(S_n; 12 + 4*x, 12 + 4*x^2, ... ) + 8*Z(S_n; 3 + x, 3 + x^2, 12 + 4*x^3, 3 + x^4, 3 + x^5, 12 + 4*x^6, ...) + 6*Z(S_n; 6 + 2*x, 12 + 4*x^2, 6 + 2*x^3, 12 + 4*x^4, ...) + 3*Z(S_n; 4, 12 + 4*x^2, 4, 12 + 4*x^4, ...) + 6*Z(S_n; 2, 4, 2, 12 + 4*x^4, 2, 4, 2, 12 + 4*x^8, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

G.f. for b(n,k), k=0,1,..,n, is 1/k!* k - th derivative of 1/24*(1/(1 - x)^12/(1 - x*t)^4 + 8/(1 - x)^3/(1 - x^3)^3/(1 - x^3*t^3)/(1 - x*t) + 6/(1 - x)^6/(1 - x^2)^3/(1 - x^2*t^2)/(1 - x*t)^2 + 3/(1 - x)^4/(1 - x^2)^4/(1 - x^2*t^2)^2 + 6/(1 - x)^2/(1 - x^2)/(1 - x^4)^2/(1 - x^4*t^4)) with respect to t, when t=0.

Links

• V. Jovovic, Generating functions

Crossrefs

Cf. A057524, A001752, A056885, A057223.

Formula

G.f.: 1/6*x*(1/(1-x)^12+2/(1-x^3)^3/(1-x)^3+3/(1-x^2)^3/(1-x)^6).

Extensions

Added more terms, Joerg Arndt, May 21 2013