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A000409 Singular n X n (0,1)-matrices: the number of n X n (0,1)-matrices having distinct, nonzero ordered rows, but having at least two equal columns or at least one zero column.
(Formerly M4306 N1801)
6
0, 6, 350, 43260, 14591171, 14657461469, 46173502811223, 474928141312623525, 16489412944755088235117, 1985178211854071817861662307, 846428472480689964807653763864449, 1299141117072945982773752362381072143359, 7268140170419155675761326840423792818571154945, 149650282980396792665043455999899697765782372693740287 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

This is a lower bound for the set of all n X n (0,1)-matrices having distinct, nonzero ordered rows and determinant 0 (compare A000410).

Here ordered means that we take only one representative from the n! matrices obtained by all permutations of the distinct rows of an n X n matrix.

a(n) is also the number of sets of n distinct nonzero (0,1)-vectors in R^n that do not span R^n.

REFERENCES

J. Kahn, J. Komlos, E. Szemeredi: On the probability that a random $\pm1$-matrix is singular, J. AMS 8 (1995), 223-240.

J. Komlos, On the determinant of (0,1)-matrices, Studia Math. Hungarica 2 (1967), 7-21.

N. Metropolis and P. R. Stein, On a class of (0,1) matrices with vanishing determinants, J. Combin. Theory, 3 (1967), 191-198.

G. Kilibarda and V. Jovovic, "Enumeration of some classes of T_0-hypergraphs", in preparation, 2004.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=2..15.

Index entries for sequences related to binary matrices

FORMULA

a(n) = -sum(stirling1(n+1, k+1)*binomial(2^k-1, n), k=0..n-1).

a(n) = binomial(2^n-1, n) - A094000(n). - Vladeta Jovovic, Nov 27 2005

MAPLE

with(combinat): T := proc(n) -sum(stirling1(n+1, k+1)*binomial(2^k-1, n), k=0..n-1); end proc:

MATHEMATICA

a[n_] := -Sum[ StirlingS1[n+1, k+1]*Binomial[2^k-1, n], {k, 0, n-1}]; Table[a[n], {n, 2, 15}] (* Jean-Fran├žois Alcover, Nov 21 2012, from formula *)

CROSSREFS

Cf. A000410, A002884, A046747.

Sequence in context: A289738 A211089 A221923 * A214445 A059415 A246112

Adjacent sequences:  A000406 A000407 A000408 * A000410 A000411 A000412

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by W. Edwin Clark, Nov 02 2003

STATUS

approved

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Last modified September 23 15:54 EDT 2017. Contains 292361 sequences.