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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)
0, 6, 350, 43260, 14591171, 14657461469, 46173502811223, 474928141312623525, 16489412944755088235117, 1985178211854071817861662307, 846428472480689964807653763864449, 1299141117072945982773752362381072143359, 7268140170419155675761326840423792818571154945, 149650282980396792665043455999899697765782372693740287 (list; graph; refs; listen; history; text; internal format)



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.


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).


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

J. Kahn, J. Komlos and E. Szemeredi, On the probability that a random ±1-matrix is singular, J. AMS 8 (1995), 223-240.

Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

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.

Index entries for sequences related to binary matrices


a(n) = (-1)*Sum_{k=0..n-1} Stirling1(n+1, k+1)*binomial(2^k-1, n).

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


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


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


(PARI) a(n) = -sum(k=0, n-1, stirling(n+1, k+1, 1)*binomial(2^k-1, n)); \\ Michel Marcus, Jun 05 2020

(Magma) [ -(&+[StirlingFirst(n+1, k+1)*Binomial(2^k-1, n): k in [0..n-1]]): n in [2..15]]; // G. C. Greubel, Jun 05 2020

(Sage) [sum((-1)^(n+k+1)*stirling_number1(n+1, k+1)*binomial(2^k-1, n) for k in (0..n-1)) for n in (2..15)] # G. C. Greubel, Jun 05 2020


Cf. A000410, A002884, A046747.

Sequence in context: A289738 A211089 A221923 * A214445 A059415 A246112

Adjacent sequences: A000406 A000407 A000408 * A000410 A000411 A000412




N. J. A. Sloane


Edited by W. Edwin Clark, Nov 02 2003



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Last modified March 28 03:48 EDT 2023. Contains 361577 sequences. (Running on oeis4.)