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.

0, 6, 350, 43260, 14591171, 14657461469, 46173502811223, 474928141312623525, 16489412944755088235117, 1985178211854071817861662307, 846428472480689964807653763864449, 1299141117072945982773752362381072143359, 7268140170419155675761326840423792818571154945, 149650282980396792665043455999899697765782372693740287

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

• 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

• 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

Crossrefs

Cf. A000410, A002884, A046747.

Programs

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

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-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

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

Formula

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

Extensions

Edited by W. Edwin Clark, Nov 02 2003