cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A089674 a(n) = number of n X n (0,1) matrices A such that the 2n+2 vectors consisting of the rows and the columns of the matrix A, as well as the main diagonal read in the upward direction and the main antidiagonal, are all distinct.

Original entry on oeis.org

0, 0, 0, 1692, 2329280, 13441654352, 190945826194432
Offset: 1

Views

Author

Vladeta Jovovic, Jan 04 2004

Keywords

Crossrefs

Cf. A088310, A088616, A089673.
Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

Extensions

a(6)-a(7) from Bert Dobbelaere, May 05 2025

A094223 Number of binary n X n matrices with all rows (columns) distinct, up to permutation of columns (rows).

Original entry on oeis.org

1, 2, 7, 68, 2251, 247016, 89254228, 108168781424, 451141297789858, 6625037125817801312, 348562672319990399962384, 66545827618461283102105245248, 46543235997095840080425299916917968, 120155975713532210671953821005746669185792, 1152009540439950050422144845158703009569109376384
Offset: 0

Views

Author

Goran Kilibarda and Vladeta Jovovic, May 28 2004

Keywords

Links

Crossrefs

Main diagonal of A059584 and A059587, A060690, A088309.
Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n - k)*StirlingS1[n, k]*Binomial[2^k, n], {k, 0, n}]; (* or *) a[n_] := Sum[ StirlingS1[n, k]*Binomial[2^k + n - 1, n], {k, 0, n}]; Table[ a[n], {n, 0, 12}] (* Robert G. Wilson v, May 29 2004 *)
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k+n-1, n)); \\ Michel Marcus, Dec 17 2022

Formula

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

Extensions

More terms from Robert G. Wilson v, May 29 2004
a(13) onwards from Andrew Howroyd, Jan 20 2024

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.

Original entry on oeis.org

0, 6, 350, 43260, 14591171, 14657461469, 46173502811223, 474928141312623525, 16489412944755088235117, 1985178211854071817861662307, 846428472480689964807653763864449, 1299141117072945982773752362381072143359, 7268140170419155675761326840423792818571154945, 149650282980396792665043455999899697765782372693740287
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

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

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

A116527 Number of singular n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct and all columns distinct, up to permutation of rows.

Original entry on oeis.org

0, 0, 0, 75, 22365, 13303500, 21058940420, 98692672142610
Offset: 1

Views

Author

Vladeta Jovovic, Apr 03 2006

Keywords

Crossrefs

Cf. A000409, A000410, A116506, A116507, A116532.

Formula

a(n) = A094000(n) - A088389(n).
Conjecture: a(n) = A000410(n) - A000409(n-1) for n>1. - Jean-François Alcover, Jan 08 2020

A318538 Number of n X n (0,1)-matrices with nonzero pairwise distinct rows and nonzero pairwise distinct columns.

Original entry on oeis.org

1, 1, 6, 174, 24360, 15198120, 38415132000, 376482729702240, 14139748304132048640, 2040859528996474439366400, 1141301651605590355550899891200, 2494751188402618305982805631973248000, 21474225685319103561274021904272069843353600
Offset: 0

Views

Author

Max Alekseyev, Aug 28 2018

Keywords

Crossrefs

Main diagonal of A318537.
Cf. A094000, A059202.

Programs

  • PARI
    { A318538(n) = n! * sum(i=0, n, stirling(n+1, i+1) * binomial(2^i - 1, n) ); }

Formula

a(n) = n! * Sum_{i=0..n} Stirling1(n+1,i+1) * binomial(2^i-1,n).
a(n) = A318537(n,n).
a(n) = A094000(n) * n!.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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