A089673
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 and the main antidiagonal, are all distinct.
Original entry on oeis.org
0, 0, 0, 652, 1658784, 10726929248, 172790068546048
Offset: 1
Binary matrices with distinct rows and columns, various versions:
A059202,
A088309,
A088310,
A088616,
A089673,
A089674,
A093466,
A094000,
A094223,
A116532,
A116539,
A181230,
A259763
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
Binary matrices with distinct rows and columns, various versions:
A059202,
A088309,
A088310,
A088616,
A089673,
A089674,
A093466,
A094000,
A094223,
A116532,
A116539,
A181230,
A259763
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
Binary matrices with distinct rows and columns, various versions:
A059202,
A088309,
A088310,
A088616,
A089673,
A089674,
A093466,
A094000,
A094223,
A116532,
A116539,
A181230,
A259763
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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 *)
-
a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k+n-1, n)); \\ Michel Marcus, Dec 17 2022
A059203
Number of n-block T_0-covers of a labeled set.
Original entry on oeis.org
1, 1, 6, 2270, 148109472315, 186266607433353989829111737621541, 7485122439882901107741903784218892557452456923078744798141861944074340339271507786827
Offset: 0
a(4) = 1 + (1/4!)*( - 50*[1!*e] + 35*[3!*e] - 10*[7!*e] + [15!*e]) = 1 + (1/4!)*( - 50*2 + 35*16 - 10*13700 + 3554627472076) = 148109472315, where [k!*e] := floor(k!*exp(1)).
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with(combinat): Digits := 1500: f := n->(-1)^n+(1/n!)*sum(stirling1(n+1,i)*floor((2^(i-1)-1)!*exp(1)), i=2..n+1): for n from 1 to 10 do printf(`%d,`,f(n)) od:
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a[0] := 1; a[n_] := (-1)^n + (1/n!)*Sum[StirlingS1[n + 1, k]*Floor[(2^(k - 1) - 1)!*E], {k, 2, n + 1}]; Table[a[n], {n, 0, 5}] (* G. C. Greubel, Dec 28 2016 *)
A318537
Irregular triangle read by rows: T(n,m) is the number of n X m (0,1)-matrices with pairwise distinct nonzero columns and pairwise distinct nonzero rows, n >= 0, m = 0..2^n-1.
Original entry on oeis.org
1, 0, 1, 0, 0, 6, 6, 0, 0, 6, 174, 840, 2520, 5040, 5040, 0, 0, 0, 840, 24360, 335160, 3553200, 32382000, 259459200, 1816214400, 10897286400, 54486432000, 217945728000, 653837184000, 1307674368000, 1307674368000, 0, 0, 0, 2520, 335160, 15198120, 476496720, 12767000400, 314181504000, 7288444800000
Offset: 0
Triangle begins:
n=0: 1;
n=1: 0, 1;
n=2: 0, 0, 6, 6;
n=3: 0, 0, 6, 174, 840, 2520, 5040, 5040;
...
-
{ A318537(n,m) = m! * sum(i=0,n, stirling(n+1,i+1)*binomial(2^i - 1,m)); }
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
-
{ A318538(n) = n! * sum(i=0, n, stirling(n+1, i+1) * binomial(2^i - 1, n) ); }
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