A086699 -id:A086699 - cp's OEIS frontend

A086752 Number of n X n matrices over GF(3) with rank n-1.

1, 32, 8112, 17971200, 355207057920, 63010655570903040, 100505356319291594711040, 1442361950110091891786121216000, 186276322602412236974585775503690956800, 216505458700483736766078241517019274701019545600, 2264736353104098912130003755084217747715114856943819161600

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 31 2003

Cf. A086699, A053290, A060868.

Table[3^(n^2) * (1 - 1/3^n) * QPochhammer[1/3^n, 3, n-1]/2, {n, 1, 10}] (* Vaclav Kotesovec, Apr 14 2024 *)

For n>=2: a(n) = Product_{j=0..n-2} (3^n - 3^j)^2 / (3^(n-1)- 3^j).

a(n) = ((3^n-1)/2)*Product_{j=0..n-2} (3^n-3^j). - David Wasserman, Mar 28 2005

More terms from David Wasserman, Mar 28 2005

A379105 Triangular array read by rows. T(n,k) is the number of n X n matrices T over GF(2) such that there are exactly 2^k vectors v in GF(2)^n with Tv=v, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 6, 9, 1, 168, 294, 49, 1, 20160, 37800, 7350, 225, 1, 9999360, 19373760, 4036200, 144150, 961, 1, 20158709760, 39687459840, 8543828160, 326932200, 2542806, 3969, 1, 163849992929280, 325139829719040, 71124337751040, 2812314375360, 23435953128, 42677334, 16129, 1

Offset: 0

Geoffrey Critzer, Dec 15 2024

Sum_{k=0..n} T(n,k)*2^k = (2^(n+1)-1)*2^(n^2-n) so that as n->oo the average number of fixed points is 2.

T(n,k) is also the number of n X n matrices over GF(2) with nullity k. As n->oo, the probability that a random n X n matrix over GF(q) has nullity k is 1/|GL_k(F_q)|*Product_{i>=k+1} (1 - 1/q^i). - Geoffrey Critzer, Dec 31 2024

			Triangle T(n,k) begins:
        1;
        1,        1;
        6,        9,       1;
      168,      294,      49,      1;
    20160,    37800,    7350,    225,   1;
  9999360, 19373760, 4036200, 144150, 961, 1;
  ...

Cf. A060867 (T(n,n-1)), A002884 (column k=0), A086699 (column k=1), A346381, A286331.

Row sums give A002416.

nn = 5; b[p_, i_] := Count[p, i];d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];aut[deg_, p_] :=Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; L=Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]; g[u_, v_, deg_] :=  Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &, L]]; Map[Select[#, # > 0 &] &,  Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0,nn}] CoefficientList[Series[g[u, 1, 1] g[u, v, 1] Product[g[u, 1, deg]^\[Nu][[deg]], {deg, 2, nn}], {u, 0, nn}], {u,v}]] // Grid

T(n,k)=Product_{j=0..n-k-1} (2^n - 2^j)^2/(2^(n-k)-2^j). - Geoffrey Critzer, Dec 31 2024

cp's OEIS Frontend

A086752 Number of n X n matrices over GF(3) with rank n-1.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions