A358433 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with index k, n>=1, 1<=k<=n.

2, 13, 3, 365, 105, 42, 43801, 12915, 6300, 2520, 21725297, 6412815, 3228960, 1562400, 624960, 43798198753, 12928608063, 6533019360, 3254791680, 1574899200, 629959680, 355991759464385, 105083758588095, 53109556520832, 26576858972160, 13227473387520, 6400390348800, 2560156139520

Offset

1,1

Comments

The index of a matrix A is the smallest positive integer such that rank(A^k) = rank(A^(k+1)).

Links

Table of n, a(n) for n=1..28.

Example

      2,
      13,       3,
     365,     105,      42,
   43801,   12915,    6300,    2520,
21725297, 6412815, 3228960, 1562400, 624960,

Mathematica

nn = 6; q = 2; b[p_, i_] := Count[p, i]; s[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^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], {n, 1, nn}];
l[greatestpart_] :=Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0, nn}], {2}]; g1[u_, v_, deg_] := Total[Map[v ^(If[ Max[Prepend[#, 0]] == 0, 1, Max[Prepend[#, 0]]]) u^(deg Total[#])/aut[deg, #] &, l[nn]]]; Map[Select[#, # > 0 &] &, Drop[Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}]CoefficientList[
  Series[g1[u, v, 1] g1[u, 1, 1]^(q - 1) Product[g1[u, 1, d]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], {u, v}], 1]] // Grid

Crossrefs

Cf. A002416 (row sums), A348015 (column k=1), A083402 (main diagonal for n>1), A346214.

Sequence in context: A337838 A120863 A286459 * A344540 A338785 A335817

Adjacent sequences: A358430 A358431 A358432 * A358434 A358435 A358436

Keyword

nonn,tabl

Author

Geoffrey Critzer, Nov 15 2022

Status

approved