A346381 Triangle read by rows. T(n,k) is the number of invertible n X n matrices over GF(2) such that the dimension of the eigenspace corresponding to eigenvalue 1 is k, 0 <= k <= n, n >= 0.

1, 0, 1, 2, 3, 1, 48, 98, 21, 1, 5824, 11640, 2590, 105, 1, 2887680, 5775424, 1283400, 52390, 465, 1, 5821595648, 11643190272, 2587376064, 105607080, 938742, 1953, 1, 47317927329792, 94635854692352, 21030189917184, 858375102144, 7630000488, 15879318, 8001, 1

Offset

0,4

Links

Table of n, a(n) for n=0..35.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Formula

For n>=1, Sum_{k=0..n} T(n,k)*2^k = 2*A002884(n). - Geoffrey Critzer, Jan 10 2025

Example

Triangle begins:
        1;
        0,       1;
        2,       3,      1;
       48,      98,     21,      1;
     5824,   11640,   2590,    105,   1;
  2887680, 5775424, 1283400, 52390, 465, 1;
  ...
T(2,0) = 2 because {{0, 1}, {1, 1}}, {{1, 1}, {1, 0}} do not have 1 as an eigenvalue.
T(2,1) = 3 because {{0, 1}, {1, 0}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}} have 1 as an eigenvalue with corresponding eigenspace of dimension 1.
T(2,2) = 1 because {{1, 0}, {0, 1}} fixes the entire space.

Mathematica

nn = 15; q = 2; 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]}]; A001037 =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];
g[u_, v_] := Total[Map[v^Length[#] u^Total[#]/aut[1, #] &, Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]]; Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v] Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}] // Grid

Crossrefs

Cf. A002820 (column k=0), A002884 (row sums).

Sequence in context: A350624 A009108 A016537 * A106385 A363841 A291293

Adjacent sequences: A346378 A346379 A346380 * A346382 A346383 A346384

Keyword

nonn,tabl,changed

Author

Geoffrey Critzer, Jul 14 2021

Status

approved