A346381 - OEIS

%I #23 Jan 11 2025 18:50:07

%S 1,0,1,2,3,1,48,98,21,1,5824,11640,2590,105,1,2887680,5775424,1283400,

%T 52390,465,1,5821595648,11643190272,2587376064,105607080,938742,1953,

%U 1,47317927329792,94635854692352,21030189917184,858375102144,7630000488,15879318,8001,1

%N 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.

%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

%F For n>=1, Sum_{k=0..n} T(n,k)*2^k = 2*A002884(n). - _Geoffrey Critzer_, Jan 10 2025

%e Triangle begins:

%e         1;

%e         0,       1;

%e         2,       3,      1;

%e        48,      98,     21,      1;

%e      5824,   11640,   2590,    105,   1;

%e   2887680, 5775424, 1283400, 52390, 465, 1;

%e   ...

%e T(2,0) = 2 because {{0, 1}, {1, 1}}, {{1, 1}, {1, 0}} do not have 1 as an eigenvalue.

%e 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.

%e T(2,2) = 1 because {{1, 0}, {0, 1}} fixes the entire space.

%t 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 =

%t Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];

%t 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

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

%K nonn,tabl,changed

%O 0,4

%A _Geoffrey Critzer_, Jul 14 2021