A346209 -id:A346209 - cp's OEIS frontend

A346421 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(3) such that the sum of the dimensions of its eigenspaces taken over all its eigenvalues is k, 0 <= k <= n, n >= 0.

1, 0, 3, 18, 24, 39, 3456, 8190, 5928, 2109, 7619508, 17094240, 13700700, 4215120, 417153, 149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179, 26394940582090344, 59388527912287392, 47325384827973252, 15262273318168800, 1648005959253654, 74268805562952, 1233891662727

Offset: 0

Geoffrey Critzer, Jul 16 2021

			             1;
             0,            3;
            18,           24,           39;
          3456,         8190,         5928,        2109;
       7619508,     17094240,     13700700,     4215120,     417153;
  149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179;

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

Cf. A346209 (column k=0), A290516 (main diagonal), A060722 (row sums).

nn = 7; q = 3; 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[Take[(Table[ Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v]^3 Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}])[[n]],n], {n, 1, nn}] // Grid

A364886 Number of n X n (-1, 1)-matrices which have only eigenvalues with strictly negative real part (which implies that the matrix has all nonzero eigenvalues).

Original entry on oeis.org

1, 2, 20, 640, 97824, 47545088

Offset: 1

Thomas Scheuerle, Aug 12 2023

As this problem is symmetric with sign we can get the same numbers for strictly positive real parts.
All values for n > 1 are even, because a transposed matrix has the same spectrum of eigenvalues.
Matrices with determinant 0 are not counted.
Let M be such a matrix then the limit of ||exp(t*M)*y|| if t goes to infinity will be zero.
n = 5 is the first case where not all entries on the main diagonal are -1. 93984 matrices with 5 times -1 on the main diagonal and 5*768 with 4 times -1 on the main diagonal have only eigenvalues with strictly negative real part.
In the case n = 6, 43586048 matrices with 6 times -1 on the main diagonal, 6*656000 matrices with 5 times -1 on the main diagonal and 15*1536  matrices with 5 times -1 on the main diagonal have only eigenvalues with strictly negative real part.

			For n = 2 the matrices are:
.
    -1,  1
    -1, -1
.
    -1, -1
     1, -1.

Index entries for sequences related to binary matrices

Cf. A056990.
Cf. A083058, A085506, A087488, A098148, A086510, A207259.
Cf. A219736, A346210, A271570, A271588, A296605, A306002.
Cf. A306791, A306792, A306793, A306794, A306795, A326928.
Cf. A346209.

cp's OEIS Frontend

A346421 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(3) such that the sum of the dimensions of its eigenspaces taken over all its eigenvalues is k, 0 <= k <= n, n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica