A346214 Triangular array read by rows. T(n,k) is the number of nilpotent n X n matrices over GF(2) with index k, 1 <= k <= n, n >= 1.

1, 1, 3, 1, 21, 42, 1, 315, 1260, 2520, 1, 6975, 104160, 312480, 624960, 1, 373023, 23436000, 104993280, 314979840, 629959680, 1, 32252031, 9175162752, 121912197120, 426692689920, 1280078069760, 2560156139520

Offset

1,3

Comments

The index of a nilpotent matrix A is the smallest positive integer k such that A^k = 0.

Links

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

Example

  1,
  1,    3,
  1,   21,     42,
  1,  315,   1260,   2520,
  1, 6975, 104160, 312480, 624960

Mathematica

  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]}] /. q -> 2; gl2 =
Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 1, 50}]; Table[Table[
   Sum[gl2[[n]]/ aut[1, Select[IntegerPartitions[n], #[[1]] == k &][[i]]], {i, 1,
     Length[Select[IntegerPartitions[n], #[[1]] == k &]]}], {k, 1, n}], {n, 1, 7}] // Grid

Crossrefs

Cf. A083402 (main diagonal), A053763 (row sums).

Sequence in context: A138354 A193632 A346412 * A190962 A010291 A370948

Adjacent sequences: A346211 A346212 A346213 * A346215 A346216 A346217

Keyword

nonn,tabl,more

Author

Geoffrey Critzer, Jul 10 2021

Status

approved