OFFSET
1,3
COMMENTS
The index of a nilpotent matrix A is the smallest positive integer k such that A^k = 0.
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
KEYWORD
AUTHOR
Geoffrey Critzer, Jul 10 2021
STATUS
approved