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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.
2
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.
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 A372167
KEYWORD
nonn,tabl,more
AUTHOR
Geoffrey Critzer, Jul 10 2021
STATUS
approved