The On-Line Encyclopedia of Integer Sequences (OEIS)

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A346412

Triangular array read by rows: T(n,k) is the number of nilpotent n X n matrices over GF(2) having rank k, 0 <= k <= n-1, n >= 1.
(history; published version)

#9 by N. J. A. Sloane at Wed Jul 21 09:29:03 EDT 2021

STATUS

proposed

approved

#8 by Jon E. Schoenfield at Fri Jul 16 12:37:30 EDT 2021

STATUS

editing

proposed

#7 by Jon E. Schoenfield at Fri Jul 16 12:37:28 EDT 2021

NAME

Triangular array read by rows. : T(n,k) is the number of nilpotent n X n matrices over GF(2) having rank k, 0 <= k <= n-1, n >= 1.

LINKS

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.

EXAMPLE

Array begins

1,;

1, 3,;

1, 21, 42,;

1, 105, 1470, 2520,;

1, 465, 32550, 390600, 624960,;

T(2,0) = 1 because the zero matrix has rank 0.

T(2,1) = 3 because {{0,0},{1,0}}, {{0,1},{0,0}}, {{1,1},{1,1}} have rank 1.

STATUS

proposed

editing

#6 by Geoffrey Critzer at Fri Jul 16 11:52:02 EDT 2021

STATUS

editing

proposed

#5 by Geoffrey Critzer at Thu Jul 15 16:24:03 EDT 2021

LINKS

#4 by Geoffrey Critzer at Thu Jul 15 16:22:43 EDT 2021

NAME

Triangular array read by rows. T(n,k) is the number of nilpotent n X n matrices over GF(2) having rank k, 0<=k<=n-1, n>=1.

EXAMPLE

T(2,0) = 1 because the zero matrix has rank 0.

T(2,1) = 3 because {{0,0},{1,0}}, {{0,1},{0,0}}, {{1,1},{1,1}} have rank 1.

CROSSREFS

Cf. A134057 (column k=1), A083402 (main diagonal), A053763 (row sums).

#3 by Geoffrey Critzer at Thu Jul 15 16:17:26 EDT 2021

NAME

Triangular array read by rows. T(n,k) is the number of nilpotent n X n matrices having rank k, 0<=k<=n-1, n>=1.

EXAMPLE

1,

CROSSREFS

Cf. A134057 (column k=1), A083402 (main diagonal), A053763 (row sums).

#2 by Geoffrey Critzer at Thu Jul 15 16:08:17 EDT 2021

NAME

allocated for Geoffrey Critzer Triangular array read by rows. T(n,k) is the number of nilpotent n X n matrices having rank k, 0<=k<=n-1, n>=1.

DATA

1, 1, 3, 1, 21, 42, 1, 105, 1470, 2520, 1, 465, 32550, 390600, 624960, 1, 1953, 605430, 36325800, 406848960, 629959680, 1, 8001, 10417302, 2768025960, 155009453760, 1680102466560, 2560156139520, 1, 32385, 172741590, 192779614440, 47809344381120, 2590958018073600, 27636885526118400, 41781748196966400

OFFSET

1,3

EXAMPLE

1,

1, 3,

1, 21, 42,

1, 105, 1470, 2520,

1, 465, 32550, 390600, 624960,

1, 1953, 605430, 36325800, 406848960, 629959680

MATHEMATICA

nn = 10; 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 = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];

g[u_, v_] := Total[Map[v^(Total[#] - Length[#]) u^Total[#]/aut[1, #] &,

Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]]; Map[Select[#, # > 0 &] &, Drop[Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[

Series[g[u, v], {u, 0, nn}], {u, v}], 1]] // Grid

KEYWORD

allocated

nonn,tabl

AUTHOR

Geoffrey Critzer, Jul 15 2021

STATUS

approved

editing

#1 by Geoffrey Critzer at Thu Jul 15 16:08:17 EDT 2021

NAME

allocated for Geoffrey Critzer

KEYWORD

allocated

STATUS

approved