A083402 -id:A083402 - cp's OEIS frontend

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, 1, 6619979775, 9978120069120, 421755245936640, 1989607056998400, 6963624699494400, 20890874098483200, 41781748196966400

Offset: 1

Geoffrey Critzer, Jul 10 2021

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

Define the co-index of an n X n matrix A to be n - index(A). Let X_n be the random variable that assigns to each nilpotent n X n matrix over GF(2) the value j in {0,1,...,n-1} of its co-index. Conjecture: lim_{n->inf} P(X_n = j) = Product_{i>=1}1-1/2^i * 2^((j-1)^2)/A002884(j). Moreover, for j < 2n, T(n,n-j) = A002884(n)/(A002884(j)*2^(n - (j-1)^2)). - Geoffrey Critzer, Jun 10 2025

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

Cf. A083402 (main diagonal), A053763 (row sums), A002884, A048651.

nn = 8;  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]}];
l = Level[Table[IntegerPartitions[n],  {n,  0,  nn}],  {2}];
\[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}];
g[u_,  v_,  deg_,  partitions_] := Total[Map[v^If[# == {},  0,  Max[#]] u^(deg Total[#])/aut[deg,  #] &, partitions]];
Map[Select[#,  # > 0 &] &, Drop[Table[\[Gamma][n,  q],  {n,  0,  nn}] CoefficientList[     Series[g[u,  v,  1,  l],  {u,  0,  nn}],  {u,  v}],  1]] // Grid

More terms from Geoffrey Critzer, Jun 10 2025

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.

Original entry on oeis.org

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

Geoffrey Critzer, Jul 15 2021

			Array begins
  1;
  1,    3;
  1,   21,     42;
  1,  105,   1470,     2520;
  1,  465,  32550,   390600,    624960;
  1, 1953, 605430, 36325800, 406848960, 629959680
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.

G. Lusztig, A note on counting nilpotent matrices of fixed rank, Bull. London Math. Soc. v.8 (1976), no. 1, 77--80; MR0407050.

Jason Fulman, A probabilistic approach toward the finite general linear and unitary group, arxiv.org, Dec 1997, page 12.
J. Kung, The cycle structure of a linear transformation over a finite field,Linear Algebra and its Applications, vol 36, March 1981,141-155.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

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

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]}];
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

T(n,n-k) = A002884(n)*Product_{i=k..n-1}(1-1/2^i)/(A002884(k)*2^(n-k)*Product_{i=1..n-k}(1-1/2^i)) Theorem 6 in Fulman link. - Geoffrey Critzer, Dec 23 2024

A358433 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with index k, n>=1, 1<=k<=n.

Original entry on oeis.org

2, 13, 3, 365, 105, 42, 43801, 12915, 6300, 2520, 21725297, 6412815, 3228960, 1562400, 624960, 43798198753, 12928608063, 6533019360, 3254791680, 1574899200, 629959680, 355991759464385, 105083758588095, 53109556520832, 26576858972160, 13227473387520, 6400390348800, 2560156139520

Offset: 1

Geoffrey Critzer, Nov 15 2022

The index of a matrix A is the smallest positive integer such that rank(A^k) = rank(A^(k+1)).

			      2,
      13,       3,
     365,     105,      42,
   43801,   12915,    6300,    2520,
21725297, 6412815, 3228960, 1562400, 624960,

Cf. A002416 (row sums), A348015 (column k=1), A083402 (main diagonal for n>1), A346214.

nn = 6; q = 2; b[p_, i_] := Count[p, i];s[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^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], {n, 1, nn}];
l[greatestpart_] :=Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0,nn}], {2}];g1[u_, v_, deg_] := Total[Map[v ^(If[ Max[Prepend[#, 0]] == 0, 1, Max[Prepend[#, 0]]]) u^(deg Total[#])/aut[deg, #] &, l[nn]]]; Map[Select[#, # > 0 &] &,Drop[Table[Product[q^n - q^i, {i, 0, n - 1}], {n,0,nn}]CoefficientList[
  Series[g1[u, v, 1] g1[u, 1, 1]^(q - 1) Product[g1[u, 1, d]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], {u, v}], 1]] // Grid

A358739 Triangular array read by rows. T(n,k) is the number of n X n matrices A over F_2 such that Sum_{phi} nullity(phi(A)) = k where the sum is over all monic irreducible polynomials in F_2[x] that divide the characteristic polynomial of A, n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 6, 10, 84, 210, 218, 5040, 19740, 15330, 25426, 1249920, 5780880, 6939660, 7604610, 11979362, 1259919360, 7533267840, 9297061200, 12276675180, 14280964866, 24071588290, 5120312279040, 34082078607360, 48312946523520, 78970351980240, 88215877158444, 112601184828930, 195647202043778

Offset: 1

Geoffrey Critzer, Nov 29 2022

			Triangle begins
        2;
        6,      10;
       84,     210,     218;
     5040,   19740,   15330,   25426;
  1249920, 5780880, 6939660, 7604610, 11979362;
  ...

Cf. A346222 (main diagonal), A083402 (1/2 * column 1), A002416 (row sums).

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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.

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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.

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A358433 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with index k, n>=1, 1<=k<=n.

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A358739 Triangular array read by rows. T(n,k) is the number of n X n matrices A over F_2 such that Sum_{phi} nullity(phi(A)) = k where the sum is over all monic irreducible polynomials in F_2[x] that divide the characteristic polynomial of A, n >= 1, 1 <= k <= n.

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