A348015 -id:A348015 - cp's OEIS frontend

A348622 Triangular array read by rows: T(n,k) is the number of periodic n X n matrices over GF(2) having rank k, n>=0, 0<=k<=n.

1, 1, 1, 1, 6, 6, 1, 28, 168, 168, 1, 120, 3360, 20160, 20160, 1, 496, 59520, 1666560, 9999360, 9999360, 1, 2016, 999936, 119992320, 3359784960, 20158709760, 20158709760, 1, 8128, 16386048, 8127479808, 975297576960, 27308332154880, 163849992929280, 163849992929280

Offset: 0

Geoffrey Critzer, Oct 25 2021

A matrix T is periodic if and only image(T) = image(T^2). Cf. A348015.

			Triangle begins:
  1;
  1,   1;
  1,   6,     6;
  1,  28,   168,     168;
  1, 120,  3360,   20160,   20160;
  1, 496, 59520, 1666560, 9999360, 9999360;
  ...

Alois P. Heinz, Rows n = 0..57, flattened
Eric Weisstein's World of Mathematics, Periodic Matrix

Cf. A348015 (row sums).

Main diagonal gives A002884.

b:= proc(n) option remember; mul(2^n-2^i, i=0..n-1) end:
T:= (n, k)-> b(n)/b(n-k):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Oct 30 2021

nn = 7; 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^(Length[#]) u^(deg Total[#])/aut[deg, #] &, l[1]]];
g2[u_, v_, deg_] := Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &,l[nn]]];
Map[Reverse, Map[Select[#, # > 0 &] &,Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g1[u, v, 1] g2[u, 1, 1]^(q - 1) Product[g2[u, 1, d]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], {u,v}]]] // Flatten

T(n,k) = A002884(n)/A002884(n-k).
T(n,k) = T(n,k-1)*T(n-k+1,1).
Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = e(x)*g(y*x) where e(x) = Sum_{n>=0} x^n/B(n), g(x) = Sum_{n>=0} Product_{i=0..n-1} (q^n-q^i)*x^n/B(n), B(n) = Product_{i=0..n-1} (q^n-q^i)/(q-1)^n and q=2. - Geoffrey Critzer, Jan 03 2025

Title improved by Geoffrey Critzer, Sep 16 2022

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

A373784 Number of n X n periodic matrices over GF(3).

Original entry on oeis.org

1, 3, 73, 17083, 36899281, 723299706163, 127954611219059353, 203907700711025670584683, 2925406687944886208690683289761, 377768302929213133608351637048047649123, 439058106335101098121200963942043393433721555433, 4592676942963368381676178826582121303149079572759596838683

Offset: 0

Geoffrey Critzer, Jun 18 2024

nonn

T is a periodic matrix if T = T^k for some k > 1. T is periodic iff image(T) = image(T^2) iff x^2 does not divide the minimal polynomial of T.

Eric Weisstein's World of Mathematics, Periodic Matrix

Cf. A348015, A053290.

nn = 11; B[n_] := Product[q^n - q^i, {i, 0, n - 1}]/(q - 1)^n /. q -> 3;
e[x_] := Sum[x^n/B[n], {n, 0, nn}];f[x_] := Sum[(q - 1)^n x^n, {n, 0, nn}] /. q -> 3; Table[B[n], {n, 0, nn}] CoefficientList[Series[e[x] f[ x], {x, 0, nn}], x]

Sum_{n>=0}a_n*x^n/B(n) = e(x)*f(x) where B(n) = A053290(n)/2^n, e(x) = Sum{n>=0}x^n/B(n) and f(x) = Sum_{n>=0} (2x)^n.

cp's OEIS Frontend

A348622 Triangular array read by rows: T(n,k) is the number of periodic n X n matrices over GF(2) having rank k, n>=0, 0<=k<=n.

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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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A373784 Number of n X n periodic matrices over GF(3).

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