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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.
3
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
0,5
COMMENTS
A matrix T is periodic if and only image(T) = image(T^2). Cf. A348015.
LINKS
Eric Weisstein's World of Mathematics, Periodic Matrix
FORMULA
T(n,k) = A002884(n)/A002884(n-k).
T(n,k) = T(n,k-1)*T(n-k+1,1).
EXAMPLE
Triangle begins:
1;
1, 1;
1, 6, 6;
1, 28, 168, 168;
1, 120, 3360, 20160, 20160;
1, 496, 59520, 1666560, 9999360, 9999360;
...
MAPLE
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
MATHEMATICA
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}]; Lt = Prepend[Table[{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
CROSSREFS
Cf. A348015 (row sums).
Main diagonal gives A002884.
Sequence in context: A098267 A366138 A122193 * A098369 A078740 A228708
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Oct 25 2021.
EXTENSIONS
Title improved by Geoffrey Critzer, Sep 16 2022
STATUS
approved