A346209
Number of n X n matrices over GF(3) with no eigenvalues in GF(3), i.e., neither 0 nor 1 nor 2 is an eigenvalue.
Original entry on oeis.org
1, 0, 18, 3456, 7619508, 149200289280, 26394940582090344, 42062797470468915399168, 603463180651533072058654437264, 77927374189849689541269666899007713280, 90570450400853976077932766909301405665963077152
Offset: 0
-
nn = 10; q = 3; \[Nu] = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[Product[1/(1 - u^d/q^(r d)), {r, 1, \[Infinity]}]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], u]
A378666
Triangular array read by rows: T(n,k) is the number of n X n idempotent matrices over GF(3) having rank k, n>=0, 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 1, 12, 1, 1, 117, 117, 1, 1, 1080, 10530, 1080, 1, 1, 9801, 882090, 882090, 9801, 1, 1, 88452, 72243171, 666860040, 72243171, 88452, 1, 1, 796797, 5873190687, 491992666011, 491992666011, 5873190687, 796797, 1, 1, 7173360, 476309310660, 360089838858960, 3267815287645062, 360089838858960, 476309310660, 7173360, 1
Offset: 0
Triangle T(n,k) begins:
1;
1, 1;
1, 12, 1;
1, 117, 117, 1;
1, 1080, 10530, 1080, 1;
1, 9801, 882090, 882090, 9801, 1;
...
-
b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, b(n-1, k-1)+3^k*b(n-1, k)))
end:
T:= (n,k)-> 3^(k*(n-k))*b(n, k):
seq(seq(T(n,k), k=0..n), n=0..8); # Alois P. Heinz, Dec 02 2024
-
nn = 8; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; B[n_, q_] := \[Gamma][n, q]/(q - 1)^n; \[Zeta][x_] := Sum[x^n/B[n, 3], {n, 0, nn}];Map[Select[#, # > 0 &] &, Table[B[n, 3], {n,0,nn}]*CoefficientList[Series[\[Zeta][x] \[Zeta][y x], {x, 0, nn}], {x, y}]] // Flatten
A086087
a(n) is the minimal m such that the group GL(m,3) has an element of order n.
Original entry on oeis.org
1, 2, 2, 4, 2, 6, 2, 4, 4, 5, 4, 3, 6, 6, 4, 16, 4, 18, 4, 8, 5, 11, 4
Offset: 2
Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 24 2003
-
A086087 := function(n) local m; if IsPrime(n) and n>3 then return Order(3*Z(n)^0); fi; m := 1; while true do if ForAny(ConjugacyClasses(GL(m, 3)), cc->Order(Representative(cc))=n) then return m; fi; m := m + 1; od; end; # Eric M. Schmidt, May 18 2013
A086752
Number of n X n matrices over GF(3) with rank n-1.
Original entry on oeis.org
1, 32, 8112, 17971200, 355207057920, 63010655570903040, 100505356319291594711040, 1442361950110091891786121216000, 186276322602412236974585775503690956800, 216505458700483736766078241517019274701019545600, 2264736353104098912130003755084217747715114856943819161600
Offset: 1
Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 31 2003
-
Table[3^(n^2) * (1 - 1/3^n) * QPochhammer[1/3^n, 3, n-1]/2, {n, 1, 10}] (* Vaclav Kotesovec, Apr 14 2024 *)
A220790
Product(6^n - 6^k, k=0..n-1).
Original entry on oeis.org
1, 5, 1050, 8127000, 2273284440000, 22906523331216000000, 8310241106635054164480000000, 108537128570336598656772717772800000000, 51032497739317419104816901041614046625792000000000
Offset: 0
-
[1] cat [&*[(6^n - 6^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013
-
/* By the second formula: */
m:=9;
A109354 := [6^(n*(n-1) div 2): n in [0..m-1]];
A027873 := [1] cat [&*[6^i-1: i in [1..n]]: n in [1..m]];
[A109354[i]*A027873[i]: i in [1..m]]; // Bruno Berselli, Jan 30 2013
-
Table[Product[6^n - 6^k, {k, 0, n-1}], {n, 0, 60}]
A220791
Number of nonsingular n X n matrices over GF(13).
Original entry on oeis.org
1, 12, 26208, 9726417792, 610296923230525440, 6471875909051511775903457280, 11598637276362103019770723830073032376320, 3512938445418644176053176560741858449740612202579886080
Offset: 0
-
[1] cat [&*[(13^n - 13^k): k in [0..n-1]]: n in [1..8]];
-
Table[Product[13^n - 13^k, {k, 0, n-1}], {n, 0, 8}]
A053995
Bases of n-dimensional vector space over GF(3).
Original entry on oeis.org
1, 2, 24, 1872, 1010880, 3963053952, 116846682720768, 26600881786245651456, 47704425342545776582103040, 684471586202557775548246032261120, 79552147744537053441626504660608566263808, 75648987817302293231984664268437958161057494925312
Offset: 0
A297892
Triangle read by rows. T(n,k) is the number of n X n diagonalizable matrices over GF(3) that have rank k, 0 <= k <= n, n >= 0.
Original entry on oeis.org
1, 1, 2, 1, 24, 14, 1, 234, 1638, 236, 1, 2160, 147420, 254880, 12692, 1, 19602, 12349260, 208173240, 124394292, 1783784, 1, 176904, 1011404394, 157378969440, 916910326332, 157779262368, 811523288
Offset: 0
Triangle begins
1;
1, 2;
1, 24, 14;
1, 234, 1638, 236;
1, 2160, 147420, 254880, 12692;
1, 19602, 12349260, 208173240, 124394292, 1783784;
-
nn = 5; g[n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 3;G[n] := Sum[u z^r/g[r], {r, 0, nn}]; Grid[Map[Select[#, # > 0 &] &,Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}]^2 Sum[
z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]]
A335384
Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.
Original entry on oeis.org
6, 48, 168, 180, 480, 2016, 3528, 5760, 11232, 13200, 20160, 26208, 61200, 78336, 123120, 181440, 267168, 374400, 511056, 682080, 892800, 1014816, 1488000, 1822176, 2755200, 3337488, 4773696, 5644800, 7738848, 9999360, 11908560, 13615200, 16511040, 19845936, 24261120, 25048800, 28003968
Offset: 1
a(1) = #GL(2,2) = (2^2-1)*(2^2-2) = 3*2 = 6 and the 6 elements of GL(2,2) that is isomorphic to S_3 are the 6 following 2 X 2 invertible matrices with entries in F_2:
(1 0) (1 1) (1 0) (0 1) (0 1) (1 1)
(0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
a(2) = #GL(2,3) = (3^2-1)*(3^2-3) = 8*6 = 48.
a(3) = #GL(3,2) = (2^3-1)*(2^3-2)*(2^3-2^2) = 168.
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
- Daniel Perrin, Cours d'Algèbre, Maths Agreg, Ellipses, 1996, pages 95-115.
Cf.
A002884 [GL(m,2)],
A053290 [GL(m,3)],
A053291 [GL(m,4)],
A053292 [GL(m,5)],
A053293 [GL(m,7)],
A052496 [GL(m,8)],
A052497 [GL(m,9)],
A052498 [GL(m,11)].
A346384
Triangle read by rows. T(n,k) is the number of invertible n X n matrices over GF(3) such that the dimension of the eigenspace corresponding to the eigenvalue 1 is k, 0 <= k <= n, n >= 0.
Original entry on oeis.org
1, 1, 1, 27, 20, 1, 6291, 4719, 221, 1, 13589289, 10191960, 477750, 2120, 1, 266377183929, 199782888129, 9364822830, 41559870, 19481, 1, 47123189360124723, 35342392020078780, 1656674625945339, 7352106327720, 3446299857, 176540, 1
Offset: 0
1;
1, 1;
27, 20, 1;
6291, 4719, 221, 1;
13589289, 10191960, 477750, 2120, 1;
266377183929, 199782888129, 9364822830, 41559870, 19481, 1;
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nn = 6; q = 3; 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]}]; A027376 = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];
g[u_, v_] := Total[Map[v^Length[#] u^Total[#]/aut[1, #] &, Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]]; Map[Select[#, # > 0 &] &, Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
Series[(g[u, v] /. v -> 1)*g[u, v]* Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A027376[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}]] // Grid
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