A058001 Number of 3 X 3 matrices with entries mod n, up to row and column permutation.

1, 36, 738, 8240, 57675, 289716, 1144836, 3780288, 10865205, 27969700, 65834406, 143887536, 295467263, 575308020, 1069960200, 1911933696, 3298486761, 5516122788, 8972008810, 14233690800, 22078652211, 33555443636, 50058302988, 73417387200, 106006948125

Offset: 1

Author

Vladeta Jovovic, Nov 04 2000

Keywords

• easy
• nonn

Comments

Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Marko R. Riedel, Number of equivalence classes of matrices, Math Stackexchange.
• Marko R. Riedel, Computing the cycle index for arbitary k x l matrices using Maple
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Cf. A058002, A058003, A058004, A002724, A052271, A052272.

Row n=3 of A246106.

Programs

Mathematica

CoefficientList[Series[x (12x^7+369x^6+2514x^5+4375x^4+2360x^3+423x^2+26x+1)/(x-1)^10,{x,0,30}],x] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,1,36,738,8240,57675,289716,1144836,3780288,10865205},30] (* Harvey P. Dale, Nov 23 2024 *)

Formula

a(n) = (1/3!^2)*(n^9 + 6*n^6 + 9*n^5 + 8*n^3 + 12*n^2).

G.f.: x*(12*x^7+369*x^6+2514*x^5+4375*x^4+2360*x^3+423*x^2+26*x+1) / (x-1)^10. - Colin Barker, Jul 09 2013