A052365 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, under row and column permutations.

1, 1, 4, 10, 24, 51, 114, 219, 424, 768, 1352, 2278, 3759, 5978, 9328, 14181, 21164, 30943, 44560, 63063, 88088, 121321, 165152, 222157, 295857, 389948, 509456, 659697, 847552, 1080452, 1367814, 1719652, 2148596, 2668107, 3294676, 4046069

Offset: 0

Author

Vladeta Jovovic, Mar 08 2000

Keywords

• nonn

Comments

Also Molien series for group of structure S_3 X S_3 = (Z_3 X Z_3).O_2^+(3) and order 36, corresponding to complete weight enumerators of Hermitian self-dual GF(3)-linear codes over GF(9) containing the all-ones vector.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• Index entries for Molien series

Crossrefs

Row 3 of A318795.

Cf. A002724, A053307, A052366, A052267, A092091.

Programs

Mathematica

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
c[p_List, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
M[m_, n_, k_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
a[n_] := M[3, 3, n];
a /@ Range[0, 40] (* Jean-François Alcover, Sep 03 2019, after Andrew Howroyd in A318795 *)

Formula

G.f.: -(x^10+2*x^8+x^7+7*x^6-3*x^5+4*x^3+x^2-2*x+1) / ((x^4-x^3+x-1)*(x^3-1)^3*(x+1)^3*(x-1)^5).

Another form for g.f.: u1/u2, where u1 := 1 + x + 2*x^3 + 10*x^4 + 17*x^5 + 19*x^6 + 20*x^7 + 29*x^8 + 37*x^9 + 34*x^10 + 23*x^11 + 12*x^12 + 7*x^13 + 3*x^14 + x^15 u2 := (1-x^2)^4*(1-x^3)^4*(1-x^6);