A287206 Triangle read by rows: T(n,k) is the number of direct sum decompositions of a finite vector space of n dimensions over GF(2) that have exactly k subspaces of dimension 1, n>=0, 0<=k<=n.

1, 0, 1, 1, 0, 3, 1, 28, 0, 28, 281, 120, 1680, 0, 840, 9921, 139376, 29760, 277760, 0, 83328, 16078337, 20000736, 140491008, 19998720, 139991040, 0, 27998208, 13596908545, 130684723136, 81282991104, 380636971008, 40637399040, 227569434624, 0, 32509919232, 191426147495937, 443803094908800, 2132774681579520, 884358943211520, 3105997683425280, 265280940933120, 1237977724354560, 0, 132640470466560

Offset: 0

Author

Geoffrey Critzer, May 21 2017

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
     1;
     0,      1;
     1,      0,     3;
     1,     28,     0,     28,
   281,    120,  1680,      0,  840;
  9921, 139376, 29760, 277760,    0, 83328;
  ...
		

Links

• David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1

Crossrefs

Cf. A270881 (row sums), A053601 (main diagonal), A289543 (column k=0).

Programs

Mathematica

nn = 8; g[n_] := QFactorial[n, q]*(q - 1)^n*q^Binomial[n, 2] /. q -> 2; e[u_] := Sum[u^r/g[r], {r, 0, nn}];
Table[Table[(Table[g[n], {n, 0, nn}] CoefficientList[  Series[Exp[e[u] - 1 - u + u t], {u, 0, nn}], {u, t}])[[n,
     k]], {k, 1, n}], {n, 1, nn + 1}] // Grid

Formula

Sum_{n>=0} T(n,k)*u^n/g(n)*t^k = exp(Sum_{r>=0} u^r/g(r) - 1 - u + t*u) where g = A002884.