A298561 Triangle read by rows. T(n,k) is the number of direct sum decompositions of GF(2)^n into subspaces of dimension at most k, 1<=k<=n.

1, 3, 4, 28, 56, 57, 840, 2800, 2920, 2921, 83328, 499968, 539648, 540144, 540145, 27998208, 323534848, 363889408, 364556032, 364558048, 364558049, 32509919232, 765789208576, 904149876736, 906907414528, 906918338560, 906918346688, 906918346689

Offset

1,2

Links

Table of n, a(n) for n=1..28.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.

Formula

exp(Sum_{j=0...k} x^j/A002884(j)) = Sum_{n>=0} T(n,k)/A002884(n)*x^n.

Example

  1
  3,     4,
  28,    56,     57,
  840,   2800,   2920,   2921,
  83328, 499968, 539648, 540144, 540145,

Mathematica

nn = 7; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 2; Flatten[Table[Table[Transpose[
     Map[Drop[#, 1] &, Table[Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[Exp[Sum[z^i/\[Gamma][i], {i, 1, k}]], {z, 0, nn}], z], {k, 1, nn}]]][[j, k]], {k, 1, j}], {j, 1, nn}]]

Crossrefs

Cf. A270881 (main diagonal), A053601 (column 1), A298339.

Sequence in context: A232110 A140896 A005326 * A226049 A354844 A100600

Adjacent sequences: A298558 A298559 A298560 * A298562 A298563 A298564

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jan 21 2018

Status

approved