A298399 Triangle read by rows: T(n,k) is the number of direct sum decompositions of GF(2)^n whose maximal subspace has dimension k, 1<=k<=n, n>=1.

1, 3, 1, 28, 28, 1, 840, 1960, 120, 1, 83328, 416640, 39680, 496, 1, 27998208, 295536640, 40354560, 666624, 2016, 1, 32509919232, 733279289344, 138360668160, 2757537792, 10924032, 8128, 1, 132640470466560, 6568159593103360, 1654847774392320, 38430207737856, 181463777280, 176865280, 32640, 1

Offset

1,2

Links

Table of n, a(n) for n=1..36.

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.

Example

Triangle begins:
      1;
      3,      1;
     28,     28,     1;
    840,   1960,   120,   1;
  83328, 416640, 39680, 496,  1;
  ...

Mathematica

nn = 7; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[ QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,
  Drop[Transpose[Table[Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[Exp[Sum[z^i/\[Gamma][i], {i, 1, k + 1}]] -
Exp[Sum[z^i/\[Gamma][i], {i, 1, k}]], {z, 0, nn}], z], {k, 0, 4}]], 1]]]

Crossrefs

Cf. A053601 (column 1), A270881 (row sums), A298561.

Sequence in context: A288517 A188110 A358165 * A365087 A120066 A326797

Adjacent sequences: A298396 A298397 A298398 * A298400 A298401 A298402

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jan 18 2018

Status

approved