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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.
0

%I #19 Aug 02 2018 15:28:10

%S 1,3,1,28,28,1,840,1960,120,1,83328,416640,39680,496,1,27998208,

%T 295536640,40354560,666624,2016,1,32509919232,733279289344,

%U 138360668160,2757537792,10924032,8128,1,132640470466560,6568159593103360,1654847774392320,38430207737856,181463777280,176865280,32640,1

%N 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.

%H Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.

%H David Ellerman, <a href="http://arxiv.org/abs/1603.07619">The number of direct-sum decompositions of a finite vector space</a>, arXiv:1603.07619 [math.CO], 2016.

%e Triangle begins:

%e 1;

%e 3, 1;

%e 28, 28, 1;

%e 840, 1960, 120, 1;

%e 83328, 416640, 39680, 496, 1;

%e ...

%t nn = 7; \[Gamma][n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[ QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,

%t Drop[Transpose[Table[Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[Exp[Sum[z^i/\[Gamma][i], {i, 1, k + 1}]] -

%t Exp[Sum[z^i/\[Gamma][i], {i, 1, k}]], {z, 0, nn}], z], {k, 0, 4}]], 1]]]

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

%K nonn,tabl

%O 1,2

%A _Geoffrey Critzer_, Jan 18 2018