A270881 - OEIS

%I #32 Aug 02 2018 15:00:47

%S 1,1,4,57,2921,540145,364558049,906918346689,8394259686375297,

%T 291375477821572448001,38187935488350036891532801,

%U 19005446750755761952317881973761,36091267618694510017592440805677594625,262587035725176662374187801686523815760228353,7345273837043092730077580223639933697831592435638273

%N Row sums of triangle A270880. Number of direct-sum decompositions of a finite vector space of n dimensions over GF(2).

%C The generating function for these numbers was first derived in Bender & Goldman. My paper derives the direct formula for the numbers for any finite vector space over GF(q) so that when q = 1, the formula gives the Bell numbers--since a direct-sum decomposition is the vector space version of a set partition. This sequence gives the numbers for q = 2. - _David P. Ellerman_, Mar 26 2016

%H Edward A. Bender, and Jay R. Goldman, <a href="https://www.jstor.org/stable/24890130">Enumerative Uses of Generating Functions</a>, Indiana University Mathematics Journal 20 (8) (1971) 753-65.

%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.

%H David Ellerman, <a href="http://arxiv.org/abs/1604.01087">The Quantum Logic of Direct-Sum Decompositions</a>, arXiv preprint arXiv:1604.01087 [quant-ph], 2016. See Section 7.5.

%t g[n_] := q^Binomial[n, 2] * FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2; Table[Total[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {m}]]], {m, 1, n}]], {n, 1, 15}] (* _Geoffrey Critzer_, May 18 2017 *)

%Y Cf. A270880.

%K nonn

%O 0,3

%A _Michel Marcus_, Mar 25 2016

%E Name extended by _David P. Ellerman_, Mar 26 2016

%E a(8)-a(14) from _Geoffrey Critzer_, May 18 2017