OFFSET
0,3
COMMENTS
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
LINKS
Edward A. Bender, and Jay R. Goldman, Enumerative Uses of Generating Functions, Indiana University Mathematics Journal 20 (8) (1971) 753-65.
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.
David Ellerman, The Quantum Logic of Direct-Sum Decompositions, arXiv preprint arXiv:1604.01087 [quant-ph], 2016. See Section 7.5.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Mar 25 2016
EXTENSIONS
Name extended by David P. Ellerman, Mar 26 2016
a(8)-a(14) from Geoffrey Critzer, May 18 2017
STATUS
approved