A270881 Row sums of triangle A270880. Number of direct-sum decompositions of a finite vector space of n dimensions over GF(2).

1, 1, 4, 57, 2921, 540145, 364558049, 906918346689, 8394259686375297, 291375477821572448001, 38187935488350036891532801, 19005446750755761952317881973761, 36091267618694510017592440805677594625, 262587035725176662374187801686523815760228353, 7345273837043092730077580223639933697831592435638273

Offset: 0

Author

Michel Marcus, Mar 25 2016

Keywords

• nonn

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.

Crossrefs

Cf. A270880.

Programs

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 *)

Extensions

Name extended by David P. Ellerman, Mar 26 2016

a(8)-a(14) from Geoffrey Critzer, May 18 2017