A270880 - OEIS

%I #28 Aug 02 2018 15:13:44

%S 1,0,1,0,1,3,0,1,28,28,0,1,400,1680,840,0,1,10416,168640,277760,83328,

%T 0,1,525792,36053248,159989760,139991040,27998208,0,1,51116992,

%U 17811244032,209056841728,419919790080,227569434624,32509919232

%N Triangle read by rows: T(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2).

%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.

%F T(n,m) = Sum_ g(n)/(g(n_1)*g(n_2)***g(n_m))/(a_1!*a_2!***a_n!) where the sum is over all partitions of n into m parts and a_1,a_2,...,a_n is the part count signature of the partition and g(n) = A002884(n). - _Geoffrey Critzer_, May 18 2017 (after formula given in first Ellerman link above).

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 1, 3;

%e 0, 1, 28, 28;

%e 0, 1, 400, 1680, 840;

%e 0, 1, 10416, 168640, 277760, 83328;

%e ...

%t g[n_] := q^Binomial[n, 2] *FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2;Table[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {m}]]], {m, 1, n}], {n, 1, 6}] // Grid (* _Geoffrey Critzer_, May 18 2017 *)

%Y Cf. A053601 (right diagonal), A270881 (row sums), A270882.

%K nonn,tabl

%O 0,6

%A _Michel Marcus_, Mar 25 2016