A270880 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).

1, 0, 1, 0, 1, 3, 0, 1, 28, 28, 0, 1, 400, 1680, 840, 0, 1, 10416, 168640, 277760, 83328, 0, 1, 525792, 36053248, 159989760, 139991040, 27998208, 0, 1, 51116992, 17811244032, 209056841728, 419919790080, 227569434624, 32509919232

Offset

0,6

Links

Table of n, a(n) for n=0..35.

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.

Formula

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).

Example

Triangle begins:
1;
0, 1;
0, 1, 3;
0, 1, 28, 28;
0, 1, 400, 1680, 840;
0, 1, 10416, 168640, 277760, 83328;
...

Mathematica

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 *)

Crossrefs

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

Sequence in context: A278325 A364527 A226780 * A217580 A197858 A060861

Adjacent sequences: A270877 A270878 A270879 * A270881 A270882 A270883

Keyword

nonn,tabl

Author

Michel Marcus, Mar 25 2016

Status

approved