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

Author

Michel Marcus, Mar 25 2016

Keywords

• nonn
• tabl

Examples

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

Links

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

Programs

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 *)

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).