A270882 Triangle read by rows: D*(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2) with a block containing any given nonzero vector.

1, 0, 1, 0, 1, 2, 0, 1, 16, 12, 0, 1, 176, 560, 224, 0, 1, 3456, 40000, 53760, 13440, 0, 1, 128000, 5848832, 20951040, 15554560, 2666496, 0, 1, 9115648, 1934195712, 17826414592, 30398054400, 14335082496, 1791885312, 0, 1, 1259921408, 1510821195776, 37083513880576, 134908940386304, 133854174117888, 43693331447808, 4161269661696

Offset: 0

Author

Michel Marcus, Mar 25 2016

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,      2;
  0, 1,     16,      12;
  0, 1,    176,     560,      224;
  0, 1,   3456,   40000,    53760,    13440;
  0, 1, 128000, 5848832, 20951040, 15554560, 2666496;
  ...
		

Links

• Jinyuan Wang, Rows n = 0..10 of triangle, flattened
• 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, A270883 (row sums).

The main diagonal appears to match A377642. - Nikita Babich, Nov 17 2024

Programs

Mathematica

(* about 40 seconds on a laptop computer *) g[n_] := q^Binomial[n, 2] * FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2; d[k_, m_] :=Map[PadRight[#, 10] &,Table[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {j}]]], {j, 1, n}], {n, 1, 10}]][[k, m]];d[0, m_] := If[m == 0, 1, 0]; d[k_, 0] := If[k == 0, 1, 0];s[n_, m_] :=Sum[FunctionExpand[QBinomial[n - 1, k, 2]]*2^(k (n - k))*d[k, m - 1], {k, 0, n - 1}]; Table[Table[s[n, m], {m, 1, n}], {n, 1,7}] (* Geoffrey Critzer, May 20 2017 *)

Formula

Recurrence: a(n) = Sum_{k=0..n-1} q-binomial(n-1,k)*q^(n*(n-k))*D_q(k,m-1) where D_q(k,m-1) is given by A270880 for q = 2 and where the q-binomial for q = 2 is given by A022166. This formula is the q-analog of summation formula for the Stirling numbers of the second kind A008277 so when q = 1, it reduces to that formula. - David P. Ellerman, Mar 26 2016

Extensions

Name extended by David P. Ellerman, Mar 26 2016

Row 8 from Geoffrey Critzer, May 20 2017