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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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.

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

EXAMPLE

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;

  ...

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 *)

CROSSREFS

Cf. A270880, A270883 (row sums).

Sequence in context: A217654 A267164 A158234 * A278778 A285931 A069845

Adjacent sequences:  A270879 A270880 A270881 * A270883 A270884 A270885

KEYWORD

nonn,tabl,changed

AUTHOR

Michel Marcus, Mar 25 2016

EXTENSIONS

Name extended by David P. Ellerman, Mar 26 2016

Row 8 from Geoffrey Critzer, May 20 2017

STATUS

approved

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Last modified February 20 15:52 EST 2020. Contains 332078 sequences. (Running on oeis4.)