|
| |
|
|
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
