|
|
A365590
|
|
Number of n X n Boolean relation matrices such that each of the diagonal blocks of its Frobenius normal form is either a 1 block or a 0 block.
|
|
3
|
|
|
1, 2, 13, 243, 11998, 1477763, 436610299, 300960642300, 474171878424571, 1680899431189662775, 13241419272545722904788, 229482664065433754849099977, 8677282817864146616211588609715, 710901968198799834001047038898570250
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
A 1(0) block is such that every entry in the block is 1(0). See Gregory, Kirkland, Pullman for a description of Frobenius normal form.
a(n) is also the number of labeled digraphs (with loops allowed A002416) on [n] such that every strongly connected component is either complete or a single vertex without a loop.
|
|
LINKS
|
|
|
FORMULA
|
Sum_{n>=0} a(n)*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-(exp(x)-1+x))) where E(x)=Sum_{n>=0} x^n/(n!*2^binomial(n,2)) and @ is the exponential Hadamard product (see Panafieu and Dovgal).
|
|
MATHEMATICA
|
nn = 13; B[n_] := n! 2^Binomial[n, 2]; ggf[egf_] := Normal[Series[egf, {x, 0, nn}]] /. Table[x^i -> x^i/2^Binomial[i, 2], {i, 0, nn}]; Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[-(Exp[x] - 1 + x)]], {x, 0, nn}], x]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|