Ralf Hemmecke <[EMAIL PROTECTED]> writes:
> On 02/17/2007 05:47 PM, Martin Rubey wrote:
> > Dear Ralf,
> > very likely the cycle index series I gave for
> >> testBinaryForests
> >> testBinaryTreesOfSets
> >> testPartitionsViaCompose
> > are rubbish. Unfortunately, I don't have BLL with me, so I cannot reliably
> > check today. I'll do what I can tonight.
>
> You could perhaps start over the code of Hornegger and Piratsu. For some
> strange reason, I don't get it to run in Axiom right now.
Great idea! Thanks! Here are the results, gfnum and gftyp are only for
completeness, note that you have to multiply with n! to get our notation in the
testcases.
I really should have put empty lists into the testcases. Well I know I'm quite
imperfect... I keep trying to improve myself, honestly, but currently I'm low
on battery.
-- Binary Forests (SinTo is for Singleton, BinTree starts with 1...)
(9) -> (cycser subst(Emenge(), SinTo() * BinTree()))
(9)
3 2 1 19 3 1 1 193 4 3 2 1
1 + x + - x + - x + -- x + - x x + - x + --- x + - x x + - x x
1 2 1 2 2 6 1 2 1 2 3 3 24 1 4 1 2 3 1 3
+
5 2 1 907 5 19 3 1 2 5 2 1 1
- x + - x + --- x + -- x x + - x x + - x x + - x x + - x x
8 2 4 4 40 1 12 1 2 2 1 3 8 1 2 4 1 4 6 2 3
+
1 49171 6 193 4 19 3 15 2 2 3 2 1
- x + ----- x + --- x x + -- x x + -- x x + - x x + - x x x
5 5 720 1 48 1 2 18 1 3 16 1 2 8 1 4 6 1 2 3
+
1 61 3 1 7 2 1 1084483 7 907 5
- x x + -- x + - x x + -- x + - x + ------- x + --- x x
5 1 5 48 2 8 2 4 18 3 6 6 5040 1 80 1 2
+
193 4 95 3 2 19 3 1 2 3 2 61 3
--- x x + -- x x + -- x x + - x x x + -- x x + -- x x
72 1 3 48 1 2 24 1 4 4 1 2 3 10 1 5 48 1 2
+
1 7 2 1 5 2 1 1 1
- x x x + -- x x + - x x + -- x x + -- x x + -- x x + - x + ...
8 1 2 4 18 1 3 6 1 6 24 2 3 10 2 5 12 3 4 7 7
Type: ZykelIndexReihe Fraction Integer
(15) -> gfnum subst(Emenge(), SinTo() * BinTree())
(15)
3 2 19 3 193 4 907 5 49171 6 1084483 7 9415243 8
1 + x + - x + -- x + --- x + --- x + ----- x + ------- x + ------- x
2 6 24 40 720 5040 13440
+
9
O(x )
Type: UnivariateTaylorSeries(Fraction Integer,x,0)
(16) -> gftyp subst(Emenge(), SinTo() * BinTree())
2 3 4 5 6 7 8 9
(16) 1 + x + 2x + 4x + 10x + 26x + 77x + 235x + 758x + O(x )
Type: UnivariateTaylorSeries(Fraction Integer,x,0)
-- norm is the same as DropEmpty, i.e., it drops the constant term
(11) -> cycser subst(SinTo() * BinTree(), norm Emenge())
(11)
3 2 1 19 3 3 1 69 4 19 2
x + - x + - x + -- x + - x x + - x + -- x + -- x x + x x
1 2 1 2 2 6 1 2 1 2 3 3 8 1 4 1 2 1 3
+
3 2 1 3211 5 69 3 19 2 19 2 3 1
- x + - x + ---- x + -- x x + -- x x + -- x x + - x x + - x x
8 2 4 4 120 1 4 1 2 6 1 3 8 1 2 4 1 4 2 2 3
+
1 21461 6 3211 4 23 3 207 2 2 19 2
- x + ----- x + ---- x x + -- x x + --- x x + -- x x
5 5 240 1 48 1 2 2 1 3 16 1 2 8 1 4
+
19 3 19 3 3 1 2 1 1581259 7
-- x x x + - x x + -- x + - x x + - x + - x + ------- x
6 1 2 3 5 1 5 48 2 8 2 4 6 3 6 6 5040 1
+
21461 5 3211 4 3211 3 2 69 3 69 2 19 2
----- x x + ---- x x + ---- x x + -- x x + -- x x x + -- x x
80 1 2 72 1 3 48 1 2 8 1 4 4 1 2 3 10 1 5
+
69 3 19 19 2 1 19 2 3 1
-- x x + -- x x x + -- x x + - x x + -- x x + -- x x + - x x
16 1 2 8 1 2 4 18 1 3 2 1 6 24 2 3 10 2 5 4 3 4
+
1
- x + ...
7 7
Type: ZykelIndexReihe Fraction Integer
(13) -> gfnum(subst(SinTo() * BinTree(), norm Emenge()))
(13)
3 2 19 3 69 4 3211 5 21461 6 1581259 7 15316309 8
x + - x + -- x + -- x + ---- x + ----- x + ------- x + -------- x
2 6 8 120 240 5040 13440
+
9
O(x )
(14) -> gftyp(subst(SinTo() * BinTree(), norm Emenge()))
2 3 4 5 6 7 8 9
(14) x + 2x + 5x + 15x + 51x + 188x + 731x + 2950x + O(x )
Type: UnivariateTaylorSeries(Fraction Integer,x,0)
-- Partitions.
(12) -> cycser subst(Emenge(), norm Emenge())
(12)
2 5 3 3 2 5 4 7 2 7 2
1 + x + x + x + - x + - x x + - x + - x + - x x + x x + - x
1 1 2 6 1 2 1 2 3 3 8 1 4 1 2 1 3 8 2
+
3 13 5 5 3 7 2 3 2 5 2
- x + -- x + - x x + - x x + - x x + x x + - x x + - x
4 4 30 1 3 1 2 6 1 3 2 1 2 1 4 6 2 3 5 5
+
203 6 67 4 10 3 31 2 2 9 2 5 3
--- x + -- x x + -- x x + -- x x + - x x + - x x x + - x x
720 1 48 1 2 9 1 3 16 1 2 8 1 4 3 1 2 3 5 1 5
+
31 3 9 4 2 2 877 7 17 5 67 4 97 3
2
-- x + - x x + - x + - x + ---- x + -- x x + -- x x + -- x x
48 2 8 2 4 9 3 3 6 5040 1 16 1 2 72 1 3 48 1 2
+
25 3 9 2 7 2 59 3 15 13 2 5
-- x x + - x x x + -- x x + -- x x + -- x x x + -- x x + - x x
24 1 4 4 1 2 3 10 1 5 48 1 2 8 1 2 4 18 1 3 6 1 6
+
19 2 1 7 2
-- x x + - x x + -- x x + - x + ...
24 2 3 2 2 5 12 3 4 7 7
Type: ZykelIndexReihe Fraction Integer
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