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FYI, Tally is still broken.
*To*: mathgroup at smc.vnet.net
*Subject*: [mg89571] FYI, Tally is still broken.
*From*: DrMajorBob <drmajorbob at att.net>
*Date*: Fri, 13 Jun 2008 06:10:25 -0400 (EDT)
*References*: <JNEIICAJLELPIHHIMDJDOEIMCEAA.michael.weyrauch@gmx.de>
*Reply-to*: drmajorbob at longhorns.com
For any who might be curious, the Tally failure detailed below (in
December 2007) is unchanged in version 6.0.3.
$Version
"6.0 for Mac OS X x86 (64-bit) (May 21, 2008)"
(Here's a repeat of the same code, without all those >> interruptions.)
Needs["Combinatorica`"];
diagrams::usage = "Calculate diagrams";
basicedges::usage;
wick[a_, b_] := pair[a, b];
wick[a_, b__] :=
Sum[Expand[pair[a, {b}[[i]]] Delete[Unevaluated[wick[b]], i]], {i,
Length[{b}]}];
ham[n_Integer] := (ns = 4*n - 3; {c[ns], c[ns + 1], d[ns + 2], d[ns + 3]});
basicedges[n_] :=
Flatten[Table[{{{i, i + 1}, EdgeColor -> Red}}, {i, 1, 2*n, 2}], 1];
hamserie[n_Integer] := Flatten[Table[ham[i], {i, n}]];
cvertices[n_Integer] := {{n, 1}, {n, 0}};
cvertexserie[n_Integer] := Flatten[Table[cvertices[i], {i, n}], 1];
pair[c[_], c[_]] := 0;
pair[d[_], d[_]] := 0;
pair[a_Integer, b_Integer] /; (a > b) := pair[b, a];
diagrams[n_] :=
Module[{wickli, rep, i, cgraph, cvertices, congraph, le, un, ta},
wickli = wick[Sequence @@ hamserie[n]] /. Plus -> List;
le = Length[wickli[[1]]];
wickli = wickli /. {pair[c[i_], d[j_]] -> pair[i, j],
pair[d[i_], c[j_]] -> pair[i, j]};
graph = {}; While[wickli =!= {},
wickli =
rempar[First[wickli], wickli, n,
le]];(*edge reduction and edgelist construction for use by> >
Combinatorica*)
rep = Dispatch[Flatten[Table[{Rule[2*i - 1, i], Rule[2*i, i]}, {i,
2*n}]]];
graph = (Take[#, -le] /. rep /. pair[a__]^_ -> pair[a]) & /@ graph;
be = basicedges[n];
cgraph = Map[List, (graph /. {pair -> List, Times -> List}), {2}];
cvertices = List /@ cvertexserie[n]; cgraph = Join[be, #] & /@ cgraph;
cgraph = Graph[#, cvertices] & /@ cgraph;(*Now I compare Union and
Tally*)
saved = cgraph; un = Union[cgraph, SameTest -> IsomorphicQ];
Print["Union: number of elements: ", Length[un]]; Print[GraphicsGrid[
Partition[ShowGraph[#] & /@ un, 3, 3, {1, 1}, {}]]];
ta = Sort@Tally[cgraph, IsomorphicQ][[All, 1]];
Print["Tally: Number of Elements: ", Length[ta]];
Print[GraphicsGrid[Partition[ShowGraph /@ ta, 3, 3, {1, 1}, {}]]];
Print[GraphicsGrid[
Partition[ShowGraph /@ ta[[{2, 4}]], 3, 3, {1, 1}, {}]]];
Print["Are 2 and 4 isomorphic? ", IsomorphicQ[ta[[2]], ta[[4]]]];
Print["Are 4 and 2 isomorphic? ", IsomorphicQ[ta[[4]], ta[[2]]]];
]
rempar[li_, wickli_List, n_Integer, le_] :=
Module[{lis, mult, gem, pre, i}, lis = {Take[li, -le]}; pre = Drop[li,
-le];
Do[lis = Join[lis, lis /. {i -> i + 1, i + 1 -> i}], {i, 1, 4*n - 1, 2}];
lis = Union[lis]; mult = Length[lis];
graph = Join[graph, {li*mult}];
Complement[wickli, pre*lis]]
diagrams[3]
Union: number of elements: 8
Tally: Number of Elements: 11
Are 2 and 4 isomorphic? True
Are 4 and 2 isomorphic? True
saved // Length
index = Thread[saved -> Range@Length@saved];
(u = Union[saved, SameTest -> IsomorphicQ]) // Length
(t = Sort@Tally[saved, IsomorphicQ][[All, 1]]) // Length
21
8
11
These are the cgraph indices returned by Union and Tally :
u /. index
t /. index
{1, 11, 4, 21, 8, 18, 15, 6}
{1, 11, 4, 2, 8, 14, 3, 18, 15, 7, 6}
Comparing cgraph[[5]] and cgraph[[10]] is irrelevant, as you can see.
(boo = Boole@Outer[IsomorphicQ, t, t, 1]) // MatrixForm;
boo == Transpose@boo
Cases[Position[boo, 1], {a_, b_} /; a < b, 1]
True
{{2, 4}, {5, 7}, {8, 10}}
Bobby
On Fri, 07 Dec 2007 12:27:32 -0600, Michael Weyrauch
<michael.weyrauch at gmx.de> wrote:
> Dear Bobby,
>
> thanks for asking.
>
> Yes, indeed, I reported this problem to WRI using official support
> channels (thanks to a service contract of my company).
>
> I got the answer from some WRI support engineer that Tally[] is indeed
> broken,
> and does not function correctly in more complicated cases. However,
> appyling
> it
> repeatedly until nothing changes any more does give the correct result.
> (The last bit I did not yet check for myself.)
>
> In practice I wrote my own Tally[], which works but is probably much
> much
> slower
> than a (correctly working) built-in Tally[].
>
> I hope that in a future version Tally will work correctly, because I
> find it
> very useful in principle.
>
>
> Regards Michael
>
> "DrMajorBob" <drmajorbob at bigfoot.com> schrieb im Newsbeitrag
> news:<fjav9f$rjb$1 at smc.vnet.net>...
>> Did we get an answer on whether this is a bug in Tally?
>>
>> Bobby
>>
>> On Sun, 18 Nov 2007 16:09:01 -0600, DrMajorBob <drmajorbob at bigfoot.com>
>> wrote:
>>
>> > There is, apparently, something wrong with Tally, but your test wasn't
>> > the right one, since cgraph's 5th and 10th elements were not returned
>> in
>
>>
>> > the Tally results. Here's a modification of your code and some tests:
>> >
>> > Needs["Combinatorica`"];
>> > diagrams::usage = "Calculate diagrams";
>> > basicedges::usage;
>> >
>> > wick[a_, b_] := pair[a, b];
>> > wick[a_, b__] :=
>> > Sum[Expand[pair[a, {b}[[i]]] Delete[Unevaluated[wick[b]], i]], {i,
>> > Length[{b}]}];
>> > ham[n_Integer] := (ns = 4*n - 3; {c[ns], c[ns + 1], d[ns + 2],
>> > d[ns + 3]});
>> > basicedges[n_] :=
>> > Flatten[Table[{{{i, i + 1}, EdgeColor -> Red}}, {i, 1, 2*n, 2}],
>> > 1];
>> > hamserie[n_Integer] := Flatten[Table[ham[i], {i, n}]];
>> > cvertices[n_Integer] := {{n, 1}, {n, 0}};
>> > cvertexserie[n_Integer] := Flatten[Table[cvertices[i], {i, n}], 1];
>> > pair[c[_], c[_]] := 0;
>> > pair[d[_], d[_]] := 0;
>> > pair[a_Integer, b_Integer] /; (a > b) := pair[b, a];
>> >
>> > diagrams[n_] :=
>> > Module[{wickli, rep, i, cgraph, cvertices, congraph, le, un, ta} ,
>> > wickli = wick[Sequence @@ hamserie[n]] /. Plus -> List;
>> > le = Length[wickli[[1]]];
>> > wickli =
>> > wickli /. {pair[c[i_], d[j_]] -> pair[i, j],
>> > pair[d[i_], c[j_]] -> pair[i, j]};
>> > graph = {};
>> > While[wickli =!= {},
>> > wickli = rempar[First[wickli], wickli, n, le]];
>> > (*edge reduction and edgelist construction for use by \
>> > Combinatorica*)
>> > rep = Dispatch[
>> > Flatten[Table[{Rule[2*i - 1, i], Rule[2*i, i]}, {i, 2*n}]]];
>> > graph = (Take[#, -le] /. rep /. pair[a__]^_ -> pair[a]) & /@
>> > graph;
>> > be = basicedges[n];
>> > cgraph = Map[List, (graph /. {pair -> List, Times -> List}), {2} >> ];
>> > cvertices = List /@ cvertexserie[n];
>> > cgraph = Join[be, #] & /@ cgraph;
>> > cgraph = Graph[#, cvertices] & /@ cgraph;
>> > (*Now I compare Union and Tally*)
>> > saved = cgraph;
>> > un = Union[cgraph, SameTest -> IsomorphicQ];
>> > Print["Union: number of elements: ", Length[un]];
>> > Print[GraphicsGrid[
>> > Partition[ShowGraph[#] & /@ un, 3, 3, {1, 1}, {}]]];
>> > ta = Sort@Tally[cgraph, IsomorphicQ][[All, 1]];
>> > Print["Tally: Number of Elements: ", Length[ta]];
>> > Print[GraphicsGrid[Partition[ShowGraph /@ ta, 3, 3, {1, 1},
>> {}]]];=
>>
>> > Print[GraphicsGrid[
>> > Partition[ShowGraph /@ ta[[{2, 4}]], 3, 3, {1, 1}, {}]]];
>> > Print["Are 2 and 4 isomorphic? ", IsomorphicQ[ta[[2]], ta[[4]]]];
>> > Print["Are 4 and 2 isomorphic? ", IsomorphicQ[ta[[4]], ta[[2]]]];
>> > ];
>> >
>> > rempar[li_, wickli_List, n_Integer, le_] :=
>> > Module[{lis, mult, gem, pre, i}, lis = {Take[li, -le]};
>> > pre = Drop[li, -le];
>> > Do[lis = Join[lis, lis /. {i -> i + 1, i + 1 -> i}], {i, 1,
>> > 4*n - 1, 2}];
>> > lis = Union[lis];
>> > mult = Length[lis];
>> > graph = Join[graph, {li*mult}];
>> > Complement[wickli, pre*lis]];
>> >
>> > diagrams[3]
>> > Union: number of elements: 8
>> > Tally: Number of Elements: 11
>> > Are 2 and 4 isomorphic? True
>> > Are 4 and 2 isomorphic? True
>> >
>> > saved // Length
>> > index = Thread[saved -> Range@Length@saved];
>> > (u = Union[saved, SameTest -> IsomorphicQ]) // Length
>> > (t = Sort@Tally[saved, IsomorphicQ][[All, 1]]) // Length
>> >
>> > 21
>> >
>> > 8
>> >
>> > 11
>> >
>> > These are the cgraph indices returned by Union and Tally:
>> >
>> > u /. index
>> > t /. index
>> >
>> > {1, 11, 4, 21, 8, 18, 15, 6}
>> >
>> > {1, 11, 4, 2, 8, 14, 3, 18, 15, 7, 6}
>> >
>> > Comparing cgraph[[5]] and cgraph[[10]] is irrelevant, as you can see.
>> >
>> > (boo = Boole@Outer[IsomorphicQ, t, t, 1]) // MatrixForm;
>> > boo == Transpose@boo
>> > Cases[Position[boo, 1], {a_, b_} /; a < b, 1]
>> >
>> > True
>> >
>> > {{2, 4}, {5, 7}, {8, 10}}
>> >
>> > boo is NOT an identity matrix, so Tally did something very odd.
>> >
>> > Bobby
>> >
>> > On Sun, 18 Nov 2007 03:53:16 -0600, Michael Weyrauch >>
>> > <michael.weyrauch at gmx.de> wrote:
>> >
>> >> Hello,
>> >>
>> >> in the Mathematica 6.0 documentation it says in the entry for
>> Tally:=
>> =
>>
>> >> Properties and Relations:
>> >>
>> >> "A sorted Tally is equivalent to a list of counts for the Union: "
>> >>
>> >> This is what I indeed expect of Tally and Union, in particular then
>> i=
>> t =
>>
>> >> holds for any list:
>> >> Length[Tally[list]] is equal to Length[Union[list]].
>> >>
>> >> Now, I have an example, where Mathematica 6.0 produces a result
>> where=
>>
>> >> Tally[list] and Union[list] are different in length, which surpris=
es =
>> =
>> me.
>> >> And in fact, the result of Tally[ ] seems wrong to me.
>> >>
>> >> You can reproduce this result using the small Mathematica package =
=
>>
>> >> enclosed, which
>> >> uses Combinatorica. (Sorry for the somewhat complicated example, =
=
>> but=
>> I =
>>
>> >> did not find
>> >> a simpler case showing the effect.)
>> >>
>> >> If you load this package into a notebook and then execute
>> >>
>> >> diagrams[2]
>> >>
>> >> Tally and Union produce the expected result: both lists have equal=
=
>>
>> >> length.
>> >> (The list elements are diagrams.)
>> >>
>> >> However, if you execute
>> >>
>> >> diagrams[3]
>> >>
>> >> Tally and Union produce lists of different length.
>> >>
>> >> To my opinion, it really should never happen that Tally and Union=
>> >> produce lists of different length. I just expect of Tally to tell =
me =
>> =
>> =
>>
>> >> the multpilicities in the equivalence classes, in addition to
>> >> the information produced by Union. (The two list to be compared a=
re =
>> =
>> =
>>
>> >> called "ta" and "un" in the package enclosed.)
>> >>
>> >> Strangely enough, the program compares list elements 5 and 10 ==
>>
>> >> explicitly, and comes to the
>> >> conclusion that element 5 and 10 belong to the same equivalence =
>> class=
>> , =
>>
>> >> nevertheless they are
>> >> both listed seperately in the Tally, but - correctly - lumped up =
in =
>> =
>> =
>>
>> >> the Union.
>> >>
>> >> Do I misinterpret something here or is there a bug in Tally? (Tal=
ly =
>> =
>> is =
>>
>> >> new in Mathematica 6, and I
>> >> would find it extremely useful, if it would do what I expect it to=
=
>> do=
>> .)
>> >>
>> >> Michael
>> >>
>> >> Here comes my little package in order to reproduce the effect....
>> >>
>> >> BeginPackage["wick`"]
>> >>
>> >> Needs["Combinatorica`"];
>> >> diagrams::usage="Calculate diagrams";
>> >> basicedges::usage;
>> >>
>> >> Begin["`Private`"]
>> >>
>> >> wick[a_, b_] := pair[a, b];
>> >> wick[a_, b__]:= Sum[Expand[pair[a, {b}[[i]]] =
>>
>> >> Delete[Unevaluated[wick[b]], i]], {i, Length[{b}]}];
>> >> ham[n_Integer]:=(ns=4*n-3;{c[ns],c[ns+1],d[ns+2],d[ns+3]});
>> >> basicedges[n_]:=Flatten[Table[{{{i,i+1}, EdgeColor->Red}}, ==
>>
>> >> {i,1,2*n,2}],1];
>> >> hamserie[n_Integer]:=Flatten[Table[ham[i],{i,n}]];
>> >> cvertices[n_Integer]:={{n,1},{n,0}};
>> >> cvertexserie[n_Integer]:=Flatten[Table[cvertices[i],{i,n}],1];
>> >> pair[c[_],c[_]]:=0;
>> >> pair[d[_],d[_]]:=0;
>> >> pair[a_Integer,b_Integer]/;(a>b):=pair[b,a];
>> >>
>> >> diagrams[n_]:=Module[{wickli, rep, i, cgraph, cvertices, congrap=
h, =
>> le, =
>>
>> >> un, ta},
>> >>
>> >> wickli=wick[Sequence@@hamserie[n]]/.Plus->List;
>> >> le=Length[wickli[[1]]];
>> >> wickli=wickli/.{pair[c[i_],d[j_]]->pair[i,j],
>> >> pair[d[i_],c[j_]]->pair[i,j]};
>> >> graph={};
>> >> While[wickli=!={},
>> >> wickli=rempar[First[wickli],wickli,n, le]];
>> >>
>> >> (*edge reduction and edgelist construction for use by =
>> Combinatorica=
>> *)
>> >> rep=Dispatch[Flatten[Table[{Rule[2*i-1,i],Rule[2*i,i]},{i,2*n}=
]]]=
>> ;
>> >> graph=(Take[#,-le]/.rep/.pair[a__]^_->pair[a])&/@graph;
>> >>
>> >> be=basicedges[n];
>> >> cgraph=Map[List,(graph/.{pair->List, Times->List}),{2}];
>> >> cvertices=List/@cvertexserie[n];
>> >> cgraph=Join[be,#]&/@cgraph;
>> >> cgraph=Graph[#,cvertices]&/@cgraph;
>> >>
>> >> (* Now I compare Union and Tally *)
>> >> un=Union[cgraph,SameTest->IsomorphicQ];
>> >> Print["Union: number of elements: ", Length[un]];
>> >> Print[GraphicsGrid[Partition[ShowGraph[#]&/@un, 3,3,{1,1},{}]]];=
>> >>
>> >> ta=Tally[cgraph,IsomorphicQ];
>> >> ta=Sort[ta];
>> >> Print["Tally: Number of Elements: ", Length[ta]];
>> >> Print[GraphicsGrid[Partition[ShowGraph[#]&/@(First/@ta), =
>>
>> >> 3,3,{1,1},{}]]];
>> >>
>> >> Print["Are 5 and 10 isomorphic? ", IsomorphicQ[cgraph[[5]], ==
>>
>> >> cgraph[[10]]]];
>> >>
>> >> ];
>> >>
>> >> rempar[li_,wickli_List,n_Integer,le_]:=Module[{lis, mult, gem, =
pre=
>> , i},
>> >> lis={Take[li,-le]}; pre=Drop[li,-le];
>> >> Do[lis=Join[lis,lis /. {i->i+1, i+1->i}], {i,1,4*n-1,2}];
>> >> lis =Union[lis];
>> >> mult=Length[lis];
>> >> graph=Join[graph,{li*mult}];
>> >> Complement[wickli,pre*lis]
>> >> ];
>> >>
>> >> End[];
>> >> EndPackage[];
>> >>
>> >>
>> >>
>> >>
>> >>
>> >>
>> >>
>> >
>> >
>> >
>>
>>
>>
>> -- =
>>
>> DrMajorBob at bigfoot.com
>>
>
>
-- =
DrMajorBob at longhorns.com
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