Spliced treble-dodging minor - clusters

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Alexander Holroyd holroyd at math.ubc.ca Wed Sep 29 12:35:12 BST 2010

Here is an idea for an automated method for analysing the plans.

For each pair of plans in the list (ie a few million pairs), do the following. "Rotate" one of the pair through all 60 possible starting rows. (Your plans are already in standard form, aren't they?) For each such rotation, compare the two plans. Specifically, look to see whether the two plans are identical except that one is obtained from the other by replacing some set of leads all of method X with all method Y. If so, say that there is a "simple splice" between the two plans. The simple splice itself may be described by saying what methods X and Y are, and what the set of leads is, in "standard form" (i.e. rotated to it's smallest version in lexicographic order).

After this is done for all pairs, the set of simple splices that arise had better be what exactly we expect, i.e. things like "2 copies of the Cambride-Beverley 6-lead splice".

We now have a graph, whose vertices (nodes) are the plans, and whose edges (links) are the simple splices. Break this graph up into its connected components, or "clusters". Each cluster is a group of plans all of which communicate with each other via simple splices.

For a start I would like to know how many clusters there are.

Now we want to analyse the clusters. Any cluster that contains a single-method extent can be removed from the game, because these we (hopefully) completely understand. For the others, it would be desirable to nominate a "standard representative" plan from each cluster - the basic plan which then gets embelished with various simple splices to form the other members of the cluster. I'm not sure whether there is a canonical way to do this in general, but one natural thing that springs to mind is to list, for each cluster, the plans that have the smallest number of methods, and see by hand which of these (if there is more than one) is the most "natural".

Then one can summarize the whole thing by listing, for each cluster, its standard representative, and the list of all simple splices it involves. I suspect such a list would be quite manageable.

Richard Smith richard at ex-parrot.com Wed Sep 29 19:57:24 BST 2010

There are 506 connected components of which 14 are simple. The simple components contain the methods:

 1.  Sg
 2.  Nm So C3 C2 Pn
 3.  Nw Mo Ak Ct Mu Ch Av Ca Ke Ce Cd Sw Va Li Co Cc We Lo
     Cu Cx Bn Pv Bt Le Md Mp C1 Cl Nb Wh Gl Fo Ml By Pm Ed
     Kh Ww Lu Cz Cy Bh Os Bw Wf
 4.  Qu Sn Ti Sh Tr Km
 5.  Kt
 6.  Bl Wk
 7.  Fr Cg
 8.  Cj Nl
 9.  Du Yo Su He Bv Bk Cm Pr Ip Nf Bs Wa
 10. Bu
 11. Es Cv
 12. Rc Bz
 13. Bm Bc El Cr Wm St Lf Ro Sa Wo Te Ev Po Dt Ck Wt Di Ms
     Do Dn Pe Wl Bg Kn Rs Ba Cs Fg Sk Ey Bp Wv Bo Hu Ki Lv
     Cn Ri Dk Cf Ny Oc Ci Ks Ls Sd Ox No Ne Ab Ws Ad Ma Br
     Ta Hm Ol Cb Ng Wi Cw Ns Sl Wr
 14. Be Me

The full list of plans grouped into clusters is here: