# [r-t] FW: A compositional question I am hoping a change-ringing theorist could help with! (re-sending)

Mark Davies mark at snowtiger.net
Wed Jun 16 22:16:58 UTC 2010

```Oh yes Mark - I forgot to say, if you just look at the deltas then you
can connect them all up with pair swaps (bellringing changes).

Here are the 14 delta sequences, with their equivalent rows:

delta sequence        row      rotated row
(a)   +1 -2 +3 +2 -1 -3  ->  6751326 = 132675
*(b)   +1 -2 -3 -1 +2 +3  ->  6752136 = 136752
(c)   +1 +3 -1 +2 -3 -2  ->  5621375 = 137562
(d)   +1 +2 -1 -3 -2 +3  ->  6721536 = 153672
(e)   +1 +3 +2 -3 -1 -2  ->  6735216 = 167352
(f)   +1 -3 -2 -1 +2 +3  ->  5631725 = 172563
(g)   +1 +3 +2 -1 -2 -3  ->  2361752 = 175236
(h)   +1 -2 -3 +2 -1 +3  ->  2315762 = 157623
*(i)   +1 +3 +2 -1 -3 -2  ->  1257631 = 125763
(j)   +1 -3 -1 +2 +3 -2  ->  1265731 = 126573
(k)   +1 -2 -1 -3 +2 +3  ->  1276351 = 127635
(l)   +1 +3 -2 -3 -1 +2  ->  1253761 = 125376
(m)   +1 +2 +3 -1 -3 -2  ->  7136527 = 136527
(n)   +1 -2 -3 -1 +3 +2  ->  7163257 = 163257

The starred rows are the "special" ones.

Anyway, I think each of the delta sequences 1-14 can be connected to at
least one of the others, and you can in fact link them all up. E.g.
numbers b and n obviously do connect just by swapping the last pair:

(b)  +1 -2 -3 -1 +2 +3
x
(n)  +1 -2 -3 -1 +3 +2

Some need two pair-swaps, e.g. e and m:

(e)   +1 +3 +2 -3 -1 -2
x     x
(m)   +1 +2 +3 -1 -3 -2

And some are a bit subtle in that they need the leading +1 swapping with
the next or last delta, e.g. d and e:

(d)   +1 +2 -1 -3 -2 +3
-> rotate ->
(d')  +3 +1 +2 -1 -3 -2
x        x
(e)   +1 +3 +2 -3 -1 -2

So there you go Mark - lots of ways of navigating through your EIRs
using bellringing changes on the deltas!

Cheers
M

```