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# Expansion of Peltier's Pivot/Loop Notation

January 11, 2009

In the Google group sprouts-theory, Josh Purinton has recently offered excellent suggestions for expanding a system of notation invented by Jeff Peltier and popularized by Jeff's landmark tables of grundy values. The expansion amounts to a "chaining" [Purinton's word] together of Peltier's "canonicalized" [Peltier's nice description] positions.

Purinton writes:

After 16-, the position would be written as "16".
After 1(17)1[2-4] it would be"12L3".
After 5(18)5[6-11] it would be "6L5L3".
After 5(19)18[12-15] it would be "6+4P1L3".
After 16(20)19 it would be "6+4+3LPP1".
After 2(21)3 it would be "6+4+A3LPP1"

(Please note that Purinton arranges his "chain" in such a way that the numbers tend to descend. Such niceties enhance the canonicalization process.)

In subsequent dialogue Purinton made the additional suggestion that the position after

10 1(11)1[2-9] 2(12)2[3] 4(13)4[5,6]

where we might say that a natural chain does not exist, can be described by the elegant

1L3(L1)(L2)

Interested readers are referred to the original discussion, entitled "Chained pivot/loop notation: a proposal."