March 16, 1965

Dr. William J. Wiswesser

Research Chemist, Crops Division
U. S. Army Biological Laboratories
Fort Detrick

Frederick, Maryland 2101

Dear Dr. Wiswesser:

Thank you for clearing up the 43 tetracyclics. (Your letter safd tricyclics;
hence the confusion.)

I have only looked explicitly at graphs with a valence _ 4, The algorithm is
to scan all the trivalent graphs, then systematically replace one or more tetra-
gonal circuits by quadrivalent centers. Thus the graphs 6-13 of your diagram
('63-8) can be obtained from the set of 8-vertex trimalent graphs as follows:

6 8L (1) 10 8D

T 8I il 8c

8 8L (2) or 86 12 8B

9 8K 13 8E (coded per p. 136 attached)

while with the reduction of 2 tetragonal circuits, 18-22 correspond to the
10-vertex graphs

14 AGECA 17 AGCDB 20 ACECC
15 ACACA 18 AGBCB 21 BDECC
16 AGECA 19 BCCCB 22 BCDDB

coded as in the sheets already sent you. Your list seems to be complete as

far as these go. I would need a computer program to search the list of 12-graphs
for each qualifying example to check against tribe X, (graphs with 3 disjoint
tetragons) but have no doubt your list is correct on’ these also. One could use
the same principle (reduction of pentagonal cirecvits) to find all possible
Vetypes, etc.

As to the theory of partitions, my experience has been that the analytical
derivations all give recursive expressions which, however elegant, amount to
precisely the same computations as, e.g., Blair and Henze used. It will take
me some time to digest your notes, but I note you may not have run into a little
known work by R. A. Fisher, which I enclose. (Fisher seems to have overlooked
or bypassed both Polya and Henze-Blair.) Your notes included several references
that were new to me and which I am very happy to have,

dIssaMslin
Dr. William J. Wiswesser March 16, 1965
Page 2

My main interest in this field is the systematic generation of hypotheses,

i.e., explicit constructions of distinct types. Combinatorial analysis of course
plays a guiding role in this, but it looks as if still more powerful methods need
to be developed and applied. Perkkgps Polya may be able to help in these steps.

Sincerely yours,

Joshua Lederberg
Professor of Genetics

3 le &
P.S. I talked to Dr. Smith over the phone. I can see no simple formulation
of the conjecture underlying old Rule 30 in its new formhatidénon.