IEEE TRANSACTIONS ON RELIABILITY, VOL. 49, NO. 3, SEPTEMBER 2000 317

Commentary: Interconnection of Massive Numbersof Paths

Alan O. Plait, Senior Member, IEEE

Index Terms—Factorial, sumerial.

NOTATION

Sumerial of kindnumber of paths from an origin point to the diago-nally opposite point in a rectilinear arraynumber of boxes in a rectilinear array in dimension.

I AM INTERESTED in the recent paper [1]. The intercon-nection of massive numbers of computer systems has really

come into its own in recent times. Witness the use of suchinterconnections in the Search for Extraterrestrial Intelligence(SETI) programs, and such problems as finding the minimumsolution to Ramsey party problems [2]. Over the years, thinkingabout related problems, I developed a designator, for a set ofspecific numbers, that I named “Sumerials” (really nothingfancy, just a new nomenclature). I define Sumerials of kindas

For factorials,

For sumerials,

Thus Sumerials are the diagonal coefficients of Pascal’sTriangle.

Among the uses of sumerials are applications to countingthe number of diagonal paths for various rectilinear arrays ofsquares. These are then relatable to the prognostications in [1].This could be used to determine the number of redundant pathsin a system.

Manuscript received November 12, 1999; revised March 13, 2000.The author, retired, was with ManTech International Corporation, Computer

Science Corporation, Armour Research Foundation (now IITRI), and the formerMagnavox Company. He is now at 8550 Park Shore Lane, Turtle Rock, Sarasota,FL 34238 USA (e-mail: APlait@ieee.org).

Publisher Item Identifier S 0018-9529(00)11756-7.

Fig. 1. 2-dimensional array of squares.

I have developed a general algorithm/equation for deter-mining the number of such diagonal paths for any dimensionalarray:

As a clarifying example, let an array of squares have 2 dimen-sions as shown in Fig. 1.

The number of paths from 0,0 to 3,4 is:

For a unit cube,

For a Tessaract (4-dimensional cube),

Of course, it is just a convenience to talk of rectilinear arrays,or squares (boxes). Right angles are not necessarily specified.

The illustrations can also be related to connections betweenneural networks of axons1 and synapses.2 The number of pathsincreases dramatically with the number of units in any dimen-sion. For a cubical array of just 10 units on a side, the numberof paths is more than .

1Axon: The part of a nerve cell which carries electrical impulses away fromthe cell body.

2Synapse: The gap between adjacent axons across which impulses are carriedby selected neuro-transmitter chemicals.

0018–9529/00$10.00 © 2000 IEEE

318 IEEE TRANSACTIONS ON RELIABILITY, VOL. 49, NO. 3, SEPTEMBER 2000

REFERENCES

[1] J. Wu, “Maximum-shortest-path (MSP): An optimal routing policy formesh-connected multicomputers,”IEEE Trans. Reliability, vol. 48, no.3, pp. 247–255, Sept. 1999.

[2] P. Hoffman,The Man Who Loved Only Numbers: Hyperion, 1998, pp.50–57.

Alan O. Plait and wife, Evelyn, are happily retired in Sarasota, FL since 1995.He has worked for ManTech International Corp, Computer Sciences Corp, Ar-mour Research Foundation (now IITRI), and the former Magnovox Company.He taught at Illinois Tech, Virginia Tech, and the Graduate School of the U.S.Department of Agriculture. For many years, he was a member of the AdCom ofthe IEEE Reliability Society, and was its President in 1985–1986; he is its Histo-rian. He is Historian of the Annual Reliability and Maintainability Symposium’sBoard of Directors. He is on the Board of Directors and is Exhibits CommitteeChairman of G.WIZ (Sarasota’s Hands-On Science and Technology Center).