362 Murray Eden 



be thought of as squares), but their positions are restricted to the points of a 

 (two-dimensional) square lattice. 



Any arrangement of cells on the lattice will be called a configuration, i.e. 

 an arrangement of k cells will be called a /c-configuration. If each cell in a 

 configuration is adjacent to at least one other cell then such a configuration 

 will be called connected. We will be interested only in connected configurations. 

 The set of all possible (connected) A'-configurations will be called the k-array. 

 The number of distinct ^-configurations, i.e. configurations that are not iso- 

 morphic under translations, reflections and rotations, will be called the order 

 of the ^'-array, symbolized N[k]. 



Each cell will have four edges, corresponding to the four nearest-neighbor 

 lattice points. An edge will be called open if its corresponding lattice point 

 is unoccupied by a cell, otherwise it is covered. Each cell also has four corners 

 corresponding to points equidistant to four lattice points. A corner will be 

 called an inner corner if it is at the center of a cluster of four cells. 



The problem of enumerating all possible A'-configurations is one that has, 

 as yet, no easy solution. Similar combinatorial problems, arising in physics 

 in what is called the order-disorder problem, have been considered by a large 

 number of workers. Of particular relevance to the above problem is the work 

 of Van der Waerden (10), Kac and Ward (11), and Humans and de Boer (12). 



Certain bounds can be set for the order of the A-array. We can determine 

 a lower bound for N[k] by enumerating all members of a certain subset of 

 [k], i.e. the subset in which all save two cells have two edges covered. Two 

 cells, i.e. the ends, have only one edge covered. It is even easier to enumerate 

 a smaller subset of this 'two-ended' set. Consider an arbitrary lattice point as 

 the origin of a random walk. Limit the choices for the first step and each 

 succeeding step in this random walk to lattice points, either above or to the 

 right. The k^^^ cell will be added after k — 1 steps are taken. At each point 

 there are exactly two possible choices, so that in all we have produced 2*^^~^ 

 configurations. Since each configuration (except those that exhibit internal 

 symmetry, in any case, a small fraction) occurs four times in 2''"^ configurations, 

 the number of distinct configurations is 2^~^. The restriction to two choice 

 points is dictated by the necessity of avoiding cross-overs in the random walk. 

 Obviously, each cross-over would have the effect of decreasing the number 

 of occupied lattice points by one. 



However, if the random walk is permitted three choice points, i.e. above, 

 to the right and to the left, one can estimate the number of such walks of length, 

 k, which contain no point adjacent to more than two occupied sites*. Such 

 walks are isomorphic to the set of 'two-ended' /c-configurations. This estimate 

 was found to be very close to (1 + \/^)'^~^- I^ consequence the lower bound 

 for the number of A'-configurations may be raised to this value. 



Upper bounds can also be computed using a somewhat diff'erent combina- 

 torial technique. Consider any A-configuration. Arbitrarily choose one cell 

 as the origin, and also arbitrarily choose one of the four possible orientations 

 of the lattice. Identify this cell by 1 if it has a cell beneath, otherwise 0. Further, 

 this cell may have a cell adjacent to it on the left; if so, assign a 1 to the next 



* The mathematical details of the results presented in the text will be the subject of a 

 separate publication. 



