These slides ( www.nist.gov/itl/cxs/upload/percolation_slides.pdf ) give a good survey. In particular it talks about your quantity under the name chi(p), and gives a recursion for the generating function that you're looking for. Victor On Tue, May 22, 2012 at 5:40 PM, Allan Wechsler <acwacw@gmail.com> wrote:
If the cells of a square grid are randomly colored black and white, with p being the probability of a cell being black, then for p small enough we can ask the expected size F(p) of the black polyomino including a given cell. (If that cell happens to be white, the size is zero.)
The reference cell is white with probability (1-p); this contributes 0*(1-p)=0 to the expected polyomino size.
It is black, and all its neighbors are white, with probability p(1-p)^4; this contributes p - 4p^2 + 6p^3 - 4p^4 + p^5 to the expected size.
All bigger polyominoes only contribute quadratic or higher terms, so we know the power series for F(p) begins 0 + p + ...
If you perform a census of the ways the starting cell can be part of a domino, you learn that the next term must be 4p^2.
Repeating this exercise with the trominoes, if I have done it right, gives us the cubic term, 12p^3.
I conjecture that all the coefficients of this power series are positive. If I did my counting right, the quartic coefficient is 24.
The sequence incipit 0, 1, 4, 12, 24, only yields one hit on OEIS, and it isn't this. I tried to calculate the quintic coefficient and made a blunder somewhere.
This must be known stuff. What are some more terms of this series?
Each oriented polyomino with one distinguished cell contributes n p^n (1-p)^m to the power series, where n is the number of cells in the polyomino, and m is the number of cells immediately adjacent to it. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun