A368388 -id:A368388 - cp's OEIS frontend

A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

1, 1, 2, 1, 8, 4, 17, 4, 2, 57, 5, 5, 5, 73, 5, 5, 73, 73, 5, 1, 5, 49321, 28165117, 20, 20, 338, 20, 246038, 63425, 28165117, 63425, 123019, 20, 49321, 20, 149998, 63425, 20, 117209258, 74999, 63425, 10, 20, 63425, 20, 74999, 10, 10, 63425, 149998, 63425, 10, 149998, 5000341, 64770, 5

Offset: 1

Pontus von Brömssen, Dec 22 2023

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. a(n)/A368387(n) is the probability that, when the appropriate number of cells have been added, those cells form the free polyomino with binary code A246521(n+1).

Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

			As an irregular triangle:
   1;
   1;
   2, 1;
   8, 4, 17, 4,  2;
  57, 5,  5, 5, 73, 5, 5, 73, 73, 5, 1, 5;
  ...
There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
For three squares, the probability for an L (or right) tromino (whose binary code is 7 = A246521(4)) is 2/3, so a(3) = 2. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 1.

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).
Persi Diaconis and William Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Semin. Mat. Univ. Politec. Torino, Vol. 49 (1991), No. 1, 95-119.
Gregory F. Lawler, Maury Bramson, and David Griffeath, Internal diffusion limited aggregation, The Annals of Probability 20 no. 4 (1992), 2117-2140.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367671, A367760, A367994, A368387 (denominators), A368388, A368390, A368392, A368393, A368660 (external diffusion-limited aggregation).

a(n)/A368387(n) = (A368392(n)/A368393(n))*A335573(n+1).

A368661 Square array read by antidiagonals; the n-th row is the decimal expansion of the minimum probability that a particular free polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 7, 5, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 6, 2, 0, 0, 0, 0, 0, 2, 2, 7, 8, 0, 0, 0

Offset: 1

Pontus von Brömssen, Jan 04 2024

The n-th row is the decimal expansion of the minimum of the numbers corresponding to rows A130866(n-1)+1..A130866(n) of A368660. See A368660 for details.

			Array begins:
  1.00000000000000000000... (1st row of A368660)
  1.00000000000000000000... (2nd row of A368660)
  0.42731251091162151298... (4th row of A368660)
  0.05462942885357382723... (6th row of A368660)
  0.01279646275569121440... (20th row of A368660)
  0.00867204327624784314... (42nd row of A368660)
  ...
The corresponding polyominoes for 1 <= n <= 6 are (all these are unique):
            _             _          _ _
       _   | |   _ _    _| |_      _|  _|
  _   | |  | |  |   |  |_   _|   _|  _|
 |_|  |_|  |_|  |_ _|    |_|    |_ _|

Index entries for sequences related to polyominoes.

Cf. A130866, A368388 (internal diffusion-limited aggregation), A368660, A368662 (maximum), A368664 (row 3), A368666 (row 4), A368864 (fixed polyominoes).

A368390 Numerator of the greatest probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 1, 2, 17, 57, 117209258, 363356390591, 292511604691740220504677006975130493521391, 2806200956345391684353279299766025388803856326541039774177, 8401384904285310565650785385525173372621364715976628525884130138767724737789789512541

Offset: 1

Pontus von Brömssen, Dec 22 2023

a(n) is the numerator of the maximum of A368386/A368387 over the n-th row. See A368386 for details.

			For 1 <= n <= 13, the following are the unique polyominoes that have the maximum probabilities for their respective sizes:
                    _      _        _
        _    _     | |_   | |_    _| |_
   _   | |  | |_   |  _|  |   |  |_    |
  |_|  |_|  |_ _|  |_|    |_ _|    |_ _|
     _          _          _ _      _ _ _
   _| |_ _    _| |_ _    _|   |_   |     |_
  |_     _|  |      _|  |      _|  |      _|
    |_ _|    |_ _ _|    |_ _ _|    |_ _ _|
                           _
              _ _ _      _| |_
   _ _ _     |     |_   |     |_
  |     |_   |       |  |       |
  |       |  |_   _ _|  |_   _ _|
  |_ _ _ _|    |_|        |_|

Pontus von Brömssen, Table of n, a(n) for n = 1..13
Index entries for sequences related to polyominoes.

Cf. A367673, A367762, A367998, A368386, A368387, A368388, A368391 (denominators), A368394, A368662 (external diffusion-limited aggregation).

A368389 Denominator of the least probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 1, 3, 35, 924, 19019, 14774760, 767191139, 2455848787392, 2993165027255, 6240848877339043968, 5018882339663051609, 238246203631345609763405422080

Offset: 1

Pontus von Brömssen, Dec 22 2023

a(n) is the denominator of the minimum of A368386/A368387 over the n-th row. See A368386 for details.

For n <= 13, the straight polyomino has the least probability of appearing among all n-celled polyominoes, and it seems likely that this is true for all n.

Index entries for sequences related to polyominoes.

Cf. A367997, A368386, A368387, A368388 (numerators), A368391, A368661 (external diffusion-limited aggregation).

cp's OEIS Frontend

A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

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A368661 Square array read by antidiagonals; the n-th row is the decimal expansion of the minimum probability that a particular free polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

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A368390 Numerator of the greatest probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

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A368389 Denominator of the least probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

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