A368391 -id:A368391 - cp's OEIS frontend

A368387 a(n) is the denominator 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, 3, 3, 35, 35, 35, 35, 35, 154, 462, 462, 231, 462, 231, 462, 924, 462, 462, 7, 924, 1846572, 492573081, 19019, 19019, 5073, 19019, 1804297, 7379372, 492573081, 7379372, 1804297, 19019, 1846572, 19019, 5534529, 7379372, 19019, 492573081, 5534529, 7379372, 19019, 19019, 7379372, 19019, 5534529, 19019, 19019, 14758744, 5534529, 7379372, 19019, 5534529, 44276232, 1844843, 19019

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. A368386(n)/a(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;
    3,   3;
   35,  35,  35,  35,  35;
  154, 462, 462, 231, 462, 231, 462, 924, 462, 462, 7, 924;
  ...
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) = 3. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 3.

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, A367672, A367761, A367995, A368386 (numerators), A368389, A368391, A368392, A368393, A368660 (external diffusion-limited aggregation).

A368386(n)/a(n) = (A368392(n)/A368393(n))*A335573(n+1).

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).

A368395 Denominator of the greatest probability that a particular fixed polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 2, 6, 140, 7, 985146162, 190373377530336, 4103913140371011711744, 1900892466087624965855151720, 615876607410431086757183559738769319966704415344178702618301644074612533917929905792000, 4113793825406037267512938546695751842153799476671563574067367370457531066126881303220587185019966105893663857242112000

Offset: 1

Pontus von Brömssen, Dec 22 2023

a(n) is the denominator of the maximum of A368392/A368393 over the n-th row.

See A368386 for details.

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

Cf. A367678, A367767, A368005, A368386, A368391, A368392, A368393, A368394 (numerators), A368865 (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

A368387 a(n) is the denominator 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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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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A368395 Denominator of the greatest probability that a particular fixed 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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