A268404 -id:A268404 - cp's OEIS frontend

A292357 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n.

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 39, 111, 39, 1, 1, 97, 649, 649, 97, 1, 1, 237, 3495, 7943, 3495, 237, 1, 1, 575, 18189, 86995, 86995, 18189, 575, 1, 1, 1391, 93231, 910667, 1890403, 910667, 93231, 1391, 1

Offset: 1

Andrew Howroyd, Oct 02 2017

Equivalently, the number of m X n binary arrays with all 1's connected and at least one 1 on each edge.

			Array begins:
===============================================================
m\n| 1   2     3       4         5           6             7
---|-----------------------------------------------------------
1  | 1   1     1       1         1           1             1...
2  | 1   5    15      39        97         237           575...
3  | 1  15   111     649      3495       18189         93231...
4  | 1  39   649    7943     86995      910667       9339937...
5  | 1  97  3495   86995   1890403    38916067     782256643...
6  | 1 237 18189  910667  38916067  1562052227   61025668579...
7  | 1 575 93231 9339937 782256643 61025668579 4617328590967...
...
T(2,2) = 5 counts 4 3-ominoes of shape 2x2 and 1 4-omino of shape 2x2.
T(3,2) = 15 counts 8 4-ominoes of shape 3x2, 6 5-ominoes of shape 3x2, and 1 6-omino of shape 3x2.
T(4,2) = 39 counts 12 5-ominoes of shape 4x2, 18 6-ominoes of shape 4x2, 8 7-ominoes of shape 4x2, and 1 8-omino of shape 4x2.

Andrew Howroyd, Table of n, a(n) for n = 1..435
Andrew Howroyd, Fixed polyominoes by width, height and number of cells
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 145.
Index entries for sequences related to polyominoes

Rows 2..4 are A034182, A034184, A034187.
Main diagonal is A268404.
Cf. A268371 (nonequivalent), A287151, A308359.

A287151 = Import["https://oeis.org/A287151/b287151.txt", "Table"][[All, 2]];
imax = Length[A287151];
mmax = Sqrt[2 imax] // Ceiling;
Clear[V]; VV = Table[V[m-n+1, n], {m, 1, mmax}, {n, 1, m}] // Flatten;
Do[Evaluate[VV[[i]]] = A287151[[i]], {i, 1, imax}];
V[0, ] = V[, 0] = 0;
T[m_, n_] := If[m == 1 || n == 1, 1, U[m, n] - 2 U[m, n-1] + U[m, n-2]];
U[m_, n_] := V[m, n] - 2 V[m-1, n] + V[m-2, n];
Table[T[m-n+1, n], {m, 1, mmax}, {n, 1, m}] // Flatten // Take[#, imax]& (* Jean-François Alcover, Sep 22 2019 *)

T(m, n) = U(m, n) - 2*U(m, n-1) + U(m, n-2) where U(m, n) = V(m, n) - 2*V(m-1, n) + V(m-2, n) and V(m, n) = A287151(m, n).

A334551 Number of fixed polyominoes with 2n-1 cells and width and height both equal to n.

Original entry on oeis.org

1, 4, 25, 120, 497, 1924, 7265, 27288, 102745, 388692, 1477721, 5643064, 21632785, 83204260, 320932177, 1240939448, 4808642313, 18668848852, 72601081385, 282762109272, 1102772229313, 4306062994148, 16832791708257, 65867445819160, 257980829463017

Offset: 1

Andrew Howroyd, Jun 06 2020

nonn

A polyomino with width and height equal to n must have at least 2n-1 cells.

			a(3) = 25. Up to rotation and reflection there are 6 possibilities:
       X       X   X         X         X     X
       X     X X   X X X   X X X   X X X   X X X
   X X X   X X     X       X       X         X

Andrew Howroyd, Table of n, a(n) for n = 1..500
Andrew Conway, Enumerating 2D percolation series by the finite-lattice method: theory, J. Phys. A: Math. Gen., 28 (1995), 335-349. See Table 4.

Main diagonal of A334552.

Cf. A268404.

Array[8Binomial[2(#-1),#-1]-3#^2+4#-8&,50] (* Paolo Xausa, Dec 21 2023 *)

a(n) = 8*binomial(2*(n-1), n-1) - 3*n^2 + 4*n - 8; \\ Peter J. Taylor, Dec 15 2020

a(n) = 2*binomial(2*(n-1), n-1) + 4*(n-2) + (n-2)^2*(2*n-5) + 2*Sum_{i=1..n-2} Sum_{j=1..n-2} ((n-2-i)*(n-2-j)+2)*binomial(i+j, i) for n > 1.
a(n) = 8*binomial(2*(n-1), n-1) - 3*n^2 + 4*n - 8. - Peter J. Taylor, Dec 15 2020
From Stefano Spezia, Sep 02 2022: (Start)
G.f.: 8*x/sqrt(1 - 4*x) - (8 - 17*x + 15*x^2)/(1 - x)^3.
a(n) ~ 2^(2*n+1)/sqrt(n*Pi). (End)

A268758 Number of polyominoes with width and height equal to 2n that are invariant under all symmetries of the square.

Original entry on oeis.org

1, 3, 17, 163, 2753, 84731, 4879497, 535376723, 112921823249, 45931435159067, 36048888105745113, 54568015172025197171, 159197415409641803530753, 894444473815989281612355579, 9671160618112663336510127727593, 201110001346886305066013828873025811

Offset: 1

Craig Knecht, Feb 14 2016

nonn

Also number of polyominoes with width and height equal to 2n - 1 that are invariant under all symmetries of the square.
Bisection of A268339.
The water retention model for mathematical surfaces is described in the link below. The definition of a "lake" in this model is related to a class of polyominoes in A268339. Percolation theory may refer to these structures as "clusters that touch all boundaries."
Transportation across the square lattice requires a path of continuous edge connected cells. Is a pattern that only connects two opposite boundaries of the square ranked differently from one that connects all four boundaries?
This sequence is part of a effort to classify water retention patterns in a square by their symmetry, their capacity to connect boundaries of the square and the number of edge cells that are connected across opposite boundaries.

			For a(2) = 3: the three polyominoes of width and height 2*2 - 1 = 3 and the corresponding three polynomial of width and height 2*2 = 4 are shown below. Note that each even-dimension polyomino is produced by duplicating the center row/column of an odd-dimension polyomino.
3 X 3:
   0 1 0     1 1 1     1 1 1
   1 1 1     1 0 1     1 1 1
   0 1 0     1 1 1     1 1 1
4 X 4:
  0 1 1 0   1 1 1 1   1 1 1 1
  1 1 1 1   1 0 0 1   1 1 1 1
  1 1 1 1   1 0 0 1   1 1 1 1
  0 1 1 0   1 1 1 1   1 1 1 1

Andrew Howroyd, Table of n, a(n) for n = 1..20
Craig Knecht, Connections between boundaries of the square
Craig Knecht Retention capacity of a random surface, arXiv:1110.6166 [cond-mat.dis-nn], 2011-2012.
Wikipedia, Water Retention on Mathematical Surfaces

Cf. A268339, A268404, A331878.

a(n) = A331878(n) - 3*A331878(n-1) + 3*A331878(n-2) - A331878(n-3) for n >= 4. - Andrew Howroyd, May 03 2020

Terms a(9) and beyond from Andrew Howroyd, May 03 2020

A293261 Number of fixed polyominoes without holes that have a width and height of n.

Original entry on oeis.org

1, 5, 106, 6074, 943340, 419355340, 554485727288, 2208574156731474, 26609978139626497670, 973224195603423767343946, 108342096917091380628767818812, 36763211016528549310068224122368860, 38044287043749436284594644308861499605492

Offset: 1

Andrew Howroyd, Oct 04 2017

nonn

Equivalently, the number of cycles on an (n+1) X (n+1) grid that touch each of the four outside edges.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..16
Anthony J. Guttmann and Iwan Jensen, Effect of Confinement: Polygons in Strips, Slabs and Rectangles, in: Polygons, Polyominoes and Polycubes, LNP 775, Springer, 2009. See Table 10.2 on p. 239.

Main diagonal of A232103.

Cf. A268404.

A377593 Number of aligned fixed polyominoes that will fit in a square of size n X n.

Original entry on oeis.org

1, 8, 151, 9472, 2081051, 1643823600, 4742607132499, 50303895480064088, 1966122506151835674303, 283294196554063138439927568, 150432366492029200690537003170367, 294212995394376069103067524948055548348, 2117957146063247996594586658579155551318256103, 56084287855193446153928896349599388059636859288133588, 5460061052459125116800111315595463810654508452342242195388707

Offset: 1

John Mason, Nov 02 2024

nonn

a(n) is the number of fixed polyominoes that have both width and height <= n. The word "aligned" in the title refers to the restriction that the polyominoes have edges parallel to the sides of the square.

			a(2) = 8 because of the monomino, 2 alignments of the domino, 4 alignments of the L-shaped tromino, and the square tetromino.

Cf. A292357, A268416, A268404.

a(n) = Sum_{i=1..n,j=1..n} A292357(i,j).

cp's OEIS Frontend

A292357 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n.

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A334551 Number of fixed polyominoes with 2n-1 cells and width and height both equal to n.

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A268758 Number of polyominoes with width and height equal to 2n that are invariant under all symmetries of the square.

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A293261 Number of fixed polyominoes without holes that have a width and height of n.

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A377593 Number of aligned fixed polyominoes that will fit in a square of size n X n.

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