A259090 -id:A259090 - cp's OEIS frontend

A006749 Number of asymmetric polyominoes with n cells.

0, 0, 0, 1, 5, 20, 84, 316, 1196, 4461, 16750, 62878, 237394, 899265, 3422111, 13069026, 50091095, 192583152, 742560511, 2870523142, 11122817672, 43191285751, 168046076423, 654997492842, 2557223459805, 9999080270766, 39152997087077, 153511067364760

Offset: 1

N. J. A. Sloane

This sequence counts polyominoes whose symmetry group has order 1.

A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 1..50 (This version corrects erroneous values from a(44) onwards in previous version.)
Tomás Oliveira e Silva, Enumeration of polyominoes
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy). See page 21.
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.

a(n) + A259090(n) = A000105(n). - R. J. Mathar, Sep 29 2021

Extended to n=28 by Tomás Oliveira e Silva.

A331621 Number of distinct structures that can be made from n cubes without overhangs.

Original entry on oeis.org

1, 1, 2, 4, 12, 35, 129, 495, 2101, 9154, 41356, 189466, 880156, 4120515, 19425037, 92038062, 438030079, 2092403558, 10027947217, 48198234188, 232261124908, 1121853426115, 5430222591596

Offset: 0

Nicholas A. Kennedy, Jan 22 2020

This is an extension of the free polyominoes (A000105) to the third dimension. Structures are considered equivalent if they can be mapped into each other by reflection in a vertical plane or rotation around the vertical axis. They are not equivalent if they can only be mapped onto each other by rotation around an axis parallel to the horizontal plane.
From John Mason, Mar 03 2025: (Start)
Equivalently, the sequence enumerates inscribed polyominoes that have a positive integer in each square, such that the size of the polyomino is considered to be the sum of the integers.
Examples.
  Size 1:
  +-+
  |1|
  +-+
.
  Size 2:
  +-+-+ +-+
  |1|1| |2|
  +-+-+ +-+
.
  Size 3:
  +-+-+-+ +-+-+ +-+-+ +-+
  |1|1|1| |2|1| |1|1| |3|
  +-+-+-+ +-+-+ +-+-+ +-+
                  |1|
                  +-+
(End)

			For n = 0, one (the empty) structure is possible.
For n = 1, only one structure is possible, a single cube.
For n = 2, two structures are possible: two cubes one on top of the other, and two next to each other.
For n = 3, four structures are possible: an L shape with the L oriented vertically, an L shape with the L laid flat, a structure with 3 cubes stacked on top of each other and a structure with 3 cubes laid flat.
For n = 4, there are 12 possible distinct structures made from 4 cubes without overhangs. These include 1 structure that is 4 cubes tall, 1 structure that is 3 cubes tall, 5 structures that are 2 cubes tall and 5 that are 1 cube tall.

Arnauld Chevallier, Javascript Node.JS program
Nicholas A. Kennedy, Buildings made from cubes, Code Golf Stack Exchange 2020
Nicholas A. Kennedy, Images of possible structures made from 4 cubes
Miles, Java program

Extension to the third dimension of A000105.

Cf. A006749, A007318, A259090, A349329, A351616, A234008.

Java
```
See Miles link
```
JavaScript
```
See Arnauld Chevallier link
```

From John Mason, Mar 03 2025: (Start)
Define a(n)=f(n)+g(n) where f(n) enumerates the structures having an asymmetrical base, and g(n) enumerates the structures having a symmetrical base.
Then for n>=4, f(n) = Sum_{i=4..n} ((A006749(i)*C(n-1,i-1)), and g(n) < Sum_{i=1..n} ((A259090(i)*C(n-1,i-1)).
For structures having a base with: reflective orthogonal symmetry about an axis that passes through cell vertices, 180 degree rotational symmetry about a point at a cell vertex or midway along an edge, the number of structures is (1) for odd n: Sum_{i=1..n} ((S(i)*C(n-1,i-1)) / 2, and (2) for even n: Sum_{i=1..n} ((S(i)*(C(n-1,i-1)+C(n/2-1,i/2-1)) / 2, for S(n) = A349329(n), A351616(n) and A234008(n) respectively.
As almost all polyominoes are asymmetrical, a(n)/f(n) tends to 1. (End)

a(15)-a(22) from John Mason, Mar 03 2025

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A006749 Number of asymmetric polyominoes with n cells.

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A331621 Number of distinct structures that can be made from n cubes without overhangs.

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