A019988 -id:A019988 - cp's OEIS frontend

A181528 Number of connected graphs with n edges embeddable into square lattice.

1, 1, 1, 2, 4, 6, 14, 28, 68, 156, 399, 1012, 2732, 7385, 20665, 58377, 168119, 488771

Offset: 0

I. E. Kashuba (kashuba(AT)bitp.kiev.ua), Oct 27 2010

a(n) <= number of connected planar graphs with n edges A046091.

			For n = 3 there are a(3) = 2 graphs: the claw graph, corresponding to a single free polystick, and the 3-path, corresponding to 4 different free polysticks.

Andreas R. Hehn, Series Expansion Methods for Quantum Lattice Models, Doctoral Thesis, ETH Zürich, 2016.

Cf. A046091, A019988 (embeddings, or free polysticks), A255539 (with n nodes, neighbors connected).

Terms a(16)-a(17) from Hehn Table 3.1 and a(0) = 1 added by Andrey Zabolotskiy, Oct 22 2022

A333233 Number of free Tangles of size n.

Original entry on oeis.org

1, 1, 2, 5, 16, 55, 221, 947, 4239, 19452, 90791, 428839, 2043548, 9807941

Offset: 0

Douglas A. Torrance, Mar 12 2020

a(n) is the number of free Tangles (smooth simple closed curves piecewise-defined by quadrants of circles) which have a dual graph containing n edges, or equivalently, enclose an area of (4*n + Pi)*r^2, where 1/r is the curvature.  By 'free', we mean that we allow rotations and reflections.
Tangles may also be 'fixed', i.e., if we do not allow rotations and reflections (A333080).
Tangles whose dual graphs are trees correspond exactly to diagonal polyominoes (A056841).
Dual graphs of Tangles are polysticks (A019988), but the only chordless cycles allowed are squares, e.g., this is *not* the dual graph of a Tangle:
o-o-o
|   |
o-o-o
but this is:
o-o-o
| | |
o-o-o

Douglas A. Torrance, Enumeration of planar Tangles, arXiv:1906.01541 [math.CO], 2020. Sums of rows from Table 4.1 (C).

Cf. A019988, A056841, A333080.

a(11)-a(13) from John Mason, Feb 14 2023

A336281 Total number of ways of embedding connected graphs with n edges in the square lattice with diagonals allowed.

Original entry on oeis.org

2, 6, 41, 318, 3108, 32243, 350575, 3896568

Offset: 1

James W. Anderson, Jul 15 2020

The embedding must map edges in the graph onto either horizontal or vertical grid lines of length 1 or diagonals of length sqrt(2). Vertices in the graph must map onto lattice points, and of course must preserve the incidence structure of the graph. A square in the lattice may have both diagonals present - their intersection does not count as an incidence.
Configurations differing only a rotation or reflection are not counted as different.
The resulting figures are variously called 'polysticks', 'polyedges' or 'polyforms'.

N. J. A. Sloane, Illustration for a(1)=2, a(2)=6, a(3)=41. [Thanks to Peter Munn for correcting errors in my first drawing.]

Without diagonal edges, we get A019988.

Cf. A052436.

a(7)-a(8) from John Mason, Aug 17 2021

A348096 Array A(n,s) read by rows: the free n-polysticks of the square lattice with symmetry group of order 2^s.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 1, 3, 1, 0, 8, 5, 1, 2, 39, 14, 2, 0, 187, 31, 4, 0, 880, 66, 4, 0, 4109, 142, 12, 2, 19274, 310, 7, 0, 90965, 694, 19, 0, 432545, 1445, 15, 0

Offset: 1

R. J. Mathar, Sep 30 2021

The array has 4 columns for symmetry groups of order 1, 2, 4 and 8 (subgroups of D_8).

Polysticks with group order 1 have no symmetry. Polysticks with group order 2 have either a mirror line (parallel to edges or along a diagonal of the lattice) or a rotation axis of order 2 (180-degree rotation). Polysticks of group order 4 have two orthogonal mirror lines and the 180-degree rotation. Polysticks of group order 8 have in addition a rotation axis or order 4 (90-degree rotations), i.e. the full symmetry of the square.

			The array starts
      0   0  1 0
      0   1  1 0
      1   3  1 0
      8   5  1 2
     39  14  2 0
    187  31  4 0
    880  66  4 0
   4109 142 12 2
  19274 310  7 0
  90965 694 19 0
A(4,3)=2 counts the fully-symmetric unit square and the cross.

Cf. A019988 (row sums), A096267 (fixed polysticks).

Sum_{s=0..3} A(n,s) = A019988(n).
8*A(n,0) + 4*A(n,1) + 2*A(n,2) + A(n,3) = A096267(n).
A(n,3) = 0 if n is not a multiple of 4.

Row n=11 added.- R. J. Mathar, Oct 05 2021

A333362 Number of free polysticks with n segments on the edges of the n-cube.

Original entry on oeis.org

1, 1, 3, 7, 27, 121, 751

Offset: 1

Peter Kagey, Mar 16 2020

A free polystick is a polystick counted up to isometries of the n-cube.

Code Golf Stack Exchange, Counting polysticks on the n-cube.
Wikipedia, Hypercube.
Wikipedia, Polystick.

Cf. A019988.

a(n) = A333333(n,n) = A333333(n+k,n) for all k >= 0.

Definition corrected by Peter Kagey, Jun 13 2023

cp's OEIS Frontend

A181528 Number of connected graphs with n edges embeddable into square lattice.

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A333233 Number of free Tangles of size n.

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A336281 Total number of ways of embedding connected graphs with n edges in the square lattice with diagonals allowed.

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A348096 Array A(n,s) read by rows: the free n-polysticks of the square lattice with symmetry group of order 2^s.

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A333362 Number of free polysticks with n segments on the edges of the n-cube.

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