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A (D,d)-polyominoid is a connected set of d-dimensional unit cubes (cells) with integer coordinates in D-dimensional space. For normal polyominoids, two cells are connected if they share a (d-1)-dimensional facet, but here we allow connections where the cells share a lower-dimensional face.

Each row is the counting sequence (by number of cells) of (D,d)-polyominoids with certain restrictions on the allowed connections between cells. Two cells have a connection of type (g,h) if they intersect in a (d-g)-dimensional unit cube and extend in d-h common dimensions. For example, d-dimensional polyominoes use connections of type (1,0), polyplets use connections of types (1,0) (edge connections) and (2,0) (corner connections), normal (3,2)-polyominoids use connections of types (1,0) ("soft" connections) and (1,1) ("hard" connections), hard polyominoids use connections of type (1,1).

Each row corresponds to a triple (D,d,C), where 1 <= d <= D and C is a set of pairs (g,h) with 1 <= g <= d and 0 <= h <= min(g, D-d). The k-th term of that row is the number of free k-celled (D,d)-polyominoids with connections of the types in C. Connections of types not in C are permitted, but the polyominoids must be connected through the specified connections only. For example, polyominoes may have cells that intersect in a point (g = 2) and hard polyominoids can have soft connections (h = 0) that are not needed to keep the polyominoids connected.

The rows are sorted first by D, then by d, and finally by a binary vector indicating which types of connections are allowed, where the connection types (g,h) are sorted lexicographically. (See table in cross-references.)

For each pair (D,d), the first row is 1, 0, 0, ..., corresponding to (D,d,{}) (no connections allowed).

The number of rows corresponding to given values of D and d is 2^((d+1)*(d+2)/2-1) if 2*d <= D and 2^((D-d+1)*(3*d-D+2)/2-1) otherwise.

This sequence and the related sequences A365650-A365655 and A365996-A366010 count polyominoids (A075679) with different rules for how the cells can be connected. In these sequences, connections other than the specified ones are permitted, but the polyominoids must be connected through the specified connections only. The polyominoids counted by this sequence, for example, are allowed to have right-angled corner-connections and flat edge-connections, as long as they are not needed for the polyominoid to be connected. A connection is flat if the two neighboring cells lie in the same plane, otherwise it is right-angled.

Also known as fixed polyplets. - David Bevan, Jul 28 2009

There are n cells, drawn on a square grid, pointwise connected; polyominoes may be rotated by 90 degrees and turned over.

Comments from Joseph Myers, Oct 27 2003. "There is a figure for n=5 (the first term this differs from A030222) on the last Vicher's link. I think the following explains this sequence, but someone should do the computations to verify it (and probably compute counts for "fixed" shapes - orientation matters - and one-sided shapes - at the same time and add those sequences if not present).

"Consider a polyplet (A030222) as made up of n components which are polyominoes, those polyominoes being joined to each other only at corners. Then sever all but n-1 of the diagonal links in such a way that a spanning tree remains. The present sequence counts such spanning trees (where different orientations of the same spanning tree do not count as distinct; note that a single symmetrical polyplet can produce multiple identical spanning trees of lesser symmetry in different orientations, which count as the same).

"Similarly, A056841 appears to count spanning trees of polyominoes (ordinary polyominoes, A000105), where the edges shared by two squares are the edges of the graph for the purposes of forming the spanning tree and A056787 may count spanning trees of polyplets where the graph has edges joining every pair of squares that share an edge or vertex (this definitely needs computations, but it does match the first three terms)."

The difference between this sequence and A030222 is illustrated through a comment and an image in A030222, also linked to here: the figures filled with identical color count as different here, but they represent the same polyplet and are counted only once in A030222. They all arise from adding one more square in three inequivalent positions (touching a corner, one side or two sides) to the (only) 4-polyplet with a hole (depicted here as not having a hole but rather a "bay", delimited to all but one (diagonal) direction). - M. F. Hasler, Sep 29 2014

These structures could be called polypletoids (or pseudo-polyominoids), because they are related to polyplets (polyominoids) as polyominoids (pseudo-polyominoes) are related to polyominoes.

A pseudo-polytan is a planar figure consisting of n isosceles right triangles joined either edge-to-edge or corner-to-corner, in such a way that the short edges of the triangles coincide with edges of the square lattice. Two figures are considered equivalent if they differ only by a rotation or reflection.

The pseudo-polytans are constructed in the same way as ordinary polytans (A006074), but allowing for corner-connections. Thus they generalize polytans in the same way that pseudo-polyominoes (aka polyplets, A030222) generalize ordinary polyominoes (A000105).

See A057787 for a description of polyarcs. The pseudo-polyarcs are constructed in the same way as ordinary polyarcs, but allowing for corner-connections. Thus they generalize polyarcs in the same way that pseudo-polyominoes (aka polyplets, A030222) generalize ordinary polyominoes (A000105). They can also be viewed as the "rounded" variant of pseudo-polytans (A354380), in the same way that ordinary polyarcs are the rounded variant of ordinary polytans (A006074).

Two figures are considered equivalent if they differ only by a rotation or reflection.

The pseudo-polyarcs grow tremendously fast, much faster than most polyforms. The initial data that have been computed suggest an asymptotic growth rate of at least 36^n.