
COMMENTS

A piece is a 2 X 2 matrix of distinct numbers, each called a label. A standard piece is a 2 X 2 matrix containing, in some order, the numbers {1,2,3,4} once each. A piece p_1 can be reduced to a standard piece p_2 if p_2 preserves the label order of p_1. For example,
617 24
  reduces to the standard piece  .
95 31
A standard puzzle of the shape 2 X k is a 2 X k matrix containing, in some order, {1,2,...,2k}. A support P for a standard puzzle Q of the shape 2 X k is a finite set of standard pieces {p_1,p_2,...} such that for any 2 X 2 submatrix T of Q, there exists a p_x in P such that T is equivalent to p_x under reduction.
A support P is connected if for any two pieces p_1, p_2 in P, there exists a standard puzzle containing p_1 and p_2 in its support. Two supports P, P' are equivalent under supportreduction if P' can be reached from P by: 1) exchanging the left and right columns of every piece in P, 2) exchanging the top and bottom row of every piece in P, and/or 3) replacing each label c of every piece in P with (5c).
Note: Han (see Links) simply calls supportreduction "reduction." It has been called "supportreduction" here to distinguish it from the reduction of pieces into standard pieces.
For further definitions and clarification, see Han reference.


REFERENCES

GuoNiu Han, Enumeration of Standard Puzzles, University of Strasbourg, May 2011, page 5.


EXAMPLE

a(1) = 6. There exist 4! = 24 standard pieces and so 24 unique supports P with 1 standard piece. Of these supports, there is at most a set of a(1) = 6 supports which cannot be supportreduced to each other, such as:
43 34 42 24 32 23
{ } , { } , { } , { } , { } , and { } .
12 12 13 13 14 14
We know these supports are connected because for any of support from this set P and any 2 standard pieces p_1, p_2 in P, there exists a standard puzzle with p_1 and p_2 in its support. (This is obvious since each support has only 1 piece.)
