A362094 Number of connected supports with n standard pieces for standard puzzles of the shape 2 X k, up to support-reduction. (See comments and reference for precise definition.)

6, 37, 259, 1391, 5460

Offset: 1

Author

Jodi Spitz, Apr 08 2023

Keywords

• nonn
• more

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,

6--17 2--4

| | reduces to the standard piece | |.

9--5 3--1

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 support-reduction 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 (5-c).

Note: Han (see Links) simply calls support-reduction "reduction." It has been called "support-reduction" here to distinguish it from the reduction of pieces into standard pieces.

For further definitions and clarification, see Han reference.

Examples

			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 support-reduced to each other, such as:
   4--3     3--4     4--2     2--4     3--2         2--3
  {|  |} , {|  |} , {|  |} , {|  |} , {|  |} , and {|  |} .
   1--2     1--2     1--3     1--3     1--4         1--4
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.)
		

References

• Guo-Niu Han, Enumeration of Standard Puzzles, University of Strasbourg, May 2011, page 5.

Links

• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Crossrefs

Cf. A196265.