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 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.) 0
 6, 37, 259, 1391, 5460 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. REFERENCES Guo-Niu Han, Enumeration of Standard Puzzles, University of Strasbourg, May 2011, page 5. LINKS Table of n, a(n) for n=1..5. Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020. 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 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.) CROSSREFS Cf. A196265. Sequence in context: A073013 A192238 A140712 * A079751 A088312 A012364 Adjacent sequences: A362091 A362092 A362093 * A362095 A362096 A362097 KEYWORD nonn,more AUTHOR Jodi Spitz, Apr 08 2023 STATUS approved

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Last modified August 15 08:18 EDT 2024. Contains 375173 sequences. (Running on oeis4.)