cp's OEIS Frontend

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.)

%I A362094 #24 Aug 05 2023 23:34:28
%S A362094 6,37,259,1391,5460
%N 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.)
%C A362094 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,
%C A362094    6--17                              2--4
%C A362094    |  | reduces to the standard piece |  |.
%C A362094    9--5                               3--1
%C A362094 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.
%C A362094 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).
%C A362094 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.
%C A362094 For further definitions and clarification, see Han reference.
%D A362094 Guo-Niu Han, Enumeration of Standard Puzzles, University of Strasbourg, May 2011, page 5.
%H A362094 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%e A362094 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:
%e A362094    4--3     3--4     4--2     2--4     3--2         2--3
%e A362094   {|  |} , {|  |} , {|  |} , {|  |} , {|  |} , and {|  |} .
%e A362094    1--2     1--2     1--3     1--3     1--4         1--4
%e A362094 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.)
%Y A362094 Cf. A196265.
%K A362094 nonn,more
%O A362094 1,1
%A A362094 _Jodi Spitz_, Apr 08 2023