This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A373692 #62 Apr 26 2025 20:28:24
%S A373692 1,3,3,15,45,15,105,1575,1575,105,945,99225,510525,99225,945,10395,
%T A373692 9823275,376473825,376473825,9823275,10395,135135,1404728325,
%U A373692 533407191975,4202869719825,533407191975,1404728325,135135,2027025,273922023375,1302400497234375,115509334438258425,115509334438258425,1302400497234375,273922023375,2027025
%N A373692 Table of the number of ways T(m,n) to partition a 2m X 2n grid into Cartesian products of size 2 X 2, read by ascending antidiagonals.
%C A373692 This is a special case of the problem to partition a Cartesian product P X Q into squares P_i X Q_i, i.e. |P_i| = |Q_i|. In our case all subsets have size 2. Using the terminology of A160911 we deal with partitions of type (2m X 2n: 2,2,2,...).
%C A373692 From _Markus Sigg_, Jul 25-26 2024: (Start)
%C A373692 T(m,n) is a multiple of (2m-1)(2n-1) as there are that many ways to place a Cartesian product with one point in the top left of the grid, and the resulting configurations are equivalent.
%C A373692 For m,n > 1, starting with the Cartesian product {1,2m} X {1,2n} and evaluating the options for adding a Cartesian product with one point in (1,2) shows that T(m,n) is a multiple of (2m-1)(2n-1)*lcm(2m-3,2n-3). (End)
%H A373692 Markus Sigg, <a href="/A373692/b373692.txt">Table of n, a(n) for n = 1..59</a>
%H A373692 Rainer Rosenthal, <a href="/A373692/a373692_1.png">Colored illustration showing that T(2,2) = 45.</a>
%H A373692 Rainer Rosenthal, <a href="/A373692/a373692.txt">Expanded Example.</a>
%H A373692 Markus Sigg, <a href="/A373692/a373692.c.txt">C program for computing the very first terms of the sequence.</a>
%H A373692 Markus Sigg, <a href="/A373692/a373692_1.c.txt">C program for computing more terms of the sequence, namely T(ROWS/2,COLS/2) where ROWS and COLS are defined in the first program lines.</a>
%e A373692 Table T(m,n) begins:
%e A373692 .
%e A373692 n 1 2 3 4 5
%e A373692 m \ ---------------------------------------------------------------------
%e A373692 1 | 1 3 15 105 945
%e A373692 2 | 3 45 1575 99225 9823275
%e A373692 3 | 15 1575 510525 376473825 533407191975
%e A373692 4 | 105 99225 376473825 4202869719825
%e A373692 5 | 945 9823275 533407191975 115509334438258425
%e A373692 6 | 10395 1404728325 1302400497234375 6907197292027901339625
%e A373692 7 | 135135 273922023375
%e A373692 8 | 2027025
%e A373692 .
%e A373692 These are the T(1,2) = 3 possible partitions:
%e A373692 .
%e A373692 |A A B B| |A B A B| |A B B A|
%e A373692 |A A B B| |A B A B| |A B B A|
%e A373692 _________________________________
%e A373692 #1 #2 #3
%e A373692 .
%e A373692 For T(2,2) = 45 consider these special partitions:
%e A373692 .
%e A373692 |A A B B| |A A B B| |A A B B|
%e A373692 |A A B B| |A A B B| |A A C C|
%e A373692 |C C D D| |C D C D| |D D B B|
%e A373692 |C C D D| |C D C D| |D D C C|
%e A373692 ___________________________________
%e A373692 Base1 Base2 Base3
%e A373692 .
%e A373692 Any partition is equivalent to exactly one of these partitions, i.e. it differs only by the order of the rows and columns. The number of equivalent partitions is respectively 9, 18, 18. Thus we have T(2,2) = 9 + 18 + 18 = 45.
%e A373692 See the picture and the expanded example in the link section.
%e A373692 .
%e A373692 Some other known terms: T(5,5) = 84250218148544569727025, T(6,4) = 6907197292027901339625, T(7,4) = 814287280679532017261528625, T(8,4) = 173936355367823940296258779550625, T(9,4) = 62626268302216078023651174787170095625, T(10,4) = 35784629301848063975515694953866493243805625.
%o A373692 (C) // See Markus Sigg link.
%Y A373692 Cf. A001147 (column 1), A079484 (column 2 - conjectured), A160911.
%K A373692 nonn,tabl,hard
%O A373692 1,2
%A A373692 _Rainer Rosenthal_ and _Markus Sigg_, Jun 13 2024
%E A373692 a(24) and beyond from _Markus Sigg_, Jul 18 2024