OFFSET
0,2
COMMENTS
Start with an n X m rectangle and cut it vertically along any set of the m-1 separators. There are binomial(m-1,c) ways of doing this with 0 <= c < m cuts. Inside each of these 1+c regions cut vertically, for which there are 2^(n-1) choices. The total number of ways of dissecting the rectangle into rectangles in this way is Sum_{c=0..m-1} binomial(m-1,c) 2^((1+c)(n-1)) = 2^(n-1)*(1+2^(n-1))^(m-1) = T(n-1,m-1).
The symmetrized version of the array is S(n,m) = T(n,m) + T(m,n) - 2^(m+n) <= A116694(n,m), which counts tilings that start with guillotine cuts either horizontally or vertically, avoiding double counting of the tilings where the order of the cuts does not matter. - R. J. Mathar, Nov 29 2015
LINKS
R. J. Mathar, Counting 2-way monotonic terrace forms over rectangular landscapes, vixra:1511.0225 (2015), Section 6.
EXAMPLE
1, 2, 4, 8, 16, 32, ...
2, 6, 18, 54, 162, 486, ...
4, 20, 100, 500, 2500, 12500, ...
8, 72, 648, 5832, 52488, 472392, ...
16, 272, 4624, 78608, 1336336, 22717712, ...
32, 1056, 34848, 1149984, 37949472, 1252332576, ...
.
The symmetrized version S(n,m) starts
1, 2, 4, 8, 16, 32, ...
2, 8, 30, 110, 402, 1478, ...
4, 30, 184, 1116, 7060, 47220, ...
8, 110, 1116, 11600, 130968, 1622120, ...
16, 402, 7060, 130968, 2672416, 60666672, ...
32, 1478, 47220, 1622120, 60666672, 2504664128, ...
MAPLE
A264872 := proc(n, m)
2^n*(1+2^n)^m ;
end proc:
seq(seq(A264872(n, d-n), n=0..d), d=0..12) ; # R. J. Mathar, Aug 14 2024
MATHEMATICA
Table[2^(n - m) (1 + 2^(n - m))^m, {n, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Nov 27 2015 *)
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Nov 27 2015
STATUS
approved