OFFSET
1,3
COMMENTS
The towers must have a contiguous base of bricks, and each brick must be at least half supported below by another brick. The stacks do not need to be stable.
Conjecture: For n > 1, T(n,2) = A000071(n+2).
A038622(n-1,k) appears to give the number of domino towers consisting of n bricks with a base of k bricks.
Conjecture: T(n,n-1) = A339252(n-2). - Peter Kagey, Nov 21 2020
Conjecture: T(n,n-2) = A339254(n-3). - Peter Luschny, Nov 29 2020
Conjecture: T(n,n-3) = A339029(n-4). - Peter Luschny, Dec 01 2020
From Peter Luschny, Dec 01 2020: (Start)
The above conjectures can be summarized as follows:
T(2*n + k, n + k) = d_{n}(n + k - 1) for k >= 1 and 0 <= n <= 3, where
d_{0}(m) = 2^(m-1)*2;
d_{1}(m) = 2^(m-3)*(10 + 6*m);
d_{2}(m) = 2^(m-5)*(70 + 43*m + 9*m^2);
d_{3}(m) = 2^(m-7)*(588 + 367*m + 84*m^2 + 9*m^3). (End)
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 25-27.
LINKS
Peter Luschny, Table of n, a(n), for row(k) for k = 1..18 (the first 14 rows by Peter Kagey).
J. Bétréma and J.-G. Penaud, Animaux et arbres guingois, Theoretical Computer Science 117, 67-89, 1993.
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two dimensional directed animals problem to a one-dimensional path problem, Adv. in Appl. Math. 9(3), 334-357, 1988.
Peter Kagey, Symmetric Brick Stacking, Mathematics Stack Exchange, 2018.
Doron Zeilberger, The amazing 3^n theorem and its even more amazing proof, arXiv:1208.2258 [math.CO], 2012.
Doron Zeilberger, The 27 towers with 4 domino pieces, illustration.
FORMULA
EXAMPLE
Table begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11
---+-----------------------------------------------------
1 | 1
2 | 1 2
3 | 1 4 4
4 | 1 7 11 8
5 | 1 12 24 28 16
6 | 1 20 52 70 68 32
7 | 1 33 110 168 193 160 64
8 | 1 54 228 401 497 510 368 128
9 | 1 88 467 944 1257 1412 1304 832 256
10 | 1 143 949 2187 3172 3736 3879 3248 1856 512
11 | 1 232 1916 5010 7946 9778 10766 10360 7920 4096 1024
.
T(3,2) = 4 because there are four domino towers of height two consisting of three bricks:
+-------+-------+ +-------+ +-------+
| | | | | | |
+---+---+---+---+, +---+---+---+---+, +-------+---+---+---+, and
| | | | | | | |
+-------+ +-------+-------+ +-------+-------+
+-------+
| |
+---+---+---+-------+.
| | |
+-------+-------+
CROSSREFS
KEYWORD
AUTHOR
Peter Kagey, Mar 31 2020
STATUS
approved