OFFSET
1,7
COMMENTS
More precisely, the preference for long pieces means that the n-th row is the colexicographically largest n-tuple (x_1, ..., x_n) such that all n-ominoes can be built by x_k pieces of size 1 X k, 1 <= k <= n (with no pieces left over). - Pontus von Brömssen, Aug 09 2023
LINKS
Michel M. Dauchez, Wood sections used for polyminoes.
FORMULA
Sum_{k=1..n} k*T(n,k) = n*A000105(n). - Pontus von Brömssen, Aug 06 2023
EXAMPLE
Trominoes (n=3) are made with 3 and 2 + 1 sections, so row 3 is (1,1,1).
Pentominoes (n=5) are 5, 2(4+1), 4(3+2), 4(3+1+1) and 2+2+1 (sections ordered giving 11, 6, 8, 2, 1 as row 5).
The table begins as follows:
1;
0, 1;
1, 1, 1;
2, 4, 2, 1;
11, 6, 8, 2, 1;
...
The table below shows how the polyomino shapes (up to pentominoes) are characterized for purposes of this sequence:
.
polyomino #squares
class shape per row characterization
-----------------------------------------------
monomino X 1 "1"
-----------------------------------------------
domino XX 2 "2"
-----------------------------------------------
tromino XXX 3 "3"
.
XX 2 "2 + 1"
X 1
-----------------------------------------------
tetromino XXXX 4 "4"
.
XXX 3 "3+1"
X 1
.
XXX 3 "3+1"
X 1
.
XX 2 "2+2"
XX 2
.
XX 2 "2+2"
XX 2
-----------------------------------------------
pentomino XXXXX 5 "5"
.
XXXX 4 "4+1"
X 1
.
XXXX 4 "4+1"
X 1
.
XXX 3 "3+2"
XX 2
.
XXX 3 "3+2"
XX 2
.
XXX 3 "3+1+1"
X X 2
.
XXX 3 "3+2"
X 1
X 1
.
XXX 3 "3+2"
X 1
X 1
.
X 1 "3+1+1"
XXX 3
X 1
.
X 1 "3+1+1"
XXX 3
X 1
.
X 1 "3+1+1"
XXX 3
X 1
.
XX 2 "2+2+1"
XX 2
X 1
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michel M. Dauchez, Jul 02 2023
EXTENSIONS
Rows 6-10 and name clarified by Pontus von Brömssen, Aug 09 2023.
STATUS
approved