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A228106
Three-dimensional table read by rows: T(n,k,r) (1 <= k <= n, r >= 1) is the number of partitions in the r-th run of strictly increasing numbers of 2 X 2 squares in the list of partitions of an n X k rectangle into integer-sided squares, considering only the list of parts.
3
1, 1, 2, 1, 2, 2, 1, 1, 3, 3, 1, 5, 1, 1, 1, 3, 3, 2, 5, 3, 1, 5, 4, 1, 1, 1, 4, 4, 2, 1, 7, 3, 1, 3, 7, 5, 4, 3, 1, 10, 6, 4, 2, 1, 6, 1, 1, 1, 4, 4, 3, 1, 7, 5, 1, 3, 1, 7, 6, 5, 3, 2, 3, 10, 8, 7, 3, 1, 6, 4, 3, 4, 1, 10, 9, 8, 5, 1, 6, 5, 4, 1, 6, 1, 1
OFFSET
1,3
COMMENTS
The sorting order for the list of partitions is ascending with larger squares taking higher precedence.
A228107 specifies the length of each row for 1 <= k <= n <= 8.
LINKS
Christopher Hunt Gribble, Rows 1..36 flattened
Christopher Hunt Gribble, C++ program
EXAMPLE
The irregular triangle begins:
. r 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
n,k
1,1 1
2,1 1
2,2 2
3,1 1
3,2 2
3,3 2 1
4,1 1
4,2 3
4,3 3 1
4,4 5 1 1
5,1 1
5,2 3
5,3 3 2
5,4 5 3 1
5,5 5 4 1 1
6,1 1
6,2 4
6,3 4 2 1
6,4 7 3 1 3
6,5 7 5 4 3 1
6,6 10 6 4 2 1 6 1 1
7,1 1
7,2 4
7,3 4 3 1
7,4 7 5 1 3 1
7,5 7 6 5 3 2 3
7,6 10 8 7 3 1 6 4 3 4 1
7,7 10 9 8 5 1 6 5 4 1 6 1 1
8,1 1
8,2 5
8,3 5 3 2
8,4 9 5 3 5 1 1
8,5 9 7 6 5 3 1 3 2
8,6 13 9 7 5 4 9 5 3 5 4 2 1 4
8,7 13 11 10 7 5 9 7 6 3 5 3 2 7 6 3 4 1
8,8 17 13 11 9 8 13 9 7 5 9 5 3 5 1 1 8 6 5 ...
...
T(5,4,1) = 5, T(5,4,2) = 3, and T(5,4,3) = 1 because the lengths of the runs of 2 X 2 squares in the list of partitions of a 5 X 4 rectangle into integer-sided squares are 5, 3 and 1, respectively. The list of partitions is:
. Square side
Run 1 2 3 4
. 1 20 0 0 0
. 16 1 0 0
. 12 2 0 0
. 8 3 0 0
. 4 4 0 0
. 2 11 0 1 0
. 7 1 1 0
. 3 2 1 0
. 3 4 0 0 1
CROSSREFS
Row sums give A224697.
Cf. A226948.
Sequence in context: A322976 A011793 A109649 * A334506 A098199 A309474
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved