A229961 T(n,k) is the number of partitions in each run k of strictly increasing numbers of 2 X 2 squares in the list of partitions of n^2 into squares, where partition sorting order is ascending with larger squares taking higher precedence; irregular triangle T(n,k), 1 <= n, 1 <= k <= A227940(n), read by rows.

1, 2, 3, 1, 5, 2, 1, 7, 5, 2, 3, 1, 1, 10, 7, 5, 3, 1, 6, 3, 1, 2, 3, 1, 1, 13, 11, 8, 6, 4, 2, 9, 7, 4, 2, 5, 3, 1, 7, 4, 2, 3, 4, 2, 1, 17, 14, 12, 10, 8, 5, 3, 1, 13, 10, 8, 6, 4, 1, 9, 6, 4, 2, 5, 2, 1, 10, 8, 6, 4, 1, 6, 4, 2, 2, 4, 2, 8, 5, 3, 1, 4, 1, 1, 4, 2, 1

Offset: 1

Author

Christopher Hunt Gribble, Oct 04 2013

Keywords

• nonn
• tabf

Comments

Row lengths are given by A227940.

Examples

			For n = 4, the 8 partitions of 16 into square parts are:
Partition  Square side
.           1  2  3  4
.
.    1     16  0  0  0
.    2     12  1  0  0
.    3      8  2  0  0
.    4      4  3  0  0
.    5      0  4  0  0
.    6      7  0  1  0
.    7      3  1  1  0
.    8      0  0  0  1
So T(4,1) = 5 as the first runs of 2 X 2 squares is (0,1,2,3,4) from partitions 1 to 5;
T(4,2) = 2 as the second run is (0,1) from partitions 6 to 7;
T(4,3) = 1 as the third run is (0) from partition 8.
The irregular triangle begins:
  \  k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  n
  1     1
  2     2
  3     3  1
  4     5  2  1
  5     7  5  2  3  1  1
  6    10  7  5  3  1  6  3  1  2  3  1  1
  7    13 11  8  6  4  2  9  7  4  2  5  3  1  7  4  2  3  4 ...
  8    17 14 12 10  8  5  3  1 13 10  8  6  4  1  9  6  4  2 ...
  9    21 19 16 14 12 10  7  5  3  1 17 15 12 10  8  6  3  1 ...
  10   26 23 21 19 17 14 12 10  8  5  3  1 22 19 17 15 13 10 ...
		

Links

• Christopher Hunt Gribble, C++ program

Crossrefs

Row sums = A037444.

Cf. A227940.