%I #15 Nov 01 2021 16:54:13

%S 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,

%T 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,

%U 2,1,10,8,6,4,1,6,4,2,2,4,2,8,5,3,1,4,1,1,4,2,1

%N 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.

%C Row lengths are given by A227940.

%H Christopher Hunt Gribble, <a href="/A229961/a229961.cpp.txt">C++ program</a>

%e For n = 4, the 8 partitions of 16 into square parts are:

%e Partition Square side

%e . 1 2 3 4

%e .

%e . 1 16 0 0 0

%e . 2 12 1 0 0

%e . 3 8 2 0 0

%e . 4 4 3 0 0

%e . 5 0 4 0 0

%e . 6 7 0 1 0

%e . 7 3 1 1 0

%e . 8 0 0 0 1

%e 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;

%e T(4,2) = 2 as the second run is (0,1) from partitions 6 to 7;

%e T(4,3) = 1 as the third run is (0) from partition 8.

%e The irregular triangle begins:

%e \ k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...

%e n

%e 1 1

%e 2 2

%e 3 3 1

%e 4 5 2 1

%e 5 7 5 2 3 1 1

%e 6 10 7 5 3 1 6 3 1 2 3 1 1

%e 7 13 11 8 6 4 2 9 7 4 2 5 3 1 7 4 2 3 4 ...

%e 8 17 14 12 10 8 5 3 1 13 10 8 6 4 1 9 6 4 2 ...

%e 9 21 19 16 14 12 10 7 5 3 1 17 15 12 10 8 6 3 1 ...

%e 10 26 23 21 19 17 14 12 10 8 5 3 1 22 19 17 15 13 10 ...

%Y Row sums = A037444.

%Y Cf. A227940.

%K nonn,tabf

%O 1,2

%A _Christopher Hunt Gribble_, Oct 04 2013