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%I #15 Sep 06 2021 08:29:08
%S 1,1,1,1,1,2,1,1,2,3,1,1,2,3,4,1,1,3,4,5,8,1,1,3,5,6,10,12,1,1,3,6,8,
%T 13,17,22
%N Triangle read by rows: T(n,k) (1 <= k <= n) = number of runs 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, where partition sorting order is ascending with larger squares taking higher precedence.
%H Christopher Hunt Gribble, <a href="/A228107/a228107.cpp.txt">C++ program</a>
%e The triangle begins:
%e \ k 1 2 3 4 5 6 7 8 ...
%e n
%e 1 1
%e 2 1 1
%e 3 1 1 2
%e 4 1 1 2 3
%e 5 1 1 2 3 4
%e 6 1 1 3 4 5 8
%e 7 1 1 3 5 6 10 12
%e 8 1 1 3 6 8 13 17 22
%e ...
%e T(5,4) = 3 because there are 3 runs of strictly increasing numbers of 2 X 2 squares in the list of partitions of a 5 X 4 rectangle into integer-sided squares. The list of partitions is:
%e . Square side
%e Run 1 2 3 4
%e . 1 20 0 0 0
%e . 16 1 0 0
%e . 12 2 0 0
%e . 8 3 0 0
%e . 4 4 0 0
%e . 2 11 0 1 0
%e . 7 1 1 0
%e . 3 2 1 0
%e . 3 4 0 0 1
%Y Diagonal is A226947.
%Y Cf. A228106
%K nonn,tabl,more
%O 1,6
%A _Christopher Hunt Gribble_, Aug 10 2013