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%I #20 Sep 06 2021 08:28:56
%S 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,
%T 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,
%U 1,6,4,3,4,1,10,9,8,5,1,6,5,4,1,6,1,1
%N 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.
%C The sorting order for the list of partitions is ascending with larger squares taking higher precedence.
%C A228107 specifies the length of each row for 1 <= k <= n <= 8.
%H Christopher Hunt Gribble, <a href="/A228106/b228106.txt">Rows 1..36 flattened</a>
%H Christopher Hunt Gribble, <a href="/A228106/a228106.cpp.txt">C++ program</a>
%e The irregular triangle begins:
%e . r 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
%e n,k
%e 1,1 1
%e 2,1 1
%e 2,2 2
%e 3,1 1
%e 3,2 2
%e 3,3 2 1
%e 4,1 1
%e 4,2 3
%e 4,3 3 1
%e 4,4 5 1 1
%e 5,1 1
%e 5,2 3
%e 5,3 3 2
%e 5,4 5 3 1
%e 5,5 5 4 1 1
%e 6,1 1
%e 6,2 4
%e 6,3 4 2 1
%e 6,4 7 3 1 3
%e 6,5 7 5 4 3 1
%e 6,6 10 6 4 2 1 6 1 1
%e 7,1 1
%e 7,2 4
%e 7,3 4 3 1
%e 7,4 7 5 1 3 1
%e 7,5 7 6 5 3 2 3
%e 7,6 10 8 7 3 1 6 4 3 4 1
%e 7,7 10 9 8 5 1 6 5 4 1 6 1 1
%e 8,1 1
%e 8,2 5
%e 8,3 5 3 2
%e 8,4 9 5 3 5 1 1
%e 8,5 9 7 6 5 3 1 3 2
%e 8,6 13 9 7 5 4 9 5 3 5 4 2 1 4
%e 8,7 13 11 10 7 5 9 7 6 3 5 3 2 7 6 3 4 1
%e 8,8 17 13 11 9 8 13 9 7 5 9 5 3 5 1 1 8 6 5 ...
%e ...
%e 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:
%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 Row sums give A224697.
%Y Cf. A226948.
%K nonn,tabf
%O 1,3
%A _Christopher Hunt Gribble_, Aug 10 2013
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