A227796 - OEIS

%I #36 Sep 06 2021 08:27:32

%S 1,1,1,2,1,1,2,1,2,2,1,1,1,3,1,3,3,1,1,5,5,1,9,1,1,1,1,3,1,3,3,2,1,5,

%T 5,3,9,5,1,1,5,5,4,9,7,1

%N T(n,k,r,s) is the number of partitions in the s-th run of strictly increasing numbers of 2 X 2 X 2 cubes in the list of partitions of an n X k X r rectangular cuboid into integer-sided cubes, considering only the list of parts; irregular triangle T(n,k,r,s), n >= k >= r >= 1, s >= 1. The sorting order for the list of partitions is ascending with larger squares taking higher precedence.

%H Christopher Hunt Gribble, <a href="/A227796/a227796.cpp.txt">C++ program</a>

%e The irregular triangle begins:

%e       s   1   2   3

%e n k r

%e 1,1,1     1

%e 2,1,1     1

%e 2,2,1     1

%e 2,2,2     2

%e 3,1,1     1

%e 3,2,1     1

%e 3,2,2     2

%e 3,3,1     1

%e 3,3,2     2

%e 3,3,3     2   1

%e 4,1,1     1

%e 4,2,1     1

%e 4,2,2     3

%e 4,3,1     1

%e 4,3,2     3

%e 4,3,3     3   1

%e 4,4,1     1

%e 4,4,2     5

%e 4,4,3     5   1

%e 4,4,4     9   1   1

%e 5,1,1     1

%e 5,2,1     1

%e 5,2,2     3

%e 5,3,1     1

%e 5,3,2     3

%e 5,3,3     3   2

%e 5,4,1     1

%e 5,4,2     5

%e 5,4,3     5   3

%e 5,4,4     9   5   1

%e 5,5,1     1

%e 5,5,2     5

%e 5,5,3     5   4

%e 5,5,4     9   7   1

%e ...

%e T(3,3,2,1) = 2 because there are 2 partitions in the 1st run of strictly increasing numbers of 2 X 2 X 2 cubes in the list of partitions of a 3 X 3 X 2 rectangular cuboid into integer-sided cubes. The 2 partitions are (18 1 X 1 X 1 cubes and 0 2 X 2 X 2 cubes) and (10 1 X 1 X 1 cubes and 1 2 X 2 X 2 cube).

%Y Row sums = A228202(n,k,r).

%Y Cf. A228106.

%K nonn,tabf

%O 1,4

%A _Christopher Hunt Gribble_, Sep 03 2013