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A225542 Number T(n,k,u) of partitions of an n X k rectangle into integer-sided square parts containing u nodes that are unconnected to any of their neighbors, considering only the number of parts; irregular triangle T(n,k,u), 1 <= k <= n, u >= 0, read by rows.

%I A225542 #26 Sep 06 2021 05:03:22
%S A225542 1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,2,0,0,0,0,1,1,1,1,
%T A225542 1,1,1,1,0,1,1,1,1,1,1,2,1,1,0,0,1,1,1,1,1,2,1,1,1,0,1,0,0,0,0,0,0,1
%N A225542 Number T(n,k,u) of partitions of an n X k rectangle into integer-sided square parts containing u nodes that are unconnected to any of their neighbors, considering only the number of parts; irregular triangle T(n,k,u), 1 <= k <= n, u >= 0, read by rows.
%C A225542 The number of entries per row is given by A225568.
%H A225542 Christopher Hunt Gribble, <a href="/A225542/b225542.txt">Rows 1..36 for n = 1..8 and k = 1..n flattened</a>
%H A225542 Christopher Hunt Gribble, <a href="/A225542/a225542.cpp.txt">C++ program</a>
%F A225542 T(n,n,u) = A227009(n,u).
%F A225542 Sum_{u=1..(n-1)^2} T(n,n,u) = A034295(n).
%e A225542 The irregular triangle begins:
%e A225542 n,k\u 0   1   2   3   4   5   6   7   8   9  10  11  12 ...
%e A225542 1,1   1
%e A225542 2,1   1
%e A225542 2,2   1   1
%e A225542 3,1   1
%e A225542 3,2   1   1
%e A225542 3,3   1   1   0   0   1
%e A225542 4,1   1
%e A225542 4,2   1   1   1
%e A225542 4,3   1   1   1   0   1
%e A225542 4,4   1   1   1   1   2   0   0   0   0   1
%e A225542 5,1   1
%e A225542 5,2   1   1   1
%e A225542 5,3   1   1   1   0   1   1
%e A225542 5,4   1   1   1   1   2   1   1   0   0   1
%e A225542 5,5   1   1   1   1   2   1   1   1   0   1   0   0   0 ...
%e A225542 ...
%e A225542 For n = 5 and k = 4 there are 2 partitions that contain 4 isolated nodes, so T(5,4,4) = 2.
%e A225542 Consider that each partition is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors.  Then the 2 partitions are:
%e A225542 1 1 1 1 1    1 1 1 1 1
%e A225542 1 0 1 0 1    1 0 0 1 1
%e A225542 1 1 1 1 1    1 0 0 1 1
%e A225542 1 0 1 0 1    1 1 1 1 1
%e A225542 1 1 1 1 1    1 1 1 1 1
%e A225542 1 1 1 1 1    1 1 1 1 1
%Y A225542 Cf. A034295, A224697, A227009, A225777, A225803, A225568.
%K A225542 nonn,tabf
%O A225542 1,26
%A A225542 _Christopher Hunt Gribble_, Jul 28 2013