A225542 - OEIS

%I #26 Sep 06 2021 05:03:22

%S 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 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 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 The number of entries per row is given by A225568.

%H Christopher Hunt Gribble, <a href="/A225542/b225542.txt">Rows 1..36 for n = 1..8 and k = 1..n flattened</a>

%H Christopher Hunt Gribble, <a href="/A225542/a225542.cpp.txt">C++ program</a>

%F T(n,n,u) = A227009(n,u).

%F Sum_{u=1..(n-1)^2} T(n,n,u) = A034295(n).

%e The irregular triangle begins:

%e n,k\u 0   1   2   3   4   5   6   7   8   9  10  11  12 ...

%e 1,1   1

%e 2,1   1

%e 2,2   1   1

%e 3,1   1

%e 3,2   1   1

%e 3,3   1   1   0   0   1

%e 4,1   1

%e 4,2   1   1   1

%e 4,3   1   1   1   0   1

%e 4,4   1   1   1   1   2   0   0   0   0   1

%e 5,1   1

%e 5,2   1   1   1

%e 5,3   1   1   1   0   1   1

%e 5,4   1   1   1   1   2   1   1   0   0   1

%e 5,5   1   1   1   1   2   1   1   1   0   1   0   0   0 ...

%e ...

%e For n = 5 and k = 4 there are 2 partitions that contain 4 isolated nodes, so T(5,4,4) = 2.

%e 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 1 1 1 1 1    1 1 1 1 1

%e 1 0 1 0 1    1 0 0 1 1

%e 1 1 1 1 1    1 0 0 1 1

%e 1 0 1 0 1    1 1 1 1 1

%e 1 1 1 1 1    1 1 1 1 1

%e 1 1 1 1 1    1 1 1 1 1

%Y Cf. A034295, A224697, A227009, A225777, A225803, A225568.

%K nonn,tabf

%O 1,26

%A _Christopher Hunt Gribble_, Jul 28 2013