A229961 - OEIS

%I #15 Nov 01 2021 16:54:13

%S 1,2,3,1,5,2,1,7,5,2,3,1,1,10,7,5,3,1,6,3,1,2,3,1,1,13,11,8,6,4,2,9,7,

%T 4,2,5,3,1,7,4,2,3,4,2,1,17,14,12,10,8,5,3,1,13,10,8,6,4,1,9,6,4,2,5,

%U 2,1,10,8,6,4,1,6,4,2,2,4,2,8,5,3,1,4,1,1,4,2,1

%N T(n,k) is the number of partitions in each run k of strictly increasing numbers of 2 X 2 squares in the list of partitions of n^2 into squares, where partition sorting order is ascending with larger squares taking higher precedence; irregular triangle T(n,k), 1 <= n, 1 <= k <= A227940(n), read by rows.

%C Row lengths are given by A227940.

%H Christopher Hunt Gribble, <a href="/A229961/a229961.cpp.txt">C++ program</a>

%e For n = 4, the 8 partitions of 16 into square parts are:

%e Partition  Square side

%e .           1  2  3  4

%e .

%e .    1     16  0  0  0

%e .    2     12  1  0  0

%e .    3      8  2  0  0

%e .    4      4  3  0  0

%e .    5      0  4  0  0

%e .    6      7  0  1  0

%e .    7      3  1  1  0

%e .    8      0  0  0  1

%e So T(4,1) = 5 as the first runs of 2 X 2 squares is (0,1,2,3,4) from partitions 1 to 5;

%e T(4,2) = 2 as the second run is (0,1) from partitions 6 to 7;

%e T(4,3) = 1 as the third run is (0) from partition 8.

%e The irregular triangle begins:

%e   \  k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...

%e   n

%e   1     1

%e   2     2

%e   3     3  1

%e   4     5  2  1

%e   5     7  5  2  3  1  1

%e   6    10  7  5  3  1  6  3  1  2  3  1  1

%e   7    13 11  8  6  4  2  9  7  4  2  5  3  1  7  4  2  3  4 ...

%e   8    17 14 12 10  8  5  3  1 13 10  8  6  4  1  9  6  4  2 ...

%e   9    21 19 16 14 12 10  7  5  3  1 17 15 12 10  8  6  3  1 ...

%e   10   26 23 21 19 17 14 12 10  8  5  3  1 22 19 17 15 13 10 ...

%Y Row sums = A037444.

%Y Cf. A227940.

%K nonn,tabf

%O 1,2

%A _Christopher Hunt Gribble_, Oct 04 2013