A229961 - cp's OEIS frontend

A229961 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.

%I A229961 #15 Nov 01 2021 16:54:13
%S A229961 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 A229961 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 A229961 2,1,10,8,6,4,1,6,4,2,2,4,2,8,5,3,1,4,1,1,4,2,1
%N A229961 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 A229961 Row lengths are given by A227940.
%H A229961 Christopher Hunt Gribble, <a href="/A229961/a229961.cpp.txt">C++ program</a>
%e A229961 For n = 4, the 8 partitions of 16 into square parts are:
%e A229961 Partition  Square side
%e A229961 .           1  2  3  4
%e A229961 .
%e A229961 .    1     16  0  0  0
%e A229961 .    2     12  1  0  0
%e A229961 .    3      8  2  0  0
%e A229961 .    4      4  3  0  0
%e A229961 .    5      0  4  0  0
%e A229961 .    6      7  0  1  0
%e A229961 .    7      3  1  1  0
%e A229961 .    8      0  0  0  1
%e A229961 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 A229961 T(4,2) = 2 as the second run is (0,1) from partitions 6 to 7;
%e A229961 T(4,3) = 1 as the third run is (0) from partition 8.
%e A229961 The irregular triangle begins:
%e A229961   \  k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
%e A229961   n
%e A229961   1     1
%e A229961   2     2
%e A229961   3     3  1
%e A229961   4     5  2  1
%e A229961   5     7  5  2  3  1  1
%e A229961   6    10  7  5  3  1  6  3  1  2  3  1  1
%e A229961   7    13 11  8  6  4  2  9  7  4  2  5  3  1  7  4  2  3  4 ...
%e A229961   8    17 14 12 10  8  5  3  1 13 10  8  6  4  1  9  6  4  2 ...
%e A229961   9    21 19 16 14 12 10  7  5  3  1 17 15 12 10  8  6  3  1 ...
%e A229961   10   26 23 21 19 17 14 12 10  8  5  3  1 22 19 17 15 13 10 ...
%Y A229961 Row sums = A037444.
%Y A229961 Cf. A227940.
%K A229961 nonn,tabf
%O A229961 1,2
%A A229961 _Christopher Hunt Gribble_, Oct 04 2013

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