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Number T(n,k,s) of partitions of an n X k rectangle into s integer-sided squares, considering only the list of parts; irregular triangle T(n,k,s), 1 <= k <= n, s >= 1, read by rows.

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`%I #34 Sep 06 2021 04:25:53
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`%S 1,0,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,1,
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`%T 0,0,1,0,0,0,1,0,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,1,0,0,1,0,0,1,0,0,0,
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`%U 0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0
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`%N Number T(n,k,s) of partitions of an n X k rectangle into s integer-sided squares, considering only the list of parts; irregular triangle T(n,k,s), 1 <= k <= n, s >= 1, read by rows.
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`%C The number of entries per row is n*k.
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`%H Christopher Hunt Gribble, <a href="/A227998/b227998.txt">Rows 1..36 for n=1..8 and k=1..n flattened</a>
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`%H Christopher Hunt Gribble, <a href="/A227998/a227998.cpp.txt">C++ program</a>
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`%F T(n,n,s) = A226912(n,s).
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`%F Sum_{s=1..n*k} T(n,k,s) = A224697(n,k), 1 <= k <= n.
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`%e T(6,4,6) = 2 because there are 2 partitions of a 6 X 4 rectangle into integer-sided squares with exactly 6 parts:
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`%e (6 2 X 2 squares) and
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`%e (4 1 X 1 squares, 1 2 X 2 square, 1 4 X 4 square).
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`%e The irregular triangle starts:
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`%e n,k Number of Square Parts s
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`%e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
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`%e 1,1 1
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`%e 2,1 0 1
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`%e 2,2 1 0 0 1
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`%e 3,1 0 0 1
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`%e 3,2 0 0 1 0 0 1
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`%e 3,3 1 0 0 0 0 1 0 0 1
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`%e 4,1 0 0 0 1
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`%e 4,2 0 1 0 0 1 0 0 1
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`%e 4,3 0 0 0 1 0 1 0 0 1 0 0 1
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`%e 4,4 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 1
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`%e 5,1 0 0 0 0 1
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`%e 5,2 0 0 0 1 0 0 1 0 0 1
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`%e 5,3 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1
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`%e 5,4 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 ...
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`%e 5,5 1 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 0 ...
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`%e 6,1 0 0 0 0 0 1
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`%e 6,2 0 0 1 0 0 1 0 0 1 0 0 1
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`%e 6,3 0 1 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 1
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`%e 6,4 0 0 1 0 0 2 0 1 2 1 0 1 1 0 1 1 0 1 ...
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`%e 6,5 0 0 0 1 1 0 1 1 1 1 2 1 1 2 1 0 1 1 ...
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`%e 6,6 1 0 0 1 0 1 0 0 3 0 1 4 1 1 2 1 1 2 ...
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`%Y Cf. A034295, A226912.
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`%K nonn,tabf
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`%O 1
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`%A _Christopher Hunt Gribble_, Aug 06 2013
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