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%I #13 Mar 21 2014 13:55:36
%S 1,1,1,2,1,1,1,1,3,1,2,1,1,1,1,1,1,3,1,1,2,2,1,2,1,1,1,1,1,1,1,1,3,2,
%T 1,3,1,1,1,2,2,1,1,2,1,1,1,1,1,1,1,1,1,1,3,3,1,3,2,1,1,3,1,1,1,1,2,2,
%U 2,1,2,2,1,1,1,2,1,1,1,1,1,1,1,1,1,1
%N Triangular array read by rows: row n gives the SE partitions of n; see Comments.
%C See Comments at A237981 for definitions of the directional partitions, NW, NE, SW, SE. The number of SE partitions of n is A122129(n) for n >=1.
%H Clark Kimberling, <a href="/A237983/b237983.txt">Table of n, a(n) for n = 1..1000</a>
%H Clark Kimberling and Peter J. C. Moses, <a href="http://faculty.evansville.edu/ck6/GalleryThree/Introduction3.html">Ferrers Matrices and Related Partitions of Integers</a>
%e The first 4 rows of the array of SE partitions:
%e 1
%e 1 .. 1
%e 2 .. 1 .. 1 .. 1 .. 1
%e 3 .. 1 .. 2 .. 1 .. 1 .. 1 .. 1 .. 1 .. 1
%e Row 4, for example, represents the 4 NE partitions of 4 as follows: [3,1], [2,1,1], [1,1,1,1], listed in "Mathematica order".
%t z = 10; ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; cornerPart[list_] := Module[{f = ferrersMatrix[list], u, l, ur, lr, nw, ne, se, sw}, {u, l} = {UpperTriangularize[#, 1], LowerTriangularize[#]} &[f]; {ur, lr} = {UpperTriangularize[#, 1], LowerTriangularize[#]} &[Reverse[f]]; {nw, ne, se, sw} = {Total[Transpose[u]] + Total[l], Total[ur] + Total[Transpose[lr]], Total[u] + Total[Transpose[l]], Total[Transpose[ur]] + Total[lr]}; Map[DeleteCases[Reverse[Sort[#]], 0] &, {nw, ne, se, sw}]]; cornerParts[n_] := Map[#[[Reverse[Ordering[PadRight[#]]]]] &, Map[DeleteDuplicates[#] &, Transpose[Map[cornerPart, IntegerPartitions[n]]]]]; cP = Map[cornerParts, Range[z]];
%t Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]](*NW corner: A237981*)
%t Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]](*NE corner: A237982*)
%t Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]](*SE corner: A237983*)
%t Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]](*SW corner: A237982*)
%t (* _Peter J. C. Moses_, Feb 25 2014 *)
%Y Cf. A237981, A237982, A237985, A238325, A238326.
%K nonn,tabf,easy
%O 1,4
%A _Clark Kimberling_ and _Peter J. C. Moses_, Feb 25 2014
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