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A237982 Triangular array read by rows: row n gives the NE partitions of n (see Comments).

%I A237982 #19 Dec 13 2015 01:12:59
%S A237982 1,2,1,1,3,2,1,1,1,1,4,3,1,2,1,1,1,1,1,1,5,4,1,3,2,3,1,1,2,2,1,2,1,1,
%T A237982 1,1,1,1,1,1,6,5,1,4,2,4,1,1,3,2,1,3,1,1,1,2,2,1,1,2,1,1,1,1,1,1,1,1,
%U A237982 1,1,7,6,1,5,2,5,1,1,4,3,4,2,1,4,1,1
%N A237982 Triangular array read by rows: row n gives the NE partitions of n (see Comments).
%C A237982 See Comments at A237981 for definitions of the directional partitions, NW, NE, SW, SE. The number of NE partitions of n, and also the number of SW partitions of n, is A237329(n), for n >=1.
%C A237982 The order is: each partition has nonincreasing parts and the partitions are ordered anti-lexicographic (called "Mathematica order" in the example). - _Wolfdieter Lang_, Mar 21 2014
%H A237982 Clark Kimberling, <a href="/A237982/b237982.txt">Table of n, a(n) for n = 1..1000</a>
%H A237982 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 A237982 The first 4 rows of the array of NW partitions:
%e A237982 1
%e A237982 2 .. 1 .. 1
%e A237982 3 .. 2 .. 1 .. 1 .. 1 .. 1
%e A237982 4 .. 3 .. 1 .. 2 .. 1 .. 1 .. 1 .. 1 .. 1 .. 1
%e A237982 Row 4, for example, represents the 4 NE partitions of 4 as follows:  [4], [3,1], [2,1,1], [1,1,1,1], listed in "Mathematica order".
%t A237982 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 A237982 Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]](*NW corner: A237981*)
%t A237982 Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]](*NE corner: A237982*)
%t A237982 Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]](*SE corner: A237983*)
%t A237982 Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]](*SW corner: A237982*)
%t A237982 (* _Peter J. C. Moses_, Feb 25 2014 *)
%Y A237982 Cf. A237981, A237329, A237983, A237985, A238325, A238326.
%K A237982 nonn,tabf,easy
%O A237982 1,2
%A A237982 _Clark Kimberling_ and _Peter J. C. Moses_, Feb 23 2014