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A237981 Array: row n gives the NW partitions of n; see Comments.

%I A237981 #26 May 17 2022 07:26:25
%S A237981 1,2,3,4,3,1,5,4,1,6,5,1,4,2,7,6,1,5,2,8,7,1,6,2,5,3,9,8,1,7,2,6,3,5,
%T A237981 3,1,10,9,1,8,2,7,3,6,4,6,3,1,11,10,1,9,2,8,3,7,4,7,3,1,6,4,1,12,11,1,
%U A237981 10,2,9,3,8,4,8,3,1,7,5,7,4,1,6,4,2,13
%N A237981 Array: row n gives the NW partitions of n; see Comments.
%C A237981 Suppose that p is a partition of n, and let m = max{greatest part of p, number of parts of p}.  Write the Ferrers graph of p with 1's as nodes, and pad the graph with 0's to form an m X m square matrix, which is introduced here as the Ferrers matrix of p, denoted by f(p).  Four kinds of partitions are defined from f(p); they will be described by referring to the example of a 3 X 3 matrix, as follows:
%C A237981 ...
%C A237981 a .. b .. c
%C A237981 d .. e .. f
%C A237981 g .. h .. i
%C A237981 ...
%C A237981 Writing summands in clockwise order, the four directional partitions of p are by
%C A237981 NW(p) = [g + d + a + b + c, h + e + f, i]
%C A237981 NE(p) = [a + b + c + f + i, d + e + h, g]
%C A237981 SE(p) = [c + f + i + h + g, b + e + d, a]
%C A237981 SW(p) = [i + h + g + d + a, f + e + b, c].
%C A237981 The order in which the parts appear does not change the partition, but it is common to list them in nondecreasing order, as in Example 1.
%C A237981 ...
%C A237981 Note that "Ferrers matrix" can be defined without reference to Ferrers graphs, as follows: an m X m matrix (x(i,j)) of 0's and 1's satisfying three properties: (1) x(1,m) = 1 or x(m,1) = 1; (2) x(i,j+1) >= x(i,j) for j=1..m-1 and i = 1..m; and (3) x(i+1,j) >= x(i,j) for i=1..m-1 and j=1..m.  The number of Ferrers matrices of order m is given by A051924.
%C A237981 The number of NW partitions of n is A003114(n) for n >=1. - _Clark Kimberling_, Mar 20 2014
%H A237981 Clark Kimberling, <a href="/A237981/b237981.txt">Table of n, a(n) for n = 1..1000</a>
%H A237981 Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, <a href="https://arxiv.org/abs/2205.07322">Hook length and symplectic content in partitions</a>, arXiv:2205.07322 [math.CO], 2022.
%H A237981 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 A237981 Example 1.  Let p = {6,3,3,3,1), a partition of 16.  Then NW(p) = [10, 4, 2], NE(p) = [6,3,3,3,1], SE(p) = [5, 4, 3, 2, 1, 1], SW(p) = [5,4,4,1,1,1].
%e A237981 ...
%e A237981 Example 2.
%e A237981 The first 9 rows of the array of NW partitions:
%e A237981 1
%e A237981 2
%e A237981 3
%e A237981 4 .. 3 .. 1
%e A237981 5 .. 4 .. 1
%e A237981 6 .. 5 .. 1 .. 4 .. 2
%e A237981 7 .. 6 .. 1 .. 5 .. 2
%e A237981 8 .. 7 .. 1 .. 6 .. 2 .. 5 .. 3
%e A237981 9 .. 8 .. 1 .. 7 .. 2 .. 6 .. 3 .. 5 .. 3 .. 1
%e A237981 Row 9, for example, represents the 5 NW partitions of 9 as follows:  [9], [8,1], [7,2], [6,3], [5,3,1], listed in "Mathematica order".
%t A237981 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} =
%t A237981 {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_] :=
%t A237981 Map[#[[Reverse[Ordering[PadRight[#]]]]] &, Map[DeleteDuplicates[#] &,    Transpose[Map[cornerPart, IntegerPartitions[n]]]]]; cP = Map[cornerParts, Range[z]];
%t A237981 Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]](*NW corner: A237981*)
%t A237981 Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]](*NE corner: A237982*)
%t A237981 Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]](*SE corner: A237983*)
%t A237981 Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]](*SW corner: A237982*)
%t A237981 (* _Peter J. C. Moses_, Feb 25 2014 *)
%Y A237981 Cf. A237982, A237983, A237985, A238325, A238326.
%K A237981 nonn,tabf,easy
%O A237981 1,2
%A A237981 _Clark Kimberling_ and _Peter J. C. Moses_, Feb 23 2014