|
%I #19 Dec 13 2015 01:12:59
%S 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 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 1,1,7,6,1,5,2,5,1,1,4,3,4,2,1,4,1,1
%N Triangular array read by rows: row n gives the NE partitions of n (see Comments).
%C 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 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 Clark Kimberling, <a href="/A237982/b237982.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 NW partitions:
%e 1
%e 2 .. 1 .. 1
%e 3 .. 2 .. 1 .. 1 .. 1 .. 1
%e 4 .. 3 .. 1 .. 2 .. 1 .. 1 .. 1 .. 1 .. 1 .. 1
%e 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 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, A237329, A237983, A237985, A238325, A238326.
%K nonn,tabf,easy
%O 1,2
%A _Clark Kimberling_ and _Peter J. C. Moses_, Feb 23 2014
|
