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A239329 The number of NE partitions of n (see Comments).

%I A239329 #10 Jul 30 2025 09:46:02
%S A239329 1,2,3,4,7,9,14,19,27,36,51,67,90,117,157,204,266,337,436,554,708,890,
%T A239329 1123,1401,1750,2172,2701,3329,4106,5026,6161,7507,9147,11095,13455,
%U A239329 16245,19597,23555,28288,33867,40514,48328,57590,68456,81286,96286,113947
%N A239329 The number of NE partitions of n (see Comments).
%C A239329 Directional partitions are defined at A237981, and NE partitions are shown at A237982. a(n) is also the number of SW partitions of n, as at A237982.
%e A239329 See A237982.
%t A239329 z = 9; 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 A239329 Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]];(*NW A237981*)
%t A239329 Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]];(*NE A237982*)
%t A239329 Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]];(*SE A237983*)
%t A239329 Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]];(*SW A237982*)
%t A239329 m1 = Map[cP[[#, 1]] &, Range[Length[cP]]];
%t A239329 Table[Length[m1[[k]]], {k, 1, z}] (* A003114, NW *)
%t A239329 m2 = Map[cP[[#, 2]] &, Range[Length[cP]]];
%t A239329 Table[Length[m2[[k]]], {k, 1, z}] (* A239329, NE *)
%t A239329 m3 = Map[cP[[#, 3]] &, Range[Length[cP]]];
%t A239329 Table[Length[m3[[k]]], {k, 1, z}] (* A122129, SE *)
%t A239329 m4 = Map[cP[[#, 4]] &, Range[Length[cP]]];
%t A239329 Table[Length[m4[[k]]], {k, 1, z}] (* A239329, SW *)
%Y A239329 Cf. A003114, A122129, A237981, A237982.
%K A239329 nonn,easy
%O A239329 1,2
%A A239329 _Clark Kimberling_, Mar 19 2014