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

1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 51, 67, 90, 117, 157, 204, 266, 337, 436, 554, 708, 890, 1123, 1401, 1750, 2172, 2701, 3329, 4106, 5026, 6161, 7507, 9147, 11095, 13455, 16245, 19597, 23555, 28288, 33867, 40514, 48328, 57590, 68456, 81286, 96286, 113947

Offset: 1

Author

Clark Kimberling, Mar 19 2014

Keywords

• nonn
• easy

Comments

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.

Examples

			See A237982.
		

Crossrefs

Cf. A003114, A122129, A237981, A237982.

Programs

Mathematica

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]];
Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]];(*NW A237981*)
Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]];(*NE A237982*)
Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]];(*SE A237983*)
Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]];(*SW A237982*)
m1 = Map[cP[[#, 1]] &, Range[Length[cP]]];
Table[Length[m1[[k]]], {k, 1, z}] (* A003114, NW *)
m2 = Map[cP[[#, 2]] &, Range[Length[cP]]];
Table[Length[m2[[k]]], {k, 1, z}] (* A239329, NE *)
m3 = Map[cP[[#, 3]] &, Range[Length[cP]]];
Table[Length[m3[[k]]], {k, 1, z}] (* A122129, SE *)
m4 = Map[cP[[#, 4]] &, Range[Length[cP]]];
Table[Length[m4[[k]]], {k, 1, z}] (* A239329, SW *)