A238326 -id:A238326 - cp's OEIS frontend

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

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, 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, 10, 2, 9, 3, 8, 4, 8, 3, 1, 7, 5, 7, 4, 1, 6, 4, 2, 13

Offset: 1

Clark Kimberling and Peter J. C. Moses, Feb 23 2014

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:
...
a .. b .. c
d .. e .. f
g .. h .. i
...
Writing summands in clockwise order, the four directional partitions of p are by
NW(p) = [g + d + a + b + c, h + e + f, i]
NE(p) = [a + b + c + f + i, d + e + h, g]
SE(p) = [c + f + i + h + g, b + e + d, a]
SW(p) = [i + h + g + d + a, f + e + b, c].
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.
...
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.
The number of NW partitions of n is A003114(n) for n >=1. - Clark Kimberling, Mar 20 2014

			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].
...
Example 2.
The first 9 rows of the array of NW partitions:
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 .. 3 .. 1
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".

Clark Kimberling, Table of n, a(n) for n = 1..1000
Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, Hook length and symplectic content in partitions, arXiv:2205.07322 [math.CO], 2022.
Clark Kimberling and Peter J. C. Moses, Ferrers Matrices and Related Partitions of Integers

Cf. A237982, A237983, A237985, A238325, A238326.

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]];
Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]](*NW corner: A237981*)
Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]](*NE corner: A237982*)
Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]](*SE corner: A237983*)
Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]](*SW corner: A237982*)
(* Peter J. C. Moses, Feb 25 2014 *)

A238325 Array: row n gives the number of occurrences of each possible antidiagonal partition of n, arranged in reverse-Mathematica order.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 6, 1, 2, 2, 4, 3, 4, 2, 2, 4, 6, 2, 6, 2, 2, 4, 4, 2, 3, 9, 4, 2, 2, 4, 4, 2, 6, 6, 3, 12, 1, 2, 2, 4, 4, 2, 4, 6, 3, 6, 6, 12, 5, 2, 2, 4, 4, 2, 4, 6, 6, 4, 6, 3, 18, 2, 4, 10, 2, 2, 4, 4, 2, 4, 6, 4, 4, 6, 3, 6, 12, 2, 6

Offset: 1

Clark Kimberling and Peter J. C. Moses, Feb 24 2014

Suppose that p is a partition of n, let F(p) be its Ferrers matrix, as defined at A237981, and let mXm be the size of F(p). The numbers of 1's in each of the 2m-1 antidiagonals of F(p) form a partition of n. Any partition which is associated with a partition of n in this manner is introduced here as an antidiagonal partition of n. A000041(n) = sum of the numbers in row n; A000009(n) = number of terms in row n, since the antidiagonal partitions of n are the conjugates of the strict partitions of n.

			The Mathematica ordering of the 6 antidiagonal partitions of 8 follows:  3221, 32111, 22211, 221111, 2111111, 11111111.  Frequencies of these among the 22 partitions of 8 are given in reverse Mathematica ordering as follows:  11111111 occurs 2 times, 2111111 occurs 2 times, 221111 occurs 4 times, 22211 occurs 6 times, 32111 occurs 2 times, and 3221 occurs 6 times, so that row 8 of the array is 2 2 4 6 2 6.
...
First 12 rows:
  1;
  2;
  2,  1;
  2,  3;
  2,  2,  3;
  2,  2,  6,  1;
  2,  2,  4,  3,  4;
  2,  2,  4,  6,  2,  6;
  2,  2,  4,  4,  2,  3,  9,  4;
  2,  2,  4,  4,  2,  6,  6,  3, 12,  1;
  2,  2,  4,  4,  2,  4,  6,  3,  6,  6, 12,  5;
  2,  2,  4,  4,  2,  4,  6,  6,  4,  6,  3, 18,  2,  4, 10;

Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling and Peter J. C. Moses, Ferrers Matrices and Related Partitions of Integers

Cf. A238326.

z = 20; ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; antiDiagPartNE[list_] := Module[{m = ferrersMatrix[list]}, Map[Diagonal[Reverse[m], #] &, Range[-#, #] &[Length[m] - 1]]]; a1[n_] :=  Last[Transpose[Tally[Map[DeleteCases[Reverse[Sort[Map[Count[#, 1] &, antiDiagPartNE[#]]]], 0] &, IntegerPartitions[n]]]]];
t = Table[a1[n], {n, 1, z}]; TableForm[Table[a1[n], {n, 1, z}]]   (* A238325, array *)
u = Flatten[t] (* A238325, sequence *)
(* Peter J. C. Moses, 18 February 2014 *)

Example corrected by Peter J. Taylor, Apr 10 2022

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

Original entry on oeis.org

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, 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, 1, 1, 7, 6, 1, 5, 2, 5, 1, 1, 4, 3, 4, 2, 1, 4, 1, 1

Offset: 1

Clark Kimberling and Peter J. C. Moses, Feb 23 2014

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.

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

			The first 4 rows of the array of NW partitions:
1
2 .. 1 .. 1
3 .. 2 .. 1 .. 1 .. 1 .. 1
4 .. 3 .. 1 .. 2 .. 1 .. 1 .. 1 .. 1 .. 1 .. 1
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".

Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling and Peter J. C. Moses, Ferrers Matrices and Related Partitions of Integers

Cf. A237981, A237329, A237983, A237985, A238325, A238326.

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]];
Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]](*NW corner: A237981*)
Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]](*NE corner: A237982*)
Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]](*SE corner: A237983*)
Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]](*SW corner: A237982*)
(* Peter J. C. Moses, Feb 25 2014 *)

A237983 Triangular array read by rows: row n gives the SE partitions of n; see Comments.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 3, 2, 1, 1, 3, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling and Peter J. C. Moses, Feb 25 2014

See Comments at A237981 for definitions of the directional partitions, NW, NE, SW, SE. The number of SE partitions of n is A122129(n) for n >=1.

			The first 4 rows of the array of SE partitions:
1
1 .. 1
2 .. 1 .. 1 .. 1 .. 1
3 .. 1 .. 2 .. 1 .. 1 .. 1 .. 1 .. 1 .. 1
Row 4, for example, represents the 4 NE partitions of 4 as follows:  [3,1], [2,1,1], [1,1,1,1], listed in "Mathematica order".

Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling and Peter J. C. Moses, Ferrers Matrices and Related Partitions of Integers

Cf. A237981, A237982, A237985, A238325, A238326.

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]];
Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]](*NW corner: A237981*)
Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]](*NE corner: A237982*)
Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]](*SE corner: A237983*)
Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]](*SW corner: A237982*)
(* Peter J. C. Moses, Feb 25 2014 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A238325 Array: row n gives the number of occurrences of each possible antidiagonal partition of n, arranged in reverse-Mathematica order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A237983 Triangular array read by rows: row n gives the SE partitions of n; see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica