A366429 -id:A366429 - cp's OEIS frontend

A366597 Irregular triangular array, read by rows: T(n,k) = number of vertices (partitions) that have degree k in the distance graph of the partitions of n, for k = 1..A366429(n).

0, 2, 2, 1, 2, 1, 2, 2, 0, 4, 1, 2, 2, 2, 4, 0, 1, 2, 0, 4, 7, 0, 0, 2, 2, 2, 2, 9, 2, 0, 4, 1, 2, 1, 4, 11, 2, 0, 4, 6, 2, 2, 2, 14, 6, 0, 4, 9, 2, 0, 0, 1, 2, 0, 4, 17, 6, 0, 2, 19, 4, 0, 0, 0, 2, 2, 4, 2, 16, 10, 1, 6, 17, 14, 0, 0, 0, 4, 1, 2, 0, 4, 23

Offset: 1

Clark Kimberling, Oct 16 2023

The distance graph of the partitions of n is defined in A366156.

			First fourteen rows:
   1
   2
   2   1
   2   1   2
   2   0   4   1
   2   2   2   4    0    1
   2   0   4   7    0    0   2
   2   2   2   9    2    0   4   1
   2   1   4   11   2    0   4   6
   2   2   2   14   6    0   4   9   2   0   0   1
   2   0   4   17   6    0   2  19   4   0   0   0   2
   2   4   2   16   10   1   6  17   14  0   0   0   4   1
   2   0   4   23   10   0   2  27   22  1   0   0   4   6
   2   2   2   22   18   2   4  27   32  4   0   0   6   12   2
Enumerate the 7 partitions (vertices) of 5 as follows:
  1: 5
  2: 4,1
  3: 3,2
  4: 3,1,1
  5: 2,2,1
  6: 2,1,1,1
  7: 1,1,1,1,1
Call q a neighbor of p if d(p,q)=2, where d is the distance function in A366156.
The set of neighbors for vertex k, for k = 1..7, is given by
  vertex 1: {2}
  vertex 2: {1,3,4}
  vertex 3: {2,4,5}
  vertex 4: {2,3,5,6}
  vertex 5: {3,4,6}
  vertex 6: {4,5,7}
  vertex 7: {6}
The number of vertices having degrees 1,2,3,4 are 2,0,4,1, respectively, so that row 5 is 2 0 4 1.

Cf. A000041 (row sums), A366429 (row lengths), A366598 (row maxima).

c[n_] := PartitionsP[n]; q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &];
s1[n_] := s1[n] = Table[s[n, k], {k, 1, c[n]}];
m[n_] := m[n] = Map[Length, s1[n]];
m1[n_] := m1[n] = Max[m[n]];  (* A366429 *)
t1 = Join[{1}, Table[Count[m[n], i], {n, 2, 15}, {i, 1, m1[n]}]]
Column[t1]
Flatten[t1]

A366461 a(n) = number of partitions of n that have the maximum number of neighbors; see Comments.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 6, 1, 2, 1, 6, 2, 1, 2, 1, 6, 2, 8, 1, 2, 1, 6, 2, 8, 1, 1, 2, 1, 6, 2, 8, 1, 6, 1, 2, 1, 6, 2, 8, 1, 6, 22, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 8, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 8, 30, 1, 2, 1, 6, 2, 8, 1, 6, 22

Offset: 1

Clark Kimberling, Oct 12 2023

nonn

Partitions p and q of n are neighbors if d(p,q) = 2, where d is the distance function in A366156.

			Refer to the Example in A366429 to see that a(5) = 1.

Cf. A000041, A366156, A366429.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := d[u, v] = Total[Abs[u - v]];
s[n_, k_] := s[n, k] = Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
t[n_] := t[n] = Table[s[n, k], {k, 1, c[n]}]
a[n_] := Max[Map[Length, t[n]]]
b[n_] := b[n] = Select[t[n], Length[#] == a[n] &]
e[n_] := Length[b[n]]
Table[e[n], {n, 1, 24}]

More terms from Pontus von Brömssen, Oct 24 2023

A366598 a(n) = greatest number of vertices having the same degree in the distance graph of the partitions of n.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 7, 9, 11, 14, 19, 17, 27, 32, 50, 62, 82, 94, 132, 138, 176, 198, 238, 288, 368

Offset: 1

Clark Kimberling, Oct 25 2023

The distance graph of the partitions of n is defined in A366156.

			Enumerate the 7 partitions (= vertices) of 5 as follows:
  1: 5
  2: 4,1
  3: 3,2
  4: 3,1,1
  5: 2,2,1
  6: 2,1,1,1
  7: 1,1,1,1,1
Call q a neighbor of p if d(p,q)=2.
The set of neighbors for vertex k, for k = 1..7, is given by
  vertex 1: {2}
  vertex 2: {1,3,4}
  vertex 3: {2,4,5}
  vertex 4: {2,3,5,6}
  vertex 5: {3,4,6}
  vertex 6: {4,5,7}
  vertex 7: {6}
The number of vertices having degrees 1,2,3,4 are 2,0,4,1, respectively; the greatest of these is 4, so that a(5) = 4.

Cf. A000041, A366156, A366429.

c[n_] := PartitionsP[n]; q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
s1[n_] := s1[n] = Table[s[n, k], {k, 1, c[n]}]
m[n_] := m[n] = Map[Length, s1[n]]
m1[n_] := m1[n] = Max[m[n]];  (* A366429 *)
t1 = Join[{1}, Table[Count[m[n], i], {n, 2, 25}, {i, 1, m1[n]}]]
Map[Max, t1]

cp's OEIS Frontend

A366597 Irregular triangular array, read by rows: T(n,k) = number of vertices (partitions) that have degree k in the distance graph of the partitions of n, for k = 1..A366429(n).

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A366461 a(n) = number of partitions of n that have the maximum number of neighbors; see Comments.

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A366598 a(n) = greatest number of vertices having the same degree in the distance graph of the partitions of n.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica