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

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

Author

Clark Kimberling, Oct 25 2023

Keywords

• nonn
• more

Comments

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

Examples

			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.
		

Crossrefs

Cf. A000041, A366156, A366429.

Programs

Mathematica

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]