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

1,2

Comments

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

Links

Table of n, a(n) for n=1..25.

Example

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.

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]

Crossrefs

Cf. A000041, A366156, A366429.

Sequence in context: A326459 A326545 A326684 * A350457 A010238 A178799

Adjacent sequences: A366595 A366596 A366597 * A366599 A366600 A366601

Keyword

nonn,more

Author

Clark Kimberling, Oct 25 2023

Status

approved