A366429 a(n) = maximum degree of vertices in the distance graph of the partitions of n.

0, 1, 2, 3, 4, 6, 7, 8, 8, 12, 13, 14, 14, 15, 20, 21, 22, 22, 23, 23, 30, 31, 32, 32, 33, 33, 34, 42, 43, 44, 44, 45, 45, 46, 46, 56, 57, 58, 58, 59, 59, 60, 60, 60, 72, 73, 74, 74, 75, 75, 76, 76, 76, 77, 90, 91, 92, 92, 93, 93, 94, 94, 94, 95, 95, 110, 111

Offset

1,3

Comments

The distance graph of the partitions of n is defined by its edges, specifically, two partitions (i.e. vertices) p and q share an edge if d(p,q) = 2, where d is defined in A366156.

Links

Table of n, a(n) for n=1..67.

Formula

a(n) = 2*binomial(A002024(n+1)-1,2) + A000267(A002262(n)) - 1. - Pontus von Brömssen, Oct 24 2023

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 maximal degree is 4, which is the degree of vertex 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_] := 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]]]
Table[a[n], {n, 1, 30}]

Crossrefs

Cf. A000041, A000097 (number of edges in distance graphs), A000267, A002024, A002262, A366156, A366461.

Sequence in context: A086163 A175059 A372429 * A071789 A131870 A004724

Adjacent sequences: A366426 A366427 A366428 * A366430 A366431 A366432

Keyword

nonn

Author

Clark Kimberling, Oct 12 2023

Extensions

More terms from Pontus von Brömssen, Oct 24 2023

Status

approved