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

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

Author

Clark Kimberling, Oct 12 2023

Keywords

• nonn

Comments

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

Examples

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

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_] := 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}]

Extensions

More terms from Pontus von Brömssen, Oct 24 2023