login
A366429
a(n) = maximum degree of vertices in the distance graph of the partitions of n.
3
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.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Oct 12 2023
EXTENSIONS
More terms from Pontus von Brömssen, Oct 24 2023
STATUS
approved