A366429 - OEIS

%I #12 Oct 24 2023 15:42:45

%S 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,

%T 34,42,43,44,44,45,45,46,46,56,57,58,58,59,59,60,60,60,72,73,74,74,75,

%U 75,76,76,76,77,90,91,92,92,93,93,94,94,94,95,95,110,111

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

%C 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.

%F a(n) = 2*binomial(A002024(n+1)-1,2) + A000267(A002262(n)) - 1. - _Pontus von Brömssen_, Oct 24 2023

%e Enumerate the 7 partitions (= vertices) of 5 as follows:

%e   1: 5

%e   2: 4,1

%e   3: 3,2

%e   4: 3,1,1

%e   5: 2,2,1

%e   6: 2,1,1,1

%e   7: 1,1,1,1,1

%e 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

%e   vertex 1: {2}

%e   vertex 2: {1,3,4}

%e   vertex 3: {2,4,5}

%e   vertex 4: {2,3,5,6}

%e   vertex 5: {3,4,6}

%e   vertex 6: {4,5,7}

%e   vertex 7: {6}

%e The maximal degree is 4, which is the degree of vertex 4, so that a(5) = 4.

%t c[n_] := PartitionsP[n];

%t q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];

%t r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];

%t d[u_, v_] := d[u, v] = Total[Abs[u - v]];

%t s[n_, k_] := s[n, k] = Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]

%t t[n_] := t[n] = Table[s[n, k], {k, 1, c[n]}]

%t a[n_] := Max[Map[Length, t[n]]]

%t Table[a[n], {n, 1, 30}]

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

%K nonn

%O 1,3

%A _Clark Kimberling_, Oct 12 2023

%E More terms from _Pontus von Brömssen_, Oct 24 2023