%I #6 Nov 12 2023 22:00:25
%S 1,2,2,2,4,4,7,9,11,14,19,17,27,32,50,62,82,94,132,138,176,198,238,
%T 288,368
%N a(n) = greatest number of vertices having the same degree in the distance graph of the partitions of n.
%C The distance graph of the partitions of n is defined in A366156.
%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.
%e 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 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.
%t c[n_] := PartitionsP[n]; 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_] := Total[Abs[u - v]];
%t s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
%t s1[n_] := s1[n] = Table[s[n, k], {k, 1, c[n]}]
%t m[n_] := m[n] = Map[Length, s1[n]]
%t m1[n_] := m1[n] = Max[m[n]]; (* A366429 *)
%t t1 = Join[{1}, Table[Count[m[n], i], {n, 2, 25}, {i, 1, m1[n]}]]
%t Map[Max, t1]
%Y Cf. A000041, A366156, A366429.
%K nonn,more
%O 1,2
%A _Clark Kimberling_, Oct 25 2023