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A366598 a(n) = greatest number of vertices having the same degree in the distance graph of the partitions of n.

%I A366598 #6 Nov 12 2023 22:00:25
%S A366598 1,2,2,2,4,4,7,9,11,14,19,17,27,32,50,62,82,94,132,138,176,198,238,
%T A366598 288,368
%N A366598 a(n) = greatest number of vertices having the same degree in the distance graph of the partitions of n.
%C A366598 The distance graph of the partitions of n is defined in A366156.
%e A366598 Enumerate the 7 partitions (= vertices) of 5 as follows:
%e A366598   1: 5
%e A366598   2: 4,1
%e A366598   3: 3,2
%e A366598   4: 3,1,1
%e A366598   5: 2,2,1
%e A366598   6: 2,1,1,1
%e A366598   7: 1,1,1,1,1
%e A366598 Call q a neighbor of p if d(p,q)=2.
%e A366598 The set of neighbors for vertex k, for k = 1..7, is given by
%e A366598   vertex 1: {2}
%e A366598   vertex 2: {1,3,4}
%e A366598   vertex 3: {2,4,5}
%e A366598   vertex 4: {2,3,5,6}
%e A366598   vertex 5: {3,4,6}
%e A366598   vertex 6: {4,5,7}
%e A366598   vertex 7: {6}
%e A366598 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 A366598 c[n_] := PartitionsP[n]; q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
%t A366598 r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
%t A366598 d[u_, v_] := Total[Abs[u - v]];
%t A366598 s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
%t A366598 s1[n_] := s1[n] = Table[s[n, k], {k, 1, c[n]}]
%t A366598 m[n_] := m[n] = Map[Length, s1[n]]
%t A366598 m1[n_] := m1[n] = Max[m[n]];  (* A366429 *)
%t A366598 t1 = Join[{1}, Table[Count[m[n], i], {n, 2, 25}, {i, 1, m1[n]}]]
%t A366598 Map[Max, t1]
%Y A366598 Cf. A000041, A366156, A366429.
%K A366598 nonn,more
%O A366598 1,2
%A A366598 _Clark Kimberling_, Oct 25 2023