A366598 - OEIS

%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