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A366597 Irregular triangular array, read by rows: T(n,k) = number of vertices (partitions) that have degree k in the distance graph of the partitions of n, for k = 1..A366429(n).

%I A366597 #9 Nov 12 2023 22:00:09
%S A366597 0,2,2,1,2,1,2,2,0,4,1,2,2,2,4,0,1,2,0,4,7,0,0,2,2,2,2,9,2,0,4,1,2,1,
%T A366597 4,11,2,0,4,6,2,2,2,14,6,0,4,9,2,0,0,1,2,0,4,17,6,0,2,19,4,0,0,0,2,2,
%U A366597 4,2,16,10,1,6,17,14,0,0,0,4,1,2,0,4,23
%N A366597 Irregular triangular array, read by rows: T(n,k) = number of vertices (partitions) that have degree k in the distance graph of the partitions of n, for k = 1..A366429(n).
%C A366597 The distance graph of the partitions of n is defined in A366156.
%e A366597 First fourteen rows:
%e A366597    1
%e A366597    2
%e A366597    2   1
%e A366597    2   1   2
%e A366597    2   0   4   1
%e A366597    2   2   2   4    0    1
%e A366597    2   0   4   7    0    0   2
%e A366597    2   2   2   9    2    0   4   1
%e A366597    2   1   4   11   2    0   4   6
%e A366597    2   2   2   14   6    0   4   9   2   0   0   1
%e A366597    2   0   4   17   6    0   2  19   4   0   0   0   2
%e A366597    2   4   2   16   10   1   6  17   14  0   0   0   4   1
%e A366597    2   0   4   23   10   0   2  27   22  1   0   0   4   6
%e A366597    2   2   2   22   18   2   4  27   32  4   0   0   6   12   2
%e A366597 Enumerate the 7 partitions (vertices) of 5 as follows:
%e A366597   1: 5
%e A366597   2: 4,1
%e A366597   3: 3,2
%e A366597   4: 3,1,1
%e A366597   5: 2,2,1
%e A366597   6: 2,1,1,1
%e A366597   7: 1,1,1,1,1
%e A366597 Call q a neighbor of p if d(p,q)=2, where d is the distance function in A366156.
%e A366597 The set of neighbors for vertex k, for k = 1..7, is given by
%e A366597   vertex 1: {2}
%e A366597   vertex 2: {1,3,4}
%e A366597   vertex 3: {2,4,5}
%e A366597   vertex 4: {2,3,5,6}
%e A366597   vertex 5: {3,4,6}
%e A366597   vertex 6: {4,5,7}
%e A366597   vertex 7: {6}
%e A366597 The number of vertices having degrees 1,2,3,4 are 2,0,4,1, respectively, so that row 5 is 2 0 4 1.
%t A366597 c[n_] := PartitionsP[n]; q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
%t A366597 r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
%t A366597 d[u_, v_] := Total[Abs[u - v]];
%t A366597 s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &];
%t A366597 s1[n_] := s1[n] = Table[s[n, k], {k, 1, c[n]}];
%t A366597 m[n_] := m[n] = Map[Length, s1[n]];
%t A366597 m1[n_] := m1[n] = Max[m[n]];  (* A366429 *)
%t A366597 t1 = Join[{1}, Table[Count[m[n], i], {n, 2, 15}, {i, 1, m1[n]}]]
%t A366597 Column[t1]
%t A366597 Flatten[t1]
%Y A366597 Cf. A000041 (row sums), A366429 (row lengths), A366598 (row maxima).
%K A366597 nonn,tabf
%O A366597 1,2
%A A366597 _Clark Kimberling_, Oct 16 2023