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).

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, 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, 4, 2, 16, 10, 1, 6, 17, 14, 0, 0, 0, 4, 1, 2, 0, 4, 23

Offset

1,2

Comments

The distance graph of the partitions of n is defined in A366156.

Links

Table of n, a(n) for n=1..83.

Example

First fourteen rows:
   1
   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   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   4   2   16   10   1   6  17   14  0   0   0   4   1
   2   0   4   23   10   0   2  27   22  1   0   0   4   6
   2   2   2   22   18   2   4  27   32  4   0   0   6   12   2
Enumerate the 7 partitions (vertices) of 5 as follows:
  1: 5
  2: 4,1
  3: 3,2
  4: 3,1,1
  5: 2,2,1
  6: 2,1,1,1
  7: 1,1,1,1,1
Call q a neighbor of p if d(p,q)=2, where d is the distance function in A366156.
The set of neighbors for vertex k, for k = 1..7, is given by
  vertex 1: {2}
  vertex 2: {1,3,4}
  vertex 3: {2,4,5}
  vertex 4: {2,3,5,6}
  vertex 5: {3,4,6}
  vertex 6: {4,5,7}
  vertex 7: {6}
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.

Mathematica

c[n_] := PartitionsP[n]; q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &];
s1[n_] := s1[n] = Table[s[n, k], {k, 1, c[n]}];
m[n_] := m[n] = Map[Length, s1[n]];
m1[n_] := m1[n] = Max[m[n]];  (* A366429 *)
t1 = Join[{1}, Table[Count[m[n], i], {n, 2, 15}, {i, 1, m1[n]}]]
Column[t1]
Flatten[t1]

Crossrefs

Cf. A000041 (row sums), A366429 (row lengths), A366598 (row maxima).

Sequence in context: A301384 A258124 A264032 * A251138 A144462 A370559

Adjacent sequences: A366594 A366595 A366596 * A366598 A366599 A366600

Keyword

nonn,tabf

Author

Clark Kimberling, Oct 16 2023

Status

approved