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
Examples
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.
Programs
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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]
Comments