A366747 Irregular triangular array, read by rows: T(n,k) = out-degree of k-th vertex in the distance graph of the strict partitions of n, where the parts of partitions and the list of partitions are in reverse-lexicographic order (Mathematica order).
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 3, 1, 2, 2, 1, 1, 1, 1, 2, 3, 1, 3, 2, 1, 2, 2, 1, 1, 1, 2, 3, 1, 3, 2, 2, 3, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 3, 2, 3, 3, 2, 1, 2, 4, 1, 2, 2, 1, 1, 1, 2, 3, 1, 3, 2, 3, 3, 2
Offset: 1
Examples
Triangle begins:
0
0
1
1
1 1
1 1 1
1 1 2 1
1 1 2 2 1
1 1 2 3 1 1 1
1 1 2 3 1 2 2 1 1
1 1 2 3 1 3 2 1 2 2 1
1 1 2 3 1 3 2 2 3 2 1 3 1 1
1 1 2 3 1 3 2 3 3 2 1 2 4 1 2 2 1
Enumerate the 6 strict partitions (= vertices) of 8 as follows:
1: 8
2: 7,1
3: 6,2
4: 5,3
5: 5,2,1
6: 4,3,1
Call q a neighbor of p if d(p,q)=2.
The set of neighbors for vertex k, for k = 1..6, is given by
vertex 1: {2} (so that vertex 1 has out-degree 1)
vertex 2: {1,3} (out-degree 1)
vertex 3: {2,4,5} (out-degree 2)
vertex 4: {3,5,6} (out-degree 2)
vertex 5: {3,4,6} (out degree 1)
vertex 6: {4,5} (out degree 0),
so that row 8 is 1,1,2,2,1.
(Out-degrees of 0 are excluded except for n = 1 and n = 2.)
Programs
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Mathematica
c[n_] := PartitionsQ[n]; q[n_, k_] := q[n, k] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &][[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 &]; t = Table[s[n, k], {n, 1, 12}, {k, 1, c[n]}]; s1[n_, k_] := Length[Select[s[n, k], # > k &]]; t1 = Join[{0, 0}, Table[s1[n, k], {n, 1, 26}, {k, 1, c[n] - 1}]]; TableForm[t1] (* array *) Flatten[t1] (* sequence *)
Comments