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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, 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
1,12
COMMENTS
See A366156 for the distance function d and A000097 for the distance graph.
Regarding reverse lexicographic order (Mathematica order, also called canonical order; see A080577).
EXAMPLE
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.)
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 *)
CROSSREFS
Cf. A000009, A096778 (row sums), A366597.
Sequence in context: A276317 A344318 A289944 * A055215 A239550 A058398
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Oct 25 2023
STATUS
approved