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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).

%I A366747 #10 Nov 12 2023 22:01:44
%S A366747 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,
%T A366747 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,
%U A366747 2,1,2,4,1,2,2,1,1,1,2,3,1,3,2,3,3,2
%N 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).
%C A366747 See A366156 for the distance function d and A000097 for the distance graph.
%C A366747 Regarding reverse lexicographic order (Mathematica order, also called canonical order; see A080577).
%e A366747 Triangle begins:
%e A366747   0
%e A366747   0
%e A366747   1
%e A366747   1
%e A366747   1   1
%e A366747   1   1   1
%e A366747   1   1   2   1
%e A366747   1   1   2   2   1
%e A366747   1   1   2   3   1   1   1
%e A366747   1   1   2   3   1   2   2   1   1
%e A366747   1   1   2   3   1   3   2   1   2   2   1
%e A366747   1   1   2   3   1   3   2   2   3   2   1   3   1   1
%e A366747   1   1   2   3   1   3   2   3   3   2   1   2   4   1   2   2   1
%e A366747 Enumerate the 6 strict partitions (= vertices) of 8 as follows:
%e A366747   1: 8
%e A366747   2: 7,1
%e A366747   3: 6,2
%e A366747   4: 5,3
%e A366747   5: 5,2,1
%e A366747   6: 4,3,1
%e A366747 Call q a neighbor of p if d(p,q)=2.
%e A366747 The set of neighbors for vertex k, for k = 1..6, is given by
%e A366747   vertex 1: {2}  (so that vertex 1 has out-degree 1)
%e A366747   vertex 2: {1,3}  (out-degree 1)
%e A366747   vertex 3: {2,4,5}  (out-degree 2)
%e A366747   vertex 4: {3,5,6} (out-degree 2)
%e A366747   vertex 5: {3,4,6} (out degree 1)
%e A366747   vertex 6: {4,5} (out degree 0),
%e A366747 so that row 8 is 1,1,2,2,1.
%e A366747 (Out-degrees of 0 are excluded except for n = 1 and n = 2.)
%t A366747 c[n_] := PartitionsQ[n]; q[n_, k_] := q[n, k] =
%t A366747 Select[IntegerPartitions[n], DeleteDuplicates[#] == # &][[k]];
%t A366747 r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
%t A366747 d[u_, v_] := Total[Abs[u - v]];
%t A366747 s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &];
%t A366747 t = Table[s[n, k], {n, 1, 12}, {k, 1, c[n]}];
%t A366747 s1[n_, k_] := Length[Select[s[n, k], # > k &]];
%t A366747 t1 = Join[{0, 0}, Table[s1[n, k], {n, 1, 26}, {k, 1, c[n] - 1}]];
%t A366747 TableForm[t1] (* array *)
%t A366747 Flatten[t1]   (* sequence *)
%Y A366747 Cf. A000009, A096778 (row sums), A366597.
%K A366747 nonn,tabf
%O A366747 1,12
%A A366747 _Clark Kimberling_, Oct 25 2023