%I #7 Jan 20 2024 09:46:38

%S 1,2,3,7,15,43,57,60,82,134,184,247,331,451,562,771,985,1277,1630,

%T 2071,2640,3344,4119,5195,6514,8062

%N a(n) = number of pairs (p,q) of partitions of n such that d(p,q) = o(p,q), where d and o are distance functions; see Comments.

%C The definition of d depends on the greedy ordering of the partitions p(i) of n; that is, p(1) >= p(2) >= ... >= p(k), where k = A000041(n); see A366156. The ordinal distance o is defined by o(p(i),p(j)) = |i-j|.

%F a(n) + A368565(n) + A368566(n) = A001255(n) for n >= 1.

%e The 5 partitions of 4 are (p(1),p(2),p(3),p(4),p(5)) = (4,21,22,211,1111). The following table shows the 25 pairs d(p(i),q(j)) and o(p(i),q(j)):

%e | 4 31 22 211 1111

%e ------------------------------------------------

%e 4 d | 0 2 4 4 6

%e o | 0 1 2 3 4

%e 31 d | 2 0 2 2 4

%e o | 1 0 1 2 3

%e 22 d | 4 2 0 2 4

%e o | 2 1 0 1 2

%e 211 d | 4 2 2 0 2

%e o | 3 2 1 0 1

%e 1111 d | 6 4 4 2 0

%e o | 4 3 2 1 0

%e The table shows 7 pairs (p,q) for which d(p,q) = o(p,q), so a(4) = 7.

%t c[n_] := PartitionsP[n];

%t q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];

%t r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];

%t d[u_, v_] := Total[Abs[u - v]];

%t p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];

%t Table[Count[p[n], 0], {n, 1, 16}] (* A368565 *)

%t Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}] (* A368566 *)

%t Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}] (* A368567 *)

%Y Cf. A000041, A001255, A366156, A368565, A368566.

%K nonn,more

%O 1,2

%A _Clark Kimberling_, Dec 31 2023