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A368564 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.

%I A368564 #7 Jan 20 2024 09:46:38
%S A368564 1,2,3,7,15,43,57,60,82,134,184,247,331,451,562,771,985,1277,1630,
%T A368564 2071,2640,3344,4119,5195,6514,8062
%N A368564 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 A368564 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 A368564 a(n) + A368565(n) + A368566(n) = A001255(n) for n >= 1.
%e A368564 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 A368564           |    4   31   22   211  1111
%e A368564 ------------------------------------------------
%e A368564    4 d    |    0    2    4    4    6
%e A368564      o    |    0    1    2    3    4
%e A368564   31 d    |    2    0    2    2    4
%e A368564      o    |    1    0    1    2    3
%e A368564   22 d    |    4    2    0    2    4
%e A368564      o    |    2    1    0    1    2
%e A368564  211 d    |    4    2    2    0    2
%e A368564      o    |    3    2    1    0    1
%e A368564 1111 d    |    6    4    4    2    0
%e A368564      o    |    4    3    2    1    0
%e A368564 The table shows 7 pairs (p,q) for which d(p,q) = o(p,q), so a(4) = 7.
%t A368564 c[n_] := PartitionsP[n];
%t A368564 q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
%t A368564 r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
%t A368564 d[u_, v_] := Total[Abs[u - v]];
%t A368564 p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
%t A368564 Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
%t A368564 Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
%t A368564 Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)
%Y A368564 Cf. A000041, A001255, A366156, A368565, A368566.
%K A368564 nonn,more
%O A368564 1,2
%A A368564 _Clark Kimberling_, Dec 31 2023