A368564 -id:A368564 - cp's OEIS frontend

A368565 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.

0, 0, 0, 0, 0, 30, 106, 316, 652, 1388, 2618, 5170, 9164, 16790, 29046, 50714, 84732, 143588, 234048, 385210, 617050, 990868, 1558310, 2459300, 3806838, 5900184

Offset: 1

Clark Kimberling, Dec 31 2023

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|.

			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)):
          |    4   31   22   211  1111
------------------------------------------------
   4 d    |    0    2    4    4    6
     o    |    0    1    2    3    4
  31 d    |    2    0    2    2    4
     o    |    1    0    1    2    3
  22 d    |    4    2    0    2    4
     o    |    2    1    0    1    2
 211 d    |    4    2    2    0    2
     o    |    3    2    1    0    1
1111 d    |    6    4    4    2    0
     o    |    4    3    2    1    0
The table shows 0 pairs (p,q) for which d(p,q) < o(p,q), so a(4) = 0.

Cf. A000041, A001255, A366156, A368564, A368566.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[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]];
p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)

A368564(n) + a(n) + A368566(n) = A001255(n) for n >= 1.

A368566 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.

Original entry on oeis.org

0, 2, 6, 18, 34, 48, 62, 108, 166, 242, 334, 512, 706, 984, 1368, 1876, 2492, 3360, 4422, 5848, 7574, 9792, 12596, 16130, 20412, 25850

Offset: 1

Clark Kimberling, Dec 31 2023

			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)):
          |    4   31   22   211  1111
------------------------------------------------
   4 d    |    0    2    4    4    6
     o    |    0    1    2    3    4
  31 d    |    2    0    2    2    4
     o    |    1    0    1    2    3
  22 d    |    4    2    0    2    4
     o    |    2    1    0    1    2
 211 d    |    4    2    2    0    2
     o    |    3    2    1    0    1
1111 d    |    6    4    4    2    0
     o    |    4    3    2    1    0
The table shows 18 pairs (p,q) for which d(p,q) > o(p,q), so a(4) = 18.

Cf. A000041, A001255, A366156, A368564, A368565.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[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]];
p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)

A368564(n) + A368565(n) + a(n) = A001255(n) for n >= 1.

cp's OEIS Frontend

A368565 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.

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