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.

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

1,2

Comments

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|.

Links

Table of n, a(n) for n=1..26.

Formula

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

Example

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.

Mathematica

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 *)

Crossrefs

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

Sequence in context: A352813 A146345 A328633 * A064842 A302647 A324580

Adjacent sequences: A368563 A368564 A368565 * A368567 A368568 A368569

Keyword

nonn,more

Author

Clark Kimberling, Dec 31 2023

Status

approved