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.

1, 2, 3, 7, 15, 43, 57, 60, 82, 134, 184, 247, 331, 451, 562, 771, 985, 1277, 1630, 2071, 2640, 3344, 4119, 5195, 6514, 8062

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

a(n) + A368565(n) + A368566(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 7 pairs (p,q) for which d(p,q) = o(p,q), so a(4) = 7.

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, A368565, A368566.

Sequence in context: A221547 A289471 A289470 * A213385 A161746 A045629

Adjacent sequences: A368561 A368562 A368563 * A368565 A368566 A368567

Keyword

nonn,more

Author

Clark Kimberling, Dec 31 2023

Status

approved