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A368564a(n) + A368565(n) + A368566(n) = A001255(n) for n >= 1.

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allocated for Clark Kimberlinga(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

1,2

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|.

A368564(n) + A368565(n) + A368566(n) = A001255(n) for n >= 1.

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

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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.

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 *)

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

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Clark Kimberling, Dec 31 2023

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allocated for Clark Kimberling

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