A368564 - OEIS

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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.

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

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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