A366745 Triangular array, read by rows: T(n,k) = number of partitions p of n such that d(p, p') = 2k, where p' = contraconjugate of p, and d is the distance function defined in A366156.

1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2, 1, 0, 6, 2, 0, 2, 1, 2, 2, 6, 2, 0, 2, 0, 6, 2, 4, 6, 2, 0, 2, 0, 2, 4, 8, 6, 6, 2, 0, 2, 0, 4, 8, 8, 8, 4, 6, 2, 0, 2, 0, 6, 10, 8, 12, 6, 4, 6, 2, 0, 2, 1, 4, 14, 8, 16, 10, 6, 6, 8, 2, 0, 2, 1, 4, 18, 10, 14, 20, 10, 6, 6, 8, 2, 0, 2, 1, 8, 8, 18, 20, 22, 16, 16, 6, 8, 8, 2, 0, 2, 0, 6, 18, 28, 30, 24, 16, 16, 10, 8, 8, 8, 2, 0, 2, 1, 4, 20, 28, 30, 36, 16, 34, 18, 16, 4, 12, 8, 2, 0, 2

Offset

1,3

Comments

Let RL denote reverse lexicographic. Write each partition p of n in RL order, and then arrange all the partitions of n in RL order, obtaining a list L as in Mathematica's IntegerPartitions[n]. Let L' be the reversal of L. Let p(i) be the i-th partition in L and let p'(i) be the i-th in L'; then p'(i) is introduced here as the contraconjugate of p(i) . (Note that RL order could be called "greedy" order, because the result of applying the greedy algorithm to p is the same as for RL, and likewise for arranging the partitions in RL order. For a discussion of various orderings, see A080577.)

Links

Table of n, a(n) for n=1..136.

Example

First 14 rows:
  1
  0    2
  1    0    2
  1    2    0    2
  1    2    2    0    2
  1    0    6    2    0    2
  1    2    2    6    2    0    2
  0    6    2    4    6    2    0    2
  0    2    4    8    6    6    2    0    2
  0    4    8    8    8    4    6    2    0    2
  0    6   10    8   12    6    4    6    2    0    2
  1    4   14    8   16   10    6    6    8    2    0    2
  1    4   18   10   14   20   10    6    6    8    2    0    2
  1    8    8   18   20   22   16   16    6    8    8    2    0    2
The list L for the partitions of 5 is:
  [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1].
The list L' of contraconjugates p' and distances d(p,p') are given by:
  p = [5]; p' = [1,1,1,1,1]; d(p,p') = 8
  p = [4,1]; p' = [2,1,1,1]; d(p,p') = 4
  p = [3,2]; p' = [2,2,1]; d(p,p') = 2
  p = [3,1,1,1]; p' = [3,1,1,1]; d(p,p') = 0
  p = [2,2,1]; p' = [3,2]; d(p,p') = 2
  p = [2,1,1,1]; p' = [4,1]; d(p,p') = 4
  p = [1,1,1,1,1], p' = [5]; d(p,p') = 8;
so that the numbers of 0s, 2s, 4s, 6s, 8s are 1, 2, 2, 0, 2, as in row 5.

Mathematica

c[n_] := PartitionsP[n];
p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[p[n, k], ConstantArray[0, n - Length[p[n, k]]]];
p1[n_, k_] := p1[n, k] = Reverse[IntegerPartitions[n]][[k]];
r1[n_, k_] := r1[n, k] = Join[p1[n, k], ConstantArray[0, n - Length[p1[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
t[n_] := Flatten[Table[d[r[n, k], r1[n, k]], {k, 1, c[n]}]];
t1 = Table[Count[t[n], m], {n, 1, 16}, {m, 0, 2 n - 1, 2}]
TableForm[t1]   (* array *)
u = Flatten[t1] (*sequence *)

Crossrefs

Cf. A000041, A366156, A366746.

Sequence in context: A287520 A130094 A337005 * A230025 A330374 A207869

Adjacent sequences: A366742 A366743 A366744 * A366746 A366747 A366748

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 25 2023

Status

approved