A366745 -id:A366745 - cp's OEIS frontend

A366746 Maximum distance between a partition of n and its contraconjugate.

1, 2, 2, 2, 2, 6, 6, 6, 8, 8, 12, 16, 20, 22, 30, 36, 42, 56, 68, 82, 118, 134, 160, 174

Offset: 1

Clark Kimberling, Oct 25 2023

See A366745 for the definition of contraconjugate.

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]]; (* distance function *)
t[n_] := Flatten[Table[d[r[n, k], r1[n, k]], {k, 1, c[n]}]];
t1 = Table[Count[t[n], m], {n, 1, 24}, {m, 0, 2 n - 1, 2}];  (* A366745 *)
Map[Max, t1]

A366856 a(n) = number of partitions p of n such that (Ferrers conjugate of p) = contraconjugate of p.

Original entry on oeis.org

1, 2, 3, 5, 7, 7, 8, 6, 10, 8, 8, 8, 8, 8, 8, 8, 8, 8, 12, 6, 10, 10, 8, 6, 6, 8, 12, 8, 12, 8, 10, 6, 8, 12, 10, 6, 10, 6, 8, 6, 8, 12, 8, 8, 8, 10, 8, 14, 14, 16, 12, 6, 8, 8, 10, 6, 12, 8, 8, 8

Offset: 1

Clark Kimberling, Dec 05 2023

See A366745 for the definition of contraconjugate.

			The partitions of 6 in reverse lexicographic order:
[6], [5,1], [4,2], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1], [1,1,1,1,1,1]
Reversed (i.e., the contraconjugates of the partitions of 6, respectively)::
[1,1,1,1,1,1], [2,1,1,1,1], [2,2,1,1], [2,2,2], [3,1,1,1], [3,2,1], [3,3], [4,1,1], [4,2], [5,1], [6]
Ferrers conjugates:
[1,1,1,1,1,1], [2,1,1,1,1], [2,2,1,1], [3,1,1,1], [2,2,2], [3,2,1], [4,1,1], [3,3], [4,2], [5,1], [6]
Comparing the 2nd and 3rd lists shows that 11-4 = 7 partitions of 6 have identical Ferrers conjugate and contraconjugate, so that a(6) = 7.

Cf. A000041, A366745, A366746.

c[n_] := PartitionsP[n]
p[n_] := p[n] = IntegerPartitions[n];
r[n_] := r[n] = Reverse[p[n]]
q1[u_] := q1[u] = Table[Count[#, _?(# >= i &)], {i, First[#]}] &[u];
(* q1[u]=conjugate of partition u *)
q[n_] := q[n] = Table[q1[p[n][[k]]], {k, 1, c[n]}]
s[n_] := s[n] = Select[Range[c[n]], r[n][[#]] == q[n][[#]] &]
Table[Length[s[n]], {n, 1, 20}]

cp's OEIS Frontend

A366746 Maximum distance between a partition of n and its contraconjugate.

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