A259465 Triangle read by rows: enumerates pairs of amicable permutations by rises.

1, 1, 1, 1, 1, 8, 1, 1, 43, 43, 1, 1, 194, 826, 194, 1, 1, 803, 11284, 11284, 803, 1, 1, 3184, 127905, 392244, 127905, 3184, 1, 1, 12367, 1297629, 10258067, 10258067, 1297629, 12367, 1, 1, 47606, 12295720, 224702858, 561134638, 224702858, 12295720, 47606, 1

Offset

0,6

Links

Alois P. Heinz, Rows n = 0..80, flattened

Carlitz, L., Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bulletin of the American Mathematical Society 80.5 (1974): 881-884. [Annotated scanned copy]

L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975

Example

Triangle begins:
  1;
  1;
  1,     1;
  1,     8,       1;
  1,    43,      43,        1;
  1,   194,     826,      194,        1;
  1,   803,   11284,    11284,      803,       1;
  1,  3184,  127905,   392244,   127905,    3184,     1;
  1, 12367, 1297629, 10258067, 10258067, 1297629, 12367, 1;
  ...

Maple

b:= proc(u, o, h) option remember; `if`(u+o=0, 1, expand(
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h)*x, j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(0, n$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 02 2015

Mathematica

b[u_, o_, h_] := b[u, o, h] = If[u+o == 0, 1, Expand[Sum[Sum[b[u-j, o+j-1, h+i-1], {i, 1, u+o-h}]*x, {j, 1, u}] + Sum[Sum[b[u+j-1, o-j, h-i], {i, 1, h}], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[0, n, n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 12 2016, after Alois P. Heinz *)

Crossrefs

Row sums give A060350.

Sequence in context: A157640 A142458 A174528 * A176227 A340560 A022171

Adjacent sequences: A259462 A259463 A259464 * A259466 A259467 A259468

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jun 30 2015, following a suggestion from L. Carlitz, Nov 30 1975.

Extensions

More terms from Alois P. Heinz, Jul 02 2015

Status

approved