A337279 Array read by antidiagonals: T(m,n) (m>=1, n>=1) is number of alpha-labelings of the complete bipartite graph K_{m,n}.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 3, 1, 1, 2, 3, 3, 2, 1, 1, 4, 2, 3, 2, 4, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 4, 2, 9, 1, 9, 2, 4, 1, 1, 3, 4, 3, 4, 4, 3, 4, 3, 1, 1, 4, 3, 10, 2, 7, 2, 10, 3, 4, 1, 1, 2, 4, 6, 4, 4, 4, 4, 6, 4, 2, 1, 1, 6, 2, 9, 3, 16, 1, 16, 3, 9, 2, 6, 1, 1, 2, 6, 3, 4, 9, 4, 4, 9, 4, 3, 6, 2, 1

Offset

1,8

Comments

(We continue from the comments in A337278)

An alpha-labeling of a bipartite graph is a graceful labeling with the further proviso that all labels of one part are less than all labels of the other part. In particular, K_{3,4} has only 3 alpha-labelings, because the third example shown in the comments in A337278 isn't alpha.

Links

Table of n, a(n) for n=1..105.

Joseph A. Gallian, Graph Labeling, Electron. J. Combin., Dynamic Survey 6.

Don Knuth, CWEB program to generate solutions

Example

The array begins:
  1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1,...
  1..1..2..3..2..4..2..4..3..4..2..6..2..4..4..5,...
  1..2..1..3..2..4..2..4..3..4..2..6..2..4..4..5,...
  1..3..3..3..3..9..3.10..6..9..3.18..3..9..9.15,...
  1..2..2..3..1..4..2..4..3..4..2..6..2..4..4..5,...
  1..4..4..9..4..7..4.16..9.14..4.30..4.14.14.25,...
  1..2..2..3..2..4..1..4..3..4..2..6..2..4..4..5,...
  1..4..4.10..4.16..4.10.10.16..4.40..4.16.16.35,...
  1..3..3..6..3..9..3.10..3..9..3.18..3..9..9.15,...
  1..4..4..9..4.14..4.16..9..7..4.30..4.14.14.25,...
  1..2..2..3..2..4..2..4..3..4..1..6..2..4..4..5,...
  1..6..6.18..6.30..6.40.18.30..6.42..6.30.30.75,...
  1..2..2..3..2..4..2..4..3..4..2..6..1..4..4..5,...
  1..4..4..9..4.14..4.16..9.14..4.30..4..7.14.25,...
  1..4..4..9..4.14..4.16..9.14..4.30..4.14..7.25,...
  1..5..5.15..5.25..5.35.15.25..5.75..5.25.25..?,...
  ...
The first few antidiagonals are:
  1,
  1,1,
  1,1,1,
  1,2,2,1,
  1,3,1,3,1,
  1,2,3,3,2,1,
  1,4,2,3,2,4,1,
  1,2,4,3,3,4,2,1,
  1,4,2,9,1,9,2,4,1,
  1,3,4,3,4,4,3,4,3,1
  ...

Crossrefs

Cf. A337278.

Sequence in context: A338117 A353468 A184305 * A157522 A059674 A367624

Adjacent sequences: A337276 A337277 A337278 * A337280 A337281 A337282

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 11 2020, based on a communication from Don Knuth, Sep 08 2020

Status

approved